Xenharmonic Reference - User contributions [en]

159edo

2026-04-07T23:51:30Z

Aura:

'''159edo''', or 159 equal divisions of the octave, is the equal tuning featuring steps of (1200/159) ~= 7.55 cents, 159 of which stack to the perfect octave [[2/1]]. Like [[53edo]], 159edo is an excellent approximation to Pythagorean tuning (stacking pure 3/2 fifths), however it includes near-just approximations of the 11th and 17th harmonics, and a slightly more accurate 7th harmonic, resulting in consistency up to the 17-odd-limit. The step-size, being slightly above the melodic just noticeable difference (JND) of 5 cents as well as more than twice the harmonic JND of the average trained musician at 3.5 cents, enables one to perform fluid modulations by means of [[comma pump]]s as well as by step substitutions. Furthermore, 159edo, like a number of higher edos, is characterized by being capable of imitating the intervals of smaller tuning systems - in this case, you get detemperings of [[10edo]], [[12edo]], [[13edo]], [[14edo]], [[17edo]], [[19edo]], [[22edo]], [[24edo]] and [[31edo]] among others with errors smaller than the melodic JND.

The interval qualities supported by 159edo are many, however, while every step of 159edo can be interpreted harmonically or subharmonically as being a 17-limit interval or simpler, some of the intervals you get have rather complex interpretations in terms of odd-limit. While the [[perfect fifth]] is really only divisible by three due to 159edo being the three-fold multiple of 53edo- resulting in [[slendric]] temperament and hence 159edo's distinction from 53 in the 7-limit- the [[perfect fourth]] has a little more to offer in terms of divisions. For starters, the perfect fourth can be divided into two instances of 15/13, giving us [[Semifourth-generated scales|island]] temperament. Dividing the perfect fourth into three instances of 11/10 gives us [[pine]] temperament. Dividing the perfect fourth into six instances of an interval which can be interpreted as 21/20 and 22/21 tempered together gives us [[sextilifourths]] temperament. The perfect fourth can also be cut into eleven intervals which, individually, are half of a Pythagorean limma, giving us a number of temperaments based on the exact interpretation of the semilimma.

On top of all that, there are a number of microtemperament-based structures also supported by 159edo, each of which provides some decent, unexpected melodic possibilities. For instance, one can split the Pythagorean minor third into three instances of 128/121 which gives us [[nexus]] temperament- a temperament which also happens to split the Pythagorean diatonic semitone into two, and the octave into three. One can also split the Ptolemaic minor third into three instances of 17/16, giving us [[archagall]] temperament, which is named for certain tunings found in other temperaments producing fractal-like acoustics. There's also the ability to split the Ptolemaic major sixth into six instances of 11/9, leading to [[parimic]] temperament. In addition, there's also the ability to split the septimal supermajor third into two instances of 17/15, leading to [[fidesmic]] temperament, which acts like a more accurate rendition of [[archy]] temperament in a different subgroup, and this can be exploited for modulation purposes. As if that weren't enough, there's the possibility of splitting the septimal subminor third into five instances of 33/32, leading to [[quartismic]] temperament. Furthermore, there's the possibility of splitting the greater tridecimal neutral tenth into three instances of 27/20, producing [[phaotismic]] temperament, and the list goes on.

== Theory ==
159edo was first used for maqams by Ozan Yarman. It was later put to use by Aura for its ability to handle near-just quartertones derived from the 2.3.11 subgroup on top of the 5-limit foundation provided by 53edo.

==== Edostep interpretations ====
159edo's edostep has the following interpretations in the 2.3.5.11.17 subgroup:

* 243/242, the difference between the 11-limit artoneutral third 11/9, and the 11-limit tendoneutral third 27/22
* 256/255, the difference between 16/15 and 17/16
* 289/288, the difference between 17/16 and 18/17

159edo tempers out the following commas in the 17-limit:

* The schisma (the difference between 5/4 and the Pythagorean diminished fourth)
* The vulture comma (the difference between four 320/243 intervals and the tritave)
* The amiton (the difference between a stack of five 10/9 intervals and 27/16)
* The kleisma (the difference between a stack of three 25/24 intervals and 9/8)
* The semicomma (the difference between a stack of three 75/64 intervals and 8/5)
* The gamelisma (the difference between a stack of three 8/7 intervals and 3/2)
* 385/384 (the difference between 77/64 and 6/5)
* The pine comma (the difference between a stack of three 11/10 intervals and 4/3)
* 625/624 (the difference between 25/24 and 26/25)
* 676/675 (the difference between a stack of two 15/13 intervals and the perfect fourth)
* The twosquare comma (the difference between a stack of two 33/32 intervals and 17/16)

==== JI approximation ====
Although 159edo inherits its approximations of the 5-limit from 53edo, the 5th harmonic can nonetheless be stacked twice without accumulating too much error, rendering it sufficient for Western Classical usage. While the 7th harmonic is technically more accurate in terms of absolute error than in 53edo, the relative error doesn't allow one to stack more than one instance of 7/4 without excessive error accumulation, and the same is true with 13/8. Regardless, the inconsistency remains less than 10 cents even when either interval is stacked three times, and 13/7 or 14/13 is tuned almost perfectly. As a whole, 159edo is characterized by its combination of accuracy in the 2.3.5.11.17 subgroup, and a series of compromises in the 7.13.19.23.29 subgroup- among the compromises are the slendric, [[marveltwin]], [[nestoria]], [[minor semivicemic]] and [[brunisimic]] temperaments.
{{Harmonics in ED|159|31|0}}

Currently, the [[ADIN]] system used for interval naming elsewhere on this site fails for 159edo, so another set of interval names will be used here, though the ADIN names will be referenced in places. Furthermore, because there are so many thirds, what is usually a single chart will be split into five.
{| class="wikitable"
|+Inframinor and Subminor Thirds in 159edo
!Quality
|Narrow Inframinor
|Inframinor
|Wide Inframinor
|Narrow Subminor
|Lesser Subminor (Septiminor)
|Greater Subminor
|Wide Subminor
|-
!Cents
|234
|242
|249
|257
|264
|272
|279
|-
!Just interpretation
|340/297
|1024/891
|15/13
|51/44
|7/6
|117/100
|20/17
|-
!Steps
|31
|32
|33
|34
|35
|36
|37
|}
{| class="wikitable"
|+Minor Thirds in 159edo
!Quality
|Narrow Minor (Gothminor)
|'''Pythagorean Minor (Triminor)'''
|Artomean Minor
|Tendomean Minor
|Ptolemaic Minor (Pentaminor)
|Wide Minor
|-
!Cents
|287
|'''294'''
|302
|309
|317
|325
|-
!Just interpretation
|33/28, 13/11
|'''32/27'''
|25/21
|153/128
|6/5
|135/112
|-
!Steps
|38
|'''39'''
|40
|41
|42
|43
|}
{| class="wikitable"
|+Supraminor, Neutral and Submajor Thirds in 159edo
!Quality
|Lesser Supraminor
|Greater Supraminor
|Artoneutral
|Tendoneutral
|Lesser Submajor
|Greater Submajor
|-
!Cents
|332
|340
|347
|355
|362
|370
|-
!Just interpretation
|40/33, 63/52
|39/32, 17/14
|11/9
|27/22
|16/13, 21/17
|99/80, 26/21
|-
!Steps
|44
|45
|46
|47
|48
|49
|}
{| class="wikitable"
|+Major Thirds in 159edo
!Quality
|Narrow Major
|Ptolemaic Major (Pentamajor)
|Artomean Major
|Tendomean Major
|'''Pythagorean Major (Trimajor)'''
|Wide Major (Gothmajor)
|-
!Cents
|377
|385
|392
|400
|'''408'''
|415
|-
!Just interpretation
|56/45
|5/4
|64/51
|63/50
|'''81/64'''
|14/11, 33/26
|-
!Steps
|50
|51
|52
|53
|'''54'''
|55
|}
{| class="wikitable"
|+Supermajor and Ultramajor Thirds in 159edo
!Quality
|Narrow Supermajor
|Lesser Supermajor
|Greater Supermajor (Septimajor)
|Wide Supermajor
|Narrow Ultramajor
|Ultramajor
|Wide Ultramajor
|-
!Cents
|423
|430
|438
|445
|453
|460
|468
|-
!Just interpretation
|51/40
|50/39
|9/7
|22/17
|13/10
|2673/2048
|891/680
|-
!Steps
|56
|57
|58
|59
|60
|61
|62
|}
Regular diatonic thirds are bolded.

=== Chords ===
159edo has a vast array of triads at its disposal, both fifth-bounded and fourth-bounded. However, the JI interpretation will tend to inform the usage of the various triads offered. The main exceptions to this rule involve chords that serve as detempers of other, smaller tuning systems.

== Notation ==
There are multiple different notation systems available for 159edo.

=== Ups and downs ===
Ups and downs notation typically represents a single step of 159edo by means of lifts and drops while using the original ups and downs to represent single steps of 53edo. One may also use ups and downs to instead represent a single 159edo step, but this makes notating the syntonic/Pythagorean comma inconvenient.

=== Syntonic-rastmic subchroma notation ===
Syntonic-rastmic subchroma notation, or SRS for short, is an alternative to Sagittal with fewer accidentals- one which was originally designed for 159edo. In 159edo, the synsharp (81/80) and the rasharp (243/242) are 3\159 (1\53) and 1\159 respectively, treating 159edo as a subdivision of 53edo wherein three rastmas make up a syntonic comma. So far, this is functionally identical to ups and downs notation (assuming the up and down are treated as 1\53), but syntonic-rastmic subchroma notation also supports halving of all of its accidentals. While 159edo divides neither the Pythagorean, syntonic, nor rastmic accidental pairs in half, this does allow for the notation of artoneutral and tendoneutral intervals via the inflection of a semisharp by half a rastma (which ends up at a full 159edo step, specifically 7 steps for the artodemisharp and 8 steps for the tendodemisharp).

The syntonic and rastmic accidentals represent their just mappings when generalized to other EDOs.

=== Formal comma systems ===
The Functional Just System and HEJI share the same accidentals for 5 and 11 (though notated with each system's respective symbol); the 11-comma is the artodemisharp and the 5-comma is the synsharp. Neutral FJS uses the standard demisharp and half-rastma as accidentals, meaning that like with SRS notation, the artodemisharp and tendodemisharp become 11-limit accidentals. FJS, Neutral FJS, and HEJI also share an accidental for 7, which in 159edo in particular is equivalent to a rastma plus a syntonic comma.

The Sagittal sequence for 159edo will also be provided.

The simplest sequence spanning an entire 159edo chromatic semitone is thus as follows:
{| class="wikitable"
|+
!Rastmas
!SRS
!FJS
!HEJI
!Neutral FJS
!Sagittal
|-
|0
| colspan="5" |natural
|-
|1
|rasharp
|17-flat 5-sharp
|17-sharp
|double 11-sharp
|5/7 kleisma up
|-
|2
|raflat synsharp
|17-sharp
|43-sharp
|17-sharp
|17 comma up
|-
|3
|synsharp
|5-sharp
|5-sharp
|5-sharp
|5 comma up
|-
|4
|rasharp synsharp
|7-sharp
|7-sharp
|7-sharp
|7 comma up
|-
|5
|raflat double synsharp
|47-sharp
|47-sharp
|47-flat semisharp
|11/5 S-diesis up
|-
|6
|demisynflat demisharp / double synsharp
|13-sharp
|13-flat sharp
|13-flat semisharp
|25 S-diesis up
|-
|7
|artodemisharp
|11-sharp
|11-sharp
|11-flat semisharp
|11 M-diesis up
|-
|8
|tendodemisharp
|11-flat sharp
|11-flat sharp
|11-sharp semisharp
|11 L-diesis up
|-
|9
|demisynsharp demisharp / double synflat sharp
|13-flat sharp
|13-sharp
|13-sharp semisharp
|25 S-diesis down sharp
|-
|10
|rasharp double synflat sharp
|47-flat sharp
|47-flat sharp
|47-sharp semisharp
|11/5 S-diesis down sharp
|-
|11
|raflat synflat sharp
|7-flat sharp
|7-flat sharp
|7-flat sharp
|7 comma down sharp
|-
|12
|synflat sharp
|5-flat sharp
|5-flat sharp
|5-flat sharp
|5 comma down sharp
|-
|13
|rasharp synflat sharp
|17-flat sharp
|43-flat sharp
|17-flat sharp
|17 comma down sharp
|-
|14
|raflat sharp
|17-sharp 5-flat sharp
|17-flat sharp
|double 11-flat
|5/7 comma down sharp
|-
|15
| colspan="5" |sharp
|}
{{Navbox EDO}}
{{Cat|Edos}}

159edo

2026-04-07T23:48:01Z

Aura:

'''159edo''', or 159 equal divisions of the octave, is the equal tuning featuring steps of (1200/159) ~= 7.55 cents, 159 of which stack to the perfect octave [[2/1]]. Like [[53edo]], 159edo is an excellent approximation to Pythagorean tuning (stacking pure 3/2 fifths), however it includes near-just approximations of the 11th and 17th harmonics, and a slightly more accurate 7th harmonic, resulting in consistency up to the 17-odd-limit. The step-size, being slightly above the melodic just noticeable difference (JND) of 5 cents as well as more than twice the harmonic JND of the average trained musician at 3.5 cents, enables one to perform fluid modulations by means of [[comma pump]]s as well as by step substitutions. Furthermore, 159edo, like a number of higher edos, is characterized by being capable of imitating the intervals of smaller tuning systems - in this case, you get detemperings of [[10edo]], [[12edo]], [[13edo]], [[14edo]], [[17edo]], [[19edo]], [[22edo]], [[24edo]] and [[31edo]] among others with errors smaller than the melodic JND.

The interval qualities supported by 159edo are many, however, while every step of 159edo can be interpreted harmonically or subharmonically as being a 17-limit interval or simpler, some of the intervals you get have rather complex interpretations in terms of odd-limit. While the [[perfect fifth]] is really only divisible by three due to 159edo being the three-fold multiple of 53edo- resulting in [[slendric]] temperament and hence 159edo's distinction from 53 in the 7-limit- the [[perfect fourth]] has a little more to offer in terms of divisions. For starters, the perfect fourth can be divided into two instances of 15/13, giving us [[Semifourth-generated scales|island]] temperament. Dividing the perfect fourth into three instances of 11/10 gives us [[pine]] temperament. Dividing the perfect fourth into six instances of an interval which can be interpreted as 21/20 and 22/21 tempered together gives us [[sextilifourths]] temperament. The perfect fourth can also be cut into eleven intervals which, individually, are half of a Pythagorean limma, giving us a number of temperaments based on the exact interpretation of the semilimma.

On top of all that, there are a number of microtemperament-based structures also supported by 159edo, each of which provides some decent, unexpected melodic possibilities. For instance, one can split the Pythagorean minor third into three instances of 128/121 which gives us [[nexus]] temperament- a temperament which also happens to split the Pythagorean diatonic semitone into two, and the octave into three. One can also split the Ptolemaic minor third into three instances of 17/16, giving us [[archagall]] temperament, which is named for certain tunings found in other temperaments producing fractal-like acoustics. There's also the ability to split the Ptolemaic major sixth into six instances of 11/9, leading to [[parimic]] temperament. In addition, there's also the ability to split the septimal supermajor third into two instances of 17/15, leading to [[fidesmic]] temperament, which acts like a more accurate rendition of [[archy]] temperament in a different subgroup, and this can be exploited for modulation purposes. As if that weren't enough, there's the possibility of splitting the septimal subminor third into five instances of 33/32, leading to [[quartismic]] temperament. Furthermore, there's the possibility of splitting the greater tridecimal neutral tenth into three instances of 27/20, producing [[phaotismic]] temperament, and the list goes on.

== Theory ==
159edo was first used for maqams by Ozan Yarman. It was later put to use by Aura for its ability to handle near-just quartertones derived from the 2.3.11 subgroup on top of the 5-limit foundation provided by 53edo.

==== Edostep interpretations ====
159edo's edostep has the following interpretations in the 2.3.5.11.17 subgroup:

* 243/242, the difference between the 11-limit artoneutral third 11/9, and the 11-limit tendoneutral third 27/22
* 256/255, the difference between 16/15 and 17/16
* 289/288, the difference between 17/16 and 18/17

159edo tempers out the following commas in the 17-limit:

* The schisma (the difference between 5/4 and the Pythagorean diminished fourth)
* The vulture comma (the difference between four 320/243 intervals and the tritave)
* The amiton (the difference between a stack of five 10/9 intervals and 27/16)
* The kleisma (the difference between a stack of three 25/24 intervals and 9/8)
* The semicomma (the difference between a stack of three 75/64 intervals and 8/5)
* The gamelisma (the difference between a stack of three 8/7 intervals and 3/2)
* 385/384 (the difference between 77/64 and 6/5)
* The pine comma (the difference between a stack of three 11/10 intervals and 4/3)
* 625/624 (the difference between 25/24 and 26/25)
* 676/675 (the difference between a stack of two 15/13 intervals and the perfect fourth)
* The twosquare comma (the difference between a stack of two 33/32 intervals and 17/16)

==== JI approximation ====
Although 159edo inherits its approximations of the 5-limit from 53edo, the 5th harmonic can nonetheless be stacked twice without accumulating too much error, rendering it sufficient for Western Classical usage. While the 7th harmonic is technically more accurate in terms of absolute error than in 53edo, the relative error doesn't allow one to stack more than one instance of 7/4 without excessive error accumulation, and the same is true with 13/8. Regardless, the inconsistency remains less than 10 cents even when either interval is stacked three times, and 13/7 or 14/13 is tuned almost perfectly. As a whole, 159edo is characterized by its combination of accuracy in the 2.3.5.11.17 subgroup, and a series of compromises in the 7.13.19.23.29 subgroup- among the compromises are the slendric, [[marveltwin]], [[nestoria]], [[minor semivicemic]] and [[brunisimic]] temperaments.
{{Harmonics in ED|159|31|0}}

Currently, the [[ADIN]] system used for interval naming elsewhere on this site fails for 159edo, so another set of interval names will be used here, though the ADIN names will be referenced in places. Furthermore, because there are so many thirds, what is usually a single chart will be split into five.
{| class="wikitable"
|+Inframinor and Subminor Thirds in 159edo
!Quality
|Narrow Inframinor
|Inframinor
|Wide Inframinor
|Narrow Subminor
|Lesser Subminor (Septiminor)
|Greater Subminor
|Wide Subminor
|-
!Cents
|234
|242
|249
|257
|264
|272
|279
|-
!Just interpretation
|340/297
|1024/891
|15/13
|51/44
|7/6
|117/100
|20/17
|-
!Steps
|31
|32
|33
|34
|35
|36
|37
|}
{| class="wikitable"
|+Minor Thirds in 159edo
!Quality
|Narrow Minor (Gothminor)
|'''Pythagorean Minor (Triminor)'''
|Artomean Minor
|Tendomean Minor
|Ptolemaic Minor (Pentaminor)
|Wide Minor
|-
!Cents
|287
|'''294'''
|302
|309
|317
|325
|-
!Just interpretation
|33/28, 13/11
|'''32/27'''
|25/21
|153/128
|6/5
|135/112
|-
!Steps
|38
|'''39'''
|40
|41
|42
|43
|}
{| class="wikitable"
|+Supraminor, Neutral and Submajor Thirds in 159edo
!Quality
|Lesser Supraminor
|Greater Supraminor
|Artoneutral
|Tendoneutral
|Lesser Submajor
|Greater Submajor
|-
!Cents
|332
|340
|347
|355
|362
|370
|-
!Just interpretation
|40/33, 63/52
|39/32, 17/14
|11/9
|27/22
|16/13, 21/17
|99/80, 26/21
|-
!Steps
|44
|45
|46
|47
|48
|49
|}
{| class="wikitable"
|+Major Thirds in 159edo
!Quality
|Narrow Major
|Ptolemaic Major (Pentamajor)
|Artomean Major
|Tendomean Major
|'''Pythagorean Major (Trimajor)'''
|Wide Major (Gothmajor)
|-
!Cents
|377
|385
|392
|400
|'''408'''
|415
|-
!Just interpretation
|56/45
|5/4
|64/51
|63/50
|'''81/64'''
|14/11, 33/26
|-
!Steps
|50
|51
|52
|53
|'''54'''
|55
|}
{| class="wikitable"
|+Supermajor and Ultramajor Thirds in 159edo
!Quality
|Narrow Supermajor
|Lesser Supermajor
|Greater Supermajor (Septimajor)
|Wide Supermajor
|Narrow Ultramajor
|Ultramajor
|Wide Ultramajor
|-
!Cents
|423
|430
|438
|445
|453
|460
|468
|-
!Just interpretation
|51/40
|50/39
|9/7
|22/17
|13/10
|2673/2048
|891/680
|-
!Steps
|56
|57
|58
|59
|60
|61
|62
|}
Regular diatonic thirds are bolded.

=== Chords ===
159edo has a vast array of triads at its disposal, both fifth-bounded and fourth-bounded. However, the JI interpretation will tend to inform the usage of the various triads offered. The main exceptions to this rule involve chords that serve as detempers of other, smaller tuning systems.

== Notation ==
There are multiple different notation systems available for 159edo.

=== Ups and downs ===
Ups and downs notation typically represents a single step of 159edo by means of lifts and drops while using the original ups and downs to represent single steps of 53edo. You may also use ups and downs to instead represent a single 159edo step, but this makes notating the syntonic/Pythagorean comma inconvenient.

=== Syntonic-rastmic subchroma notation ===
Syntonic-rastmic subchroma notation, or SRS for short, is an alternative to Sagittal with fewer accidentals- one which was originally designed for 159edo. In 159edo, the synsharp (81/80) and the rasharp (243/242) are 3\159 (1\53) and 1\159 respectively, treating 159edo as a subdivision of 53edo wherein three rastmas make up a syntonic comma. So far, this is functionally identical to ups and downs notation (assuming the up and down are treated as 1\53), but syntonic-rastmic subchroma notation also supports halving of all of its accidentals. While 159edo divides neither the Pythagorean, syntonic, nor rastmic accidental pairs in half, this does allow for the notation of artoneutral and tendoneutral intervals via the inflection of a semisharp by half a rastma (which ends up at a full 159edo step, specifically 7 steps for the artodemisharp and 8 steps for the tendodemisharp).

The syntonic and rastmic accidentals represent their just mappings when generalized to other EDOs.

=== Formal comma systems ===
The Functional Just System and HEJI share the same accidentals for 5 and 11 (though notated with each system's respective symbol); the 11-comma is the artodemisharp and the 5-comma is the synsharp. Neutral FJS uses the standard demisharp and half-rastma as accidentals, meaning that like with SRS notation, the artodemisharp and tendodemisharp become 11-limit accidentals. FJS, Neutral FJS, and HEJI also share an accidental for 7, which in 159edo in particular is equivalent to a rastma plus a syntonic comma.

The Sagittal sequence for 159edo will also be provided.

The simplest sequence spanning an entire 159edo chromatic semitone is thus as follows:
{| class="wikitable"
|+
!Rastmas
!SRS
!FJS
!HEJI
!Neutral FJS
!Sagittal
|-
|0
| colspan="5" |natural
|-
|1
|rasharp
|17-flat 5-sharp
|17-sharp
|double 11-sharp
|5/7 kleisma up
|-
|2
|raflat synsharp
|17-sharp
|43-sharp
|17-sharp
|17 comma up
|-
|3
|synsharp
|5-sharp
|5-sharp
|5-sharp
|5 comma up
|-
|4
|rasharp synsharp
|7-sharp
|7-sharp
|7-sharp
|7 comma up
|-
|5
|raflat double synsharp
|47-sharp
|47-sharp
|47-flat semisharp
|11/5 S-diesis up
|-
|6
|demisynflat demisharp / double synsharp
|13-sharp
|13-flat sharp
|13-flat semisharp
|25 S-diesis up
|-
|7
|artodemisharp
|11-sharp
|11-sharp
|11-flat semisharp
|11 M-diesis up
|-
|8
|tendodemisharp
|11-flat sharp
|11-flat sharp
|11-sharp semisharp
|11 L-diesis up
|-
|9
|demisynsharp demisharp / double synflat sharp
|13-flat sharp
|13-sharp
|13-sharp semisharp
|25 S-diesis down sharp
|-
|10
|rasharp double synflat sharp
|47-flat sharp
|47-flat sharp
|47-sharp semisharp
|11/5 S-diesis down sharp
|-
|11
|raflat synflat sharp
|7-flat sharp
|7-flat sharp
|7-flat sharp
|7 comma down sharp
|-
|12
|synflat sharp
|5-flat sharp
|5-flat sharp
|5-flat sharp
|5 comma down sharp
|-
|13
|rasharp synflat sharp
|17-flat sharp
|43-flat sharp
|17-flat sharp
|17 comma down sharp
|-
|14
|raflat sharp
|17-sharp 5-flat sharp
|17-flat sharp
|double 11-flat
|5/7 comma down sharp
|-
|15
| colspan="5" |sharp
|}
{{Navbox EDO}}
{{Cat|Edos}}

EDO

2026-04-07T06:52:31Z

Aura: Added links

An '''equal division of the octave''' ('''EDO''' or '''edo''', /ˈidoʊ/ ''EE-doh'' or /idiˈoʊ/ ''ee-dee-OH'') is a tuning system constructed by dividing the [[octave]] into a number of equal steps. It is a type of equal temperament.

The dominant modern tuning system may be called 12edo (12-EDO) because it divides the octave into 12 semitones that are all the same size. It may also be called 12-tone equal temperament or 12-TET, but this is discouraged because it does not specify which interval is being equally divided.

An edo with the same number of notes as a certain [[MOS]] will have crudely similar properties, as will one with the same number of notes as the MOS has L steps. These two edos form the boundaries of how the MOS can be tuned.

The notation ''m''\''n'' denotes ''m'' steps of ''n''-edo, i.e. the frequency ratio 2^(''m''/''n'').

== Uses ==

Edos are the most common type of tuning system in contemporary xenharmony. Unlike other types such as rank-2 temperaments and just intonation scales, equal temperaments allow for free modulation and transposition due to their uniform step size. That is, every n-step interval is the same as every other n-step interval. This comes at the expense of less freedom in approximating target intervals. It also encourages a less structured approach to composition where pitch shifts and interval quality changes can happen without much deeper meaning.

== List of edos ==
''Do not add subgroups to edos larger than 93; these are assumed to reasonably represent all prime-limits.''
{| class="wikitable"
|+ Popular edos are highlighted. Temperaments are capitalized and can be found in the [[List of regular temperaments]].
|-
!Edo
!Description
!First twelve steps (¢)
!Fifth (¢)
!Edostep interpretation
! colspan="2" |Example basic (in 2...23, primes and 9) and [[erac]] groups
|-
| rowspan="2" |1
| rowspan="2" |Equivalent to the 2-limit.
| rowspan="2" |1200
| rowspan="2" |1200
| rowspan="2" |2/1
|2
|-
|2
|-
| rowspan="2" |2
| rowspan="2" |Just a 12edo tritone.
| rowspan="2" |600, 1200
| rowspan="2" |600
| rowspan="2" |''none available in basic subgroup''
|2
|-
|2.<3.>>5.>>7.<17
|-
| rowspan="2" |3
| rowspan="2" |An augmented triad.
| rowspan="2" |400, 800, 1200
| rowspan="2" |800
| rowspan="2" |5/4
|2.5
|-
|2.>3.5.>19?
|-
| rowspan="2" |4
| rowspan="2" |A diminished tetrad.
| rowspan="2" |300, 600, 900, 1200
| rowspan="2" |600
| rowspan="2" |19/16
|2.19
|-
|2.<3.<5.<7.<17
|-
| rowspan="2" class="thl" |[[5edo|5]]
| rowspan="2" |Equalized [[pentic]], collapsed [[diatonic]], and the smallest edo to have strong melodic properties. Good approximation of 2.3.7 for its size.
| rowspan="2" |240, 480, 720, 960, 1200
| rowspan="2" |720
| rowspan="2" |9/8, 8/7, 7/6
|2.3.7
|-
|2.>>3.<7
|-
| rowspan="2" |6
| rowspan="2" |Also known as the whole-tone scale, 6edo is a subset of 12edo. Good approximation of 2.5.7 for its size.
| rowspan="2" |200, 400, 600, 800, 1000, 1200
| rowspan="2" |600, 800
| rowspan="2" |9/8, 10/9, 28/25, 8/7
|2.9.5
|-
|2.9.5.>7
|-
| rowspan="2" class="thl" |[[7edo|7]]
| rowspan="2" |Equalized [[diatonic]], and the first edo to (very vaguely) support diatonic functional harmony.
| rowspan="2" |171.4, 342.9, 514.3, 685.7, 857.1, 1028.6, 1200
| rowspan="2" |685.7
| rowspan="2" |9/8, 10/9, 16/15
|2.3.5.11.13
|-
|2.<3.<<5.>13
|-
| rowspan="2" |[[8edo|8]]
| rowspan="2" |Notable for containing few strong consonances, but still contains in-tune ratios 12/11 and 13/10.
| rowspan="2" |150, 300, 450, 600, 750, 900, 1050, 1200
| rowspan="2" |750
| rowspan="2" |''none available in basic subgroup''
|2.19
|-
|2.x3.x5.x7.x11.x13
|-
| rowspan="2" |[[9edo|9]]
| rowspan="2" |The first edo to support the [[antidiatonic]] scale, loosely resembling the pelog scale. It contains approximations to many [[Prime limit|7-limit]] intervals, but not the [[7/4]] itself (see erac group).
| rowspan="2" |133.3, 266.7, 400, 533.3, 666.7, 800, 933.3, 1066.7, 1200
| rowspan="2" |666.7
| rowspan="2" |9/8, 16/15, 25/24
|2.5.11
|-
|2.<<3.>5.<<7
|-
| rowspan="2" |[[10edo|10]]
| rowspan="2" |The doubling of 5edo, useful as an interval categorization archetype and as a melodic system in its own right, supporting [[mosh]].
| rowspan="2" |120, 240, 360, 480, 600, 720, 840, 960, 1080, 1200
| rowspan="2" |720
| rowspan="2" |16/15, 10/9, 81/80, 36/35
|2.3.5.7.13
|-
|2.>>3.<7.13
|-
| rowspan="2" |[[11edo|11]]
| rowspan="2" |Basic smitonic and checkertonic. Simplest reasonable tuning of [[Orgone]].
| rowspan="2" |109.1, 218.2, 327.3, 436.4, 545.5, 654.5, 763.6, 872.7, 981.8, 1090.9
| rowspan="2" |654.5, 763.6
| rowspan="2" |128/119, 17/16, 18/17
|2.9.7.11
|-
|2.x3.x5.7.11
|-
| rowspan="2" class="thl" |[[12edo|12]]
| rowspan="2" |The basic tuning of [[diatonic]], and consequently the most widespread EDO. Supports the 5-limit decently well.
| rowspan="2" |{{First 12 edo intervals|edo=12}}
| rowspan="2" |700
| rowspan="2" |256/243, [[chromatic semitone]], 16/15, 25/24
|2.3.5.17.19
|-
|2.3.>5.>>7.17.19
|-
|[[13edo|13]]
|Basic [[oneirotonic]], [[archeotonic]], and [[gramitonic]].
|{{First 12 edo intervals|edo=13}}
|646.2, 738.5
|17/16, 18/17, 19/18, 20/19
|2.5.11.13.17.19.23
|-
|[[14edo|14]]
|Basic [[semiquartal]].
|{{First 12 edo intervals|edo=14}}
|685.7
|28/27, 21/20, 15/14
|2.3.7.13
|-
| class="thl" |[[15edo|15]]
|The basic tuning of Zarlino's [[Diatonic|intense diatonic]], a subset of pentawood which is itself a degenerate tuning of blackdye. Supporting Porcupine temperament and dubitably the 11-limit.
|{{First 12 edo intervals|edo=15}}
|720
|81/80, 25/24, 16/15, 33/32, 36/35
|2.3.5.7.11.23
|-
|[[16edo|16]]
|The most popular antidiatonic edo, which supports [[Trismegistus]] and [[Mavila]].
|{{First 12 edo intervals|edo=16}}
|675, 750
|20/19, 133/128, 26/25
|2.5.7.13.19
|-
| class="thl" |[[17edo|17]]
|Smallest non-12 edo whose fifth is of comparable quality to 12edo's; thus, unless you're satisfied with 7edo, the first xen edo that also allows use of the MOS diatonic scale. Noted for its melodically tense third-tone, neogothic minor chords, and approximation to the 13th harmonic. The largest edo which supports a full piano range in a DAW.
|{{First 12 edo intervals|edo=17}}
|705.9
|256/243, 24/23, 27/26, 33/32
|2.3.13.23
|-
| rowspan="2" |[[18edo|18]]
| rowspan="2" |[[Straddle-3]] version of 12edo; provides the basic version of the straddle-3 diatonic 5L1m1s as well as soft smitonic, hard oneirotonic, and basic taric.
| rowspan="2" |{{First 12 edo intervals|edo=18}}
| rowspan="2" |666.6, 733.3
| rowspan="2" |
|2.9.5.21.13
|-
|2.xx3.>5.>>7.<11.<13
|-
| class="thl" |[[19edo|19]]
|A simple tuning of Meantone, with a very accurate 6/5 and a reasonably good 5/4 and 9/7. Supports Semaphore temperament.
|{{First 12 edo intervals|edo=19}}
|694.7
|25/24, [[diaschisma]], 36/35, 28/27
|2.3.5.23
|-
| |20
|Has a balzano (2L7s) MOS scale and accurate 13:16:19 triads.
|{{First 12 edo intervals|edo=20}}
|660, 720
|
|2.7.11.13.19
|-
| rowspan="2" |[[21edo|21]]
| rowspan="2" |Basic tuning of 7-limit Whitewood, favoring 7/4 over 5/4. Has soft (hardness 3/2) oneirotonic. Has an extremely accurate 23rd harmonic. Has a 12edo major third and a neogothic minor third, so major and minor triads sound somewhat like compressed neogothic triads.
| rowspan="2" |{{First 12 edo intervals|edo=21}}
| rowspan="2" |685.7, 742.9
| rowspan="2" |
|2.3.5.7.23
|-
|2.x>3.x<5.7.x<11.x<13.23
|-
| rowspan="2" class="thl" |[[22edo|22]]
| rowspan="2" |Represents the 7-limit and 11-limit decently well, serving as the primary tuning of Pajara and also a good Superpyth tuning, especially for Archy.
| rowspan="2" |{{First 12 edo intervals|edo=22}}
| rowspan="2" |709.1
| rowspan="2" |
|2.3.5.7.11.17
|-
|2.>3.<5.>>7.<11
|-
|23
|The largest edo without a diatonic, 5edo, or 7edo fifth. A straddle-3,5,7,11 edo. Has a hard armotonic and a very hard oneirotonic.
|{{First 12 edo intervals|edo=23}}
|678.3
|
|2.x3.x5.x7.x11.13.17.23
|-
| div class="thl"|[[24edo|24]]
|Regular old quarter-tones. Good at representing neutral intervals like 11/9, and tempers artoneutral and tendoneutral thirds to the same interval.
|{{First 12 edo intervals|edo=24}}
|700
|
|2.3.11.13.17.19
|-
|25
|A straddle-fifth tuning with a 672c fifth that supports Mavila, or that can be used as the generator for Trismegistus with the more accurate 720c fifth. Also supports Blackwood and Didacus. The largest edo which supports five octaves in a DAW without substantial modification.
|{{First 12 edo intervals|edo=25}}
|720
|
|2.5.7.19
|-
|[[26edo|26]]
|A simple tuning of Flattone. Has an absurdly accurate 7/4.
|{{First 12 edo intervals|edo=26}}
|692.7
|
|2.3.7.11.13
|-
|27
|A good tuning for [[Archy]] and Sensi. It has 3/2 at 16 steps.
|{{First 12 edo intervals|edo=27}}
|711.1
|
|2.3.5.7.13.23
|-
|28
|A tuning of 7-limit Whitewood favoring 5/4 over 7/4. Has a very hard [[oneirotonic]] scale converging on Buzzard temperament.
|{{First 12 edo intervals|edo=28}}
|685.7, 728.6
|
|2.3.5.7.11
|-
|[[29edo|29]]
|Another neogothic tuning, and the first edo to have a more accurate perfect fifth than 12edo, so it also functions as an approximation of Pythagorean tuning, and as is typical with small Pythagorean edos, Garibaldi. It has 4/3 at 12 steps.
|{{First 12 edo intervals|edo=29}}
|703.4
|
|2.3.7/5.11/5.13/5.19.23
|-
|30
|Doubled 15edo. Due to 15edo's ~25% error on some harmonics, this becomes a straddle-3 and -5 system, which also inherits 10edo's 13/8.
|{{First 12 edo intervals|edo=30}}
|680, 720
|
|2.x3.x5.7.11.13
|-
| class="thl" |[[31edo|31]]
|The definitive Septimal Meantone and Mohajira tuning, and the largest edo which supports four octaves in a DAW without substantial modification.
|{{First 12 edo intervals|edo=31}}
|696.8
|
|2.3.5.7.11.23
|-
|32
|A standard tuning of [[Archy#5/4 as doubly limma-flat major third (5 & 37)|Ultrapyth]] (5 & 37); also contains 16edo as a subset allowing for the use of antidiatonic. It is a 2.x3.5 Meantone tuning; otherwise it is "okay" at most primes up to 23, similarly to 15edo for 11. It has a 5-limit zarlino scale, although it is closer to mosh than to mosdiatonic.
|{{First 12 edo intervals|edo=32}}
|712.5
|
|2.3.7.11.17.19.23
|-
|33
|Contains a very flat perfect fifth, and as a result a near-7edo diatonic, supporting Deeptone and with a very well-tuned 13 and 11edo's 7/4 and 11/8. Supports Semaphore with the flat 7/4, which can be interpreted as [[Barbados]] temperament in the patent val.
|{{First 12 edo intervals|edo=33}}
|690.9
|
|2.3.11.13.17.19.23
|-
| rowspan="2" class="thl"|[[34edo|34]]
| rowspan="2" |An accurate medium-sized non-Meantone 5-limit edo. Supports Diaschismic, Tetracot, and Kleismic, alongside equally halving 3/2 and 4/3 and thus having both neutrals and interordinals. It can be notated with the 12-form and/or 10-form.
It is the double of 17edo, which it takes its circle of fifths from.
| rowspan="2" |{{First 12 edo intervals|edo=34}}
| rowspan="2" |705.9
| rowspan="2" |
|2.3.5.13.23
|-
|2.3.5.x7.13.x19.23
|-
|35
|Contains both 5edo and 7edo and is thus a direct example of a straddle-3 system.
|{{First 12 edo intervals|edo=35}}
|685.7, 720.0
|
|2.5.7.11.17
|-
|36
|Triple 12edo, which functions as an extremely accurate [[2.3.7 subgroup|septal]] Compton and Slendric system.
|{{First 12 edo intervals|edo=36}}
|700
|
|2.3.7.13.17.19.23
|-
| rowspan="2" class="thl" |[[37edo|37]]
| rowspan="2" |An extremely accurate no-3 (or straddle-3) 13-limit edo. Most temperaments in this subgroup have near-optimal tunings in 37edo. Can also be seen as having an Archy 3, as in Porcupine.
| rowspan="2" |{{First 12 edo intervals|edo=37}}
| rowspan="2" |681.1, 713.5
| rowspan="2" |
|2.9.5.7.11.13.17.19
|-
|2.<x3.5.7.11.13.17.19
|-
|38
|19edo with neutrals. Functions as a tuning of Mohajira, as it has a good (and consistently mapped) 11/9 despite tuning 11 poorly.
|{{First 12 edo intervals|edo=38}}
|694.7
|
|2.3.5.7.13.17.23
|-
|39
|Is a super-Pythagorean (though not strictly Superpyth as in the temperament) diatonic system with "gothmajor" and "gothminor" thirds in-between standard septimal and neogothic thirds.
|{{First 12 edo intervals|39|edo=39}}
|707.7
|
|2.3.11
|-
|[[40edo|40]]
|An acceptable tuning of diminished and deeptone. As a result, the 5-limit diatonic is omnidiatonic rather than zarlino or mosdiatonic. Alternatively, can be used as a straddle-3 system.
|{{First 12 edo intervals|40|edo=40}}
|690
|
|
|-
| class="thl" |[[41edo|41]]
|The first reasonably accurate [[Hemifamity]] edo (which is also a Garibaldi edo). Used for the Kite guitar. {{Adv|One of two viably small tunings of 11-limit [[penslen]].}}
|{{First 12 edo intervals|edo=41}}
|702.4
|81/80, 64/63, 49/48, 50/49, 55/54, 45/44
|2.3.5.7.11.13.19
|-
|42
|The largest EDO which supports three octaves in a DAW without substantial modification (considered a key cutoff for 'large EDOs' by Vector), and also the edo with the sharpest diatonic fifth, having a mosdiatonic chroma equivalent to a 12edo wholetone and being nearly 1/2-comma Archy.
|{{First 12 edo intervals|edo=42}}
|685.7, 714.3
|
|2.7.11.17.23
|-
|43
|A sharp-of-31edo Meantone tuning; its mapping of 11 is "[[Meantone|Huygens]]". Like all Meantone tunings that do not map 11/9 to a perfect neutral third, its 11/9 is sharp of neutral.
|{{First 12 edo intervals|edo=43}}
|697.7
|
|2.3.5.7.11.13.17
|-
|44
|A tuning which is, very prominently, straddle-7; its other prime harmonics up to 23 are within 25% error (except for 3, which is inherited from 22edo).
|{{First 12 edo intervals|edo=44}}
|709.1
|
|2.3.5.11.13.17.19.23
|-
|45
|A nearly optimal tuning of Flattone, compromising between a good 9/7 and a reasonable interseptimal diesis. Inherits 9edo's 7/6 and has 15edo as a subset.
|{{First 12 edo intervals|edo=45}}
|693.3, 720
|
|2.3.7.11.17.19
|-
| class="thl"|46
|The second reasonably accurate [[Hemifamity]] edo. Has a diatonic with neogothic thirds. {{Adv|One of two viably small tunings of 11-limit [[penslen]].}}
|{{First 12 edo intervals|edo=46}}
|704.3
|
|2.3.5.7.11.13.17.23
|-
|47
|The first edo with two distinct mosdiatonic scales. Supports Magic and Archy with its sharp fifth and Deeptone with its flat fifth. Has a very accurate 9/8 as a straddle-3 system, which generates a sort of Schismic analogue of Didacus.
|{{First 12 edo intervals|edo=47}}
|689.4, 714.9
|
|2.5.7.13.17
|-
|48
|Four times 12edo, associated with [[Buzzard]] temperament.
|25, 50, 75, 100, 125, 150, 175, 200, 225, 250, 275, 300
|700
|
|2.3.7.11.17.19.23
|-
|49
|A nearly optimal tuning of Archy which maps 5/4 to a limma-flat major third, and squeezes a 14/11 into the 2-edostep limma between 5/4 and 9/7. It also supports straddle-3 Meantone (or, more conventionally, Didacus).
|{{First 12 edo intervals|edo=49}}
|710.2
|
|2.5.17.19
|-
|50
|Approaches golden Meantone, and serves as a definitive tuning of Meanpop. Also contains 25edo as a subset, along with 10edo, and as such has an accurate 5, 7, and 13 with the latter two divisible into 5 parts.
|{{First 12 edo intervals|edo=50}}
|696
|
|2.3.5.11.13.23
|-
|51
|Straddle-5 and -11, with a val option mapping 6/5 to 11/9, and one mapping the 11-limit neutral thirds together with the 13-limit ones at the perfect neutral third. Has 17edo as a subset.
|{{First 12 edo intervals|edo=51}}
|705.9
|
|2.3.7.13
|-
|52
|Doubles 26edo, adding a sharp Archy fifth and a more accurate 5/4 which support Porcupine temperament.
|{{First 12 edo intervals|edo=52}}
|692.3, 715.4
|
|2.5.7.11.19.23
|-
|class="thl"|[[53edo|53]]
|Nearly identical to a circle of 53 Pythagorean fifths, serving as the most directly obvious tuning of Schismic temperament (which also functions as a Garibaldi temperament).
|{{First 12 edo intervals|edo=53}}
|701.9
|81/80, 64/63, 50/49, 65/64, 512/507, 91/90
|2.3.5.7.13.19
|-
|54
|Double 27edo, and the sharper end of the Pajara tuning range. Can alternatively be used as a very flat Deeptone system or combining the fifths as a straddle-fifth system.
|
|688.9, 711.1
|
|2.11.13.17.23
|-
|55
|A very sharp Meantone tuning, which is so sharp that it does not even support Septimal Meantone, and is best interpreted as Mohajira as it pertains to Meantone extensions.
|
|698.2
|
|2.3.5.11.17.23
|-
|56
|An edo with a diatonic scale in the "shrub" region, with a diatonic major third between neogothic and septimal major. Tempers 9/7 to 450c, however this is not actually an interordinal as it is distinguished from 21/16 by a single edostep.
|
|
|
|
|-
|57
|Has 19edo's 5-limit combined with better interpretations of higher limits.
|
|
|
|
|-
|58
|Double of 29edo, and is the first edo to support hemipythagorean harmony better than 24edo. Thus, it has perfect neutrals and interordinals, and is thus useful for defining categories of intervals.
|{{First 12 edo intervals|edo=58}}
|703.4
|
|2.3.7.17
|-
|59
|Has the sharpest best fifth for an edo with a 2-step diatonic semitone. It supports Porcupine with a flatter tuning of the generator than 22edo, but sharper than 37edo; it is in fact 22 + 37.
|
|
|
|
|-
|60
|5 sets of 12edo, supporting Magic temperament and having 10edo's 7 and 13, also supporting 7-limit Compton temperament and many structures associated with 10edo and 15edo with their respective mappings.
|
|
|
|
|-
|61
|Makes 8/7 - 32/27 - 6/5 - 16/13 - 5/4 - 81/64 - 21/16 equidistant.
|
|
|
|
|-
|62
|Doubled 31edo, which shares its mappings through the 11-limit.
|
|
|
|
|-
|63
|A very good general system, as it is triple 21edo, whose harmonics are generally off by about 1/3 of a step. It is also the largest edo which supports two octaves in a DAW without substantial modification.
|{{First 12 edo intervals|edo=63}}
|704.8
|
|2.3.5.7.11.13.23
|-
|64
|An edo whose intervals are generally far from just intonation, being straddle-3, -5, and -11. It can function as a tuning of Flattone with its flat 3, 5, and 7.
|{{First 12 edo intervals|edo=64}}
|693.8, 712.5
|
|2.13.19
|-
|65
|A non-Garibaldi Schismic system (in fact, it supports Sensi), and a straddle-7 and -13 system.
|{{First 12 edo intervals|edo=65}}
|701.5
|
|2.3.5.11.17.19
|-
|66
|Tripled 22edo, with an improved approximation to 7 that supports Slendric.
|{{First 12 edo intervals|edo=66}}
|709.1, 690.9
|
|2.5.7.11.13.17
|-
|67
|Approximate 1/6-comma Meantone and Slendric edo, which also supports [[Orgone]].
|{{First 12 edo intervals|edo=67}}
|698.5
|
|2.3.7.11.13.17.23
|-
|68
|Doubled 34edo, which improves its approximation to 7 while retaining 34edo's structural properties; it is similar to how 34edo retains 17edo's 2.3.13 while adding 5. 5/3 is twice 9/7, supporting Sensamagic. Additionally, there is a second diatonic fifth.
|{{First 12 edo intervals|edo=68}}
|705
|
|2.3.5.7.11.17.19
|-
|69
|A nice tuning that is approximately 2/7-comma Meantone, somewhat between standard Septimal Meantone and 19edo. As a result, it is a Mohajira system (setting 7/4 to the semiflat minor seventh) but not a Septimal Meantone system (as the augmented sixth is interordinal). Its 7/4 is, however, reached by stacking its second-best fourth twice, which means 69edo supports Archy with the sharp fifth. As a dual-fifth system, it is neogothic.
|{{First 12 edo intervals|edo=69}}
|695.7, 713
|
|2.5.7.11.13.17.19.23
|-
|70
|Double 35edo, and thus contains a diatonic scale that is exactly in the middle of the diatonic tuning range. It is a [[Hemifamity]] system, as is typical with tunings with sharpened fifths.
|{{First 12 edo intervals|edo=70}}
|702.9
|
|2.3.11.13.17
|-
|71
|A dual-fifth system. The sharp fifth is within the Superpyth tuning range (and produces the same mapping for 5 as Superpyth), despite not supporting Archy. The flat fifth, analogously, produces Flattone's mapping for 7 and is well-tuned for Flattone, but does not support Flattone.
|{{First 12 edo intervals|edo=71}}
|693. 709.9
|
|2.5.7.13.17.23
|-
|[[72edo|72]]
|A multiple of 12edo and a very good Miracle and Compton system. It is also the first multiple of 12 to have a second MOS diatonic.
|{{First 12 edo intervals|edo=72}}
|700
|
|2.3.5.7.11.17.19.23
|-
| colspan="6" |...
|-
|80
|Notable for its sharp tendency and high capacity for higher-limit harmony, particularly noted by Osmium.
|{{First 12 edo intervals|edo=80}}
|705
|
|2.3.5.11.13.17.19.23
|-
|81
|A convergent to Golden Meantone, and the last one to support Meantone in the patent val.
|{{First 12 edo intervals|edo=81}}
|696.3
|
|2.9.5.11.13.17.19
|-
| colspan="6" |...
|-
|84
|A tuning system notable for its large number of contorted mappings, and also for its tuning of [[Orwell]].
|{{First 12 edo intervals|edo=84}}
|700
|
|2.3.5.7.13.19.23
|-
| colspan="6" |...
|-
|87
|A good 17-limit system, which shares 29edo's 3-limit. Essentially optimal for 13-limit [[Rodan]] (41 & 46) temperament.
Its step size is near a significant value of approximately 13 cents where all intervals become approximated by the edo to within a reasonable degree of intonational error on free-pitch instruments.
|{{First 12 edo intervals|edo=87}}
|703.4
|
|2.3.5.7.11.13.17
|-
| colspan="6" |...
|-
|93
|The triple of 31edo. As a Meantone system, it places 11/9 sharp of the neutral third, tempered together with 16/13.
Its step size is near a significant value of approximately 13 cents where all intervals become approximated by the edo to within a reasonable degree of intonational error on free-pitch instruments. As such, it is the last edo whose subgroup is specified on this table.
|{{First 12 edo intervals|edo=93}}
|696.8, 709.7
|
|2.5.7.11.13.17.19.23
|-
|[[94edo|94]]
|A Garibaldi system, being 41 + 53 and thus having a close-to-just tuning of Garibaldi.
Its step size is near a significant value of approximately 13 cents where all intervals become approximated by the edo to within a reasonable degree of intonational error on free-pitch instruments. As such, it is the first edo whose subgroup is listed as "-" on the table.
|{{First 12 edo intervals|edo=94}}
|702.1
|
| -
|-
| colspan="6" |...
|-
|99
|Perhaps the strongest 7-limit edo below 100. Supports [[Hemifamity]], [[Didacus]], and [[Ennealimmal]].
|{{First 12 edo intervals|edo=99}}
|703.0
|126/125, 225/224, kleisma
| -
|-
| colspan="6" |...
|-
|140
|A significant edo for interval categorization, as the next resolution level up from 58edo.
|{{First 12 edo intervals|edo=140}}
|702.9
|
| -
|-
| colspan="6" |...
|-
|[[159edo|159]]
|Triple of 53edo, with the ability to represent intonational differences on specific intervals, and which has been extensively practiced and studied by Aura.
|{{First 12 edo intervals|edo=159}}
|701.9
|
| -
|-
| colspan="6" |...
|-
|171
|Has a surgically accurate approximation of 7-limit just intonation and is at the intersection of the [[Schismic]] and [[Ennealimmal]] temperaments. It is also a [[Neutral]] temperament, as [[11/9]] is mapped to exactly half of a perfect fifth.
|{{First 12 edo intervals|edo=171}}
|701.8
| 225/224, 5120/5103, [[kleisma]]
| -
|-
| colspan="6" |...
|-
|200
|Notable for its extremely good approximation of 3/2, and also for being a [[Schismic]] and [[Slendric]] system with an 8/7 of exactly 234 cents.
|{{First 12 edo intervals|edo=200}}
|702
|
| -
|-
| colspan="6" |...
|-
|270
|Notable for its very accurate approximation of the 13-limit, with all intervals in the 15-odd-limit more in-tune than out-of-tune except for 15/13 and 26/15. It also does relatively well at approximating higher prime limits.
|{{First 12 edo intervals|edo=270}}
|702.2
|385/384, 364/363, 352/351, 351/350, 325/324, 540/539, 441/440
| -
|-
| colspan="6" |...
|-
|306
|Notable for being a convergent to 3/2, and for being a multiple of 34edo (a tuning with major structural significance). Its step is the difference between a just 3/2 and 34edo's 3/2.
|{{First 12 edo intervals|edo=306}}
|702
|
| -
|-
| colspan="6" |...
|-
|311
|An edo renowned for being a good edo for the whole 41-odd-limit and quite a bit more (mainly composite) harmonics above 41.
|{{First 12 edo intervals|edo=311}}
|702.3
|
| -
|-
| colspan="6" |...
|-
|612
|Separates and accurately tunes the syntonic and Pythagorean commas, and thus also the schisma, which separates a practically just 3/2 from 12edo's approximation. Mostly notable as the double of 306edo (and thus another 34edo multiple, and consequently a 68edo multiple).
|{{First 12 edo intervals|edo=612}}
|702
|
| -
|-
| colspan="6" |...
|-
|665
|Notable for being a convergent to 3/2. Tempers out the "satanic comma", so-named because it equates 666 perfect fifths (octave-reduced) to a single perfect fifth.
|{{First 12 edo intervals|edo=665}}
|702
|
| -
|}
{{Cat|Core knowledge}}

159edo

2026-04-07T06:45:01Z

Aura: /* Notation */

'''159edo''', or 159 equal divisions of the octave, is the equal tuning featuring steps of (1200/159) ~= 7.55 cents, 159 of which stack to the perfect octave [[2/1]]. Like [[53edo]], 159edo is an excellent approximation to Pythagorean tuning (stacking pure 3/2 fifths), however it includes near-just approximations of the 11th and 17th harmonics, and a slightly more accurate 7th harmonic, resulting in consistency up to the 17-odd-limit. The step-size, being slightly above the melodic just noticeable difference (JND) of 5 cents as well as more than twice the harmonic JND of the average trained musician at 3.5 cents, enables one to perform fluid modulations by means of [[comma pump]]s as well as by step substitutions. Furthermore, 159edo, like a number of higher edos, is characterized by being capable of imitating the intervals of smaller tuning systems - in this case, you get detemperings of [[10edo]], [[12edo]], [[13edo]], [[14edo]], [[17edo]], [[19edo]], [[22edo]], [[24edo]] and [[31edo]] among others with errors smaller than the melodic JND.

The interval qualities supported by 159edo are many, however, while every step of 159edo can be interpreted harmonically or subharmonically as being a 17-limit interval or simpler, some of the intervals you get have rather complex interpretations in terms of odd-limit. While the [[perfect fifth]] is really only divisible by three due to 159edo being the three-fold multiple of 53edo- resulting in [[slendric]] temperament and hence 159edo's distinction from 53 in the 7-limit- the [[perfect fourth]] has a little more to offer in terms of divisions. For starters, the perfect fourth can be divided into two instances of 15/13, giving us [[Semifourth-generated scales|island]] temperament. Dividing the perfect fourth into three instances of 11/10 gives us [[pine]] temperament. Dividing the perfect fourth into six instances of an interval which can be interpreted as 21/20 and 22/21 tempered together gives us [[sextilifourths]] temperament. The perfect fourth can also be cut into eleven intervals which, individually, are half of a Pythagorean limma, giving us a number of temperaments based on the exact interpretation of the semilimma.

On top of all that, there are a number of microtemperament-based structures also supported by 159edo, each of which provides some decent, unexpected melodic possibilities. For instance, one can split the Pythagorean minor third into three instances of 128/121 which gives us [[nexus]] temperament- a temperament which also happens to split the Pythagorean diatonic semitone into two, and the octave into three. One can also split the Ptolemaic minor third into three instances of 17/16, giving us [[archagall]] temperament, which is named for certain tunings found in other temperaments producing fractal-like acoustics. There's also the ability to split the Ptolemaic major sixth into six instances of 11/9, leading to [[parimic]] temperament. In addition, there's also the ability to split the septimal supermajor third into two instances of 17/15, leading to [[fidesmic]] temperament, which acts like a more accurate rendition of [[archy]] temperament in a different subgroup, and this can be exploited for modulation purposes. As if that weren't enough, there's the possibility of splitting the septimal subminor third into five instances of 33/32, leading to [[quartismic]] temperament. Furthermore, there's the possibility of splitting the greater tridecimal neutral tenth into three instances of 27/20, producing [[phaotismic]] temperament, and the list goes on.

== Theory ==
159edo was first used for maqams by Ozan Yarman. It was later put to use by Aura for its ability to handle near-just quartertones derived from the 2.3.11 subgroup on top of the 5-limit foundation provided by 53edo.

==== Edostep interpretations ====
159edo's edostep has the following interpretations in the 2.3.5.11.17 subgroup:

* 243/242, the difference between the 11-limit artoneutral third 11/9, and the 11-limit tendoneutral third 27/22
* 256/255, the difference between 16/15 and 17/16
* 289/288, the difference between 17/16 and 18/17

159edo tempers out the following commas in the 17-limit:

* The schisma (the difference between 5/4 and the Pythagorean diminished fourth)
* The vulture comma (the difference between four 320/243 intervals and the tritave)
* The amiton (the difference between a stack of five 10/9 intervals and 27/16)
* The kleisma (the difference between a stack of three 25/24 intervals and 9/8)
* The semicomma (the difference between a stack of three 75/64 intervals and 8/5)
* The gamelisma (the difference between a stack of three 8/7 intervals and 3/2)
* 385/384 (the difference between 77/64 and 6/5)
* The pine comma (the difference between a stack of three 11/10 intervals and 4/3)
* 625/624 (the difference between 25/24 and 26/25)
* 676/675 (the difference between a stack of two 15/13 intervals and the perfect fourth)
* The twosquare comma (the difference between a stack of two 33/32 intervals and 17/16)

==== JI approximation ====
Although 159edo inherits its approximations of the 5-limit from 53edo, the 5th harmonic can nonetheless be stacked twice without accumulating too much error, rendering it sufficient for Western Classical usage. While the 7th harmonic is technically more accurate in terms of absolute error than in 53edo, the relative error doesn't allow one to stack more than one instance of 7/4 without excessive error accumulation, and the same is true with 13/8. Regardless, the inconsistency remains less than 10 cents even when either interval is stacked three times, and 13/7 or 14/13 is tuned almost perfectly. As a whole, 159edo is characterized by its combination of accuracy in the 2.3.5.11.17 subgroup, and a series of compromises in the 7.13.19.23.29 subgroup- among the compromises are the slendric, [[marveltwin]], [[nestoria]], [[minor semivicemic]] and [[brunisimic]] temperaments.
{{Harmonics in ED|159|31|0}}

Currently, the [[ADIN]] system used for interval naming elsewhere on this site fails for 159edo, so another set of interval names will be used here, though the ADIN names will be referenced in places. Furthermore, because there are so many thirds, what is usually a single chart will be split into five.
{| class="wikitable"
|+Inframinor and Subminor Thirds in 159edo
!Quality
|Narrow Inframinor
|Inframinor
|Wide Inframinor
|Narrow Subminor
|Lesser Subminor (Septiminor)
|Greater Subminor
|Wide Subminor
|-
!Cents
|234
|242
|249
|257
|264
|272
|279
|-
!Just interpretation
|340/297
|1024/891
|15/13
|51/44
|7/6
|117/100
|20/17
|-
!Steps
|31
|32
|33
|34
|35
|36
|37
|}
{| class="wikitable"
|+Minor Thirds in 159edo
!Quality
|Narrow Minor (Gothminor)
|'''Pythagorean Minor (Triminor)'''
|Artomean Minor
|Tendomean Minor
|Ptolemaic Minor (Pentaminor)
|Wide Minor
|-
!Cents
|287
|'''294'''
|302
|309
|317
|325
|-
!Just interpretation
|33/28, 13/11
|'''32/27'''
|25/21
|153/128
|6/5
|135/112
|-
!Steps
|38
|'''39'''
|40
|41
|42
|43
|}
{| class="wikitable"
|+Supraminor, Neutral and Submajor Thirds in 159edo
!Quality
|Lesser Supraminor
|Greater Supraminor
|Artoneutral
|Tendoneutral
|Lesser Submajor
|Greater Submajor
|-
!Cents
|332
|340
|347
|355
|362
|370
|-
!Just interpretation
|40/33, 63/52
|39/32, 17/14
|11/9
|27/22
|16/13, 21/17
|99/80, 26/21
|-
!Steps
|44
|45
|46
|47
|48
|49
|}
{| class="wikitable"
|+Major Thirds in 159edo
!Quality
|Narrow Major
|Ptolemaic Major (Pentamajor)
|Artomean Major
|Tendomean Major
|'''Pythagorean Major (Trimajor)'''
|Wide Major (Gothmajor)
|-
!Cents
|377
|385
|392
|400
|'''408'''
|415
|-
!Just interpretation
|56/45
|5/4
|64/51
|63/50
|'''81/64'''
|14/11, 33/26
|-
!Steps
|50
|51
|52
|53
|'''54'''
|55
|}
{| class="wikitable"
|+Supermajor and Ultramajor Thirds in 159edo
!Quality
|Narrow Supermajor
|Lesser Supermajor
|Greater Supermajor (Septimajor)
|Wide Supermajor
|Narrow Ultramajor
|Ultramajor
|Wide Ultramajor
|-
!Cents
|423
|430
|438
|445
|453
|460
|468
|-
!Just interpretation
|51/40
|50/39
|9/7
|22/17
|13/10
|2673/2048
|891/680
|-
!Steps
|56
|57
|58
|59
|60
|61
|62
|}
Regular diatonic thirds are bolded.

=== Chords ===
159edo has a vast array of triads at its disposal, both fifth-bounded and fourth-bounded. However, the JI interpretation will tend to inform the usage of the various triads offered. The main exceptions to this rule involve chords that serve as detempers of other, smaller tuning systems.

== Notation ==
There are multiple different notation systems available for 159edo.

=== Ups and downs ===
Ups and downs notation represents a single step of 159edo by means of lifts and drops while using the original ups and downs to represent single steps of 53edo.

=== Sagittal notation ===
<todo>

=== Syntonic-rastmic subchroma notation ===
Syntonic-rastmic subchroma notation, or SRS for short, is an alternative to Sagittal with fewer accidentals- one which was originally designed for 159edo. In 159edo, the synsharp (81/80) and the rasharp (243/242) are 3\159 (1\53) and 1\159 respectively, treating 159edo as a subdivision of 53edo wherein three rastmas make up a syntonic comma. So far, this is functionally identical to ups and downs notation (assuming the up and down are treated as 1\53), but syntonic-rastmic subchroma notation also supports halving of all of its accidentals. While 159edo divides neither the Pythagorean, syntonic, nor rastmic accidental pairs in half, this does allow for the notation of artoneutral and tendoneutral intervals via the inflection of a semisharp by half a rastma (which ends up at a full 159edo step, specifically 7 steps for the artodemisharp and 8 steps for the tendodemisharp).

The syntonic and rastmic accidentals represent their just mappings when generalized to other EDOs.{{Navbox EDO}}
{{Cat|Edos}}

159edo

2026-04-07T06:30:29Z

Aura: /* Syntonic-rastmic subchroma notation */

'''159edo''', or 159 equal divisions of the octave, is the equal tuning featuring steps of (1200/159) ~= 7.55 cents, 159 of which stack to the perfect octave [[2/1]]. Like [[53edo]], 159edo is an excellent approximation to Pythagorean tuning (stacking pure 3/2 fifths), however it includes near-just approximations of the 11th and 17th harmonics, and a slightly more accurate 7th harmonic, resulting in consistency up to the 17-odd-limit. The step-size, being slightly above the melodic just noticeable difference (JND) of 5 cents as well as more than twice the harmonic JND of the average trained musician at 3.5 cents, enables one to perform fluid modulations by means of [[comma pump]]s as well as by step substitutions. Furthermore, 159edo, like a number of higher edos, is characterized by being capable of imitating the intervals of smaller tuning systems - in this case, you get detemperings of [[10edo]], [[12edo]], [[13edo]], [[14edo]], [[17edo]], [[19edo]], [[22edo]], [[24edo]] and [[31edo]] among others with errors smaller than the melodic JND.

The interval qualities supported by 159edo are many, however, while every step of 159edo can be interpreted harmonically or subharmonically as being a 17-limit interval or simpler, some of the intervals you get have rather complex interpretations in terms of odd-limit. While the [[perfect fifth]] is really only divisible by three due to 159edo being the three-fold multiple of 53edo- resulting in [[slendric]] temperament and hence 159edo's distinction from 53 in the 7-limit- the [[perfect fourth]] has a little more to offer in terms of divisions. For starters, the perfect fourth can be divided into two instances of 15/13, giving us [[Semifourth-generated scales|island]] temperament. Dividing the perfect fourth into three instances of 11/10 gives us [[pine]] temperament. Dividing the perfect fourth into six instances of an interval which can be interpreted as 21/20 and 22/21 tempered together gives us [[sextilifourths]] temperament. The perfect fourth can also be cut into eleven intervals which, individually, are half of a Pythagorean limma, giving us a number of temperaments based on the exact interpretation of the semilimma.

On top of all that, there are a number of microtemperament-based structures also supported by 159edo, each of which provides some decent, unexpected melodic possibilities. For instance, one can split the Pythagorean minor third into three instances of 128/121 which gives us [[nexus]] temperament- a temperament which also happens to split the Pythagorean diatonic semitone into two, and the octave into three. One can also split the Ptolemaic minor third into three instances of 17/16, giving us [[archagall]] temperament, which is named for certain tunings found in other temperaments producing fractal-like acoustics. There's also the ability to split the Ptolemaic major sixth into six instances of 11/9, leading to [[parimic]] temperament. In addition, there's also the ability to split the septimal supermajor third into two instances of 17/15, leading to [[fidesmic]] temperament, which acts like a more accurate rendition of [[archy]] temperament in a different subgroup, and this can be exploited for modulation purposes. As if that weren't enough, there's the possibility of splitting the septimal subminor third into five instances of 33/32, leading to [[quartismic]] temperament. Furthermore, there's the possibility of splitting the greater tridecimal neutral tenth into three instances of 27/20, producing [[phaotismic]] temperament, and the list goes on.

== Theory ==
159edo was first used for maqams by Ozan Yarman. It was later put to use by Aura for its ability to handle near-just quartertones derived from the 2.3.11 subgroup on top of the 5-limit foundation provided by 53edo.

==== Edostep interpretations ====
159edo's edostep has the following interpretations in the 2.3.5.11.17 subgroup:

* 243/242, the difference between the 11-limit artoneutral third 11/9, and the 11-limit tendoneutral third 27/22
* 256/255, the difference between 16/15 and 17/16
* 289/288, the difference between 17/16 and 18/17

159edo tempers out the following commas in the 17-limit:

* The schisma (the difference between 5/4 and the Pythagorean diminished fourth)
* The vulture comma (the difference between four 320/243 intervals and the tritave)
* The amiton (the difference between a stack of five 10/9 intervals and 27/16)
* The kleisma (the difference between a stack of three 25/24 intervals and 9/8)
* The semicomma (the difference between a stack of three 75/64 intervals and 8/5)
* The gamelisma (the difference between a stack of three 8/7 intervals and 3/2)
* 385/384 (the difference between 77/64 and 6/5)
* The pine comma (the difference between a stack of three 11/10 intervals and 4/3)
* 625/624 (the difference between 25/24 and 26/25)
* 676/675 (the difference between a stack of two 15/13 intervals and the perfect fourth)
* The twosquare comma (the difference between a stack of two 33/32 intervals and 17/16)

==== JI approximation ====
Although 159edo inherits its approximations of the 5-limit from 53edo, the 5th harmonic can nonetheless be stacked twice without accumulating too much error, rendering it sufficient for Western Classical usage. While the 7th harmonic is technically more accurate in terms of absolute error than in 53edo, the relative error doesn't allow one to stack more than one instance of 7/4 without excessive error accumulation, and the same is true with 13/8. Regardless, the inconsistency remains less than 10 cents even when either interval is stacked three times, and 13/7 or 14/13 is tuned almost perfectly. As a whole, 159edo is characterized by its combination of accuracy in the 2.3.5.11.17 subgroup, and a series of compromises in the 7.13.19.23.29 subgroup- among the compromises are the slendric, [[marveltwin]], [[nestoria]], [[minor semivicemic]] and [[brunisimic]] temperaments.
{{Harmonics in ED|159|31|0}}

Currently, the [[ADIN]] system used for interval naming elsewhere on this site fails for 159edo, so another set of interval names will be used here, though the ADIN names will be referenced in places. Furthermore, because there are so many thirds, what is usually a single chart will be split into five.
{| class="wikitable"
|+Inframinor and Subminor Thirds in 159edo
!Quality
|Narrow Inframinor
|Inframinor
|Wide Inframinor
|Narrow Subminor
|Lesser Subminor (Septiminor)
|Greater Subminor
|Wide Subminor
|-
!Cents
|234
|242
|249
|257
|264
|272
|279
|-
!Just interpretation
|340/297
|1024/891
|15/13
|51/44
|7/6
|117/100
|20/17
|-
!Steps
|31
|32
|33
|34
|35
|36
|37
|}
{| class="wikitable"
|+Minor Thirds in 159edo
!Quality
|Narrow Minor (Gothminor)
|'''Pythagorean Minor (Triminor)'''
|Artomean Minor
|Tendomean Minor
|Ptolemaic Minor (Pentaminor)
|Wide Minor
|-
!Cents
|287
|'''294'''
|302
|309
|317
|325
|-
!Just interpretation
|33/28, 13/11
|'''32/27'''
|25/21
|153/128
|6/5
|135/112
|-
!Steps
|38
|'''39'''
|40
|41
|42
|43
|}
{| class="wikitable"
|+Supraminor, Neutral and Submajor Thirds in 159edo
!Quality
|Lesser Supraminor
|Greater Supraminor
|Artoneutral
|Tendoneutral
|Lesser Submajor
|Greater Submajor
|-
!Cents
|332
|340
|347
|355
|362
|370
|-
!Just interpretation
|40/33, 63/52
|39/32, 17/14
|11/9
|27/22
|16/13, 21/17
|99/80, 26/21
|-
!Steps
|44
|45
|46
|47
|48
|49
|}
{| class="wikitable"
|+Major Thirds in 159edo
!Quality
|Narrow Major
|Ptolemaic Major (Pentamajor)
|Artomean Major
|Tendomean Major
|'''Pythagorean Major (Trimajor)'''
|Wide Major (Gothmajor)
|-
!Cents
|377
|385
|392
|400
|'''408'''
|415
|-
!Just interpretation
|56/45
|5/4
|64/51
|63/50
|'''81/64'''
|14/11, 33/26
|-
!Steps
|50
|51
|52
|53
|'''54'''
|55
|}
{| class="wikitable"
|+Supermajor and Ultramajor Thirds in 159edo
!Quality
|Narrow Supermajor
|Lesser Supermajor
|Greater Supermajor (Septimajor)
|Wide Supermajor
|Narrow Ultramajor
|Ultramajor
|Wide Ultramajor
|-
!Cents
|423
|430
|438
|445
|453
|460
|468
|-
!Just interpretation
|51/40
|50/39
|9/7
|22/17
|13/10
|2673/2048
|891/680
|-
!Steps
|56
|57
|58
|59
|60
|61
|62
|}
Regular diatonic thirds are bolded.

=== Chords ===
159edo has a vast array of triads at its disposal, both fifth-bounded and fourth-bounded. However, the JI interpretation will tend to inform the usage of the various triads offered. The main exceptions to this rule involve chords that serve as detempers of other, smaller tuning systems.

== Notation ==
There are multiple different tuning systems available for 159edo.

=== Ups and downs ===
Ups and downs notation represents a single step of 159edo by means of lifts and drops while using the original ups and downs to represent single steps of 53edo.

=== Sagittal notation ===
<todo>

=== Syntonic-rastmic subchroma notation ===
Syntonic-rastmic subchroma notation, or SRS for short, is an alternative to Sagittal with fewer accidentals- one which was originally designed for 159edo. In 159edo, the synsharp (81/80) and the rasharp (243/242) are 3\159 (1\53) and 1\159 respectively, treating 159edo as a subdivision of 53edo wherein three rastmas make up a syntonic comma. So far, this is functionally identical to ups and downs notation (assuming the up and down are treated as 1\53), but syntonic-rastmic subchroma notation also supports halving of all of its accidentals. While 159edo divides neither the Pythagorean, syntonic, nor rastmic accidental pairs in half, this does allow for the notation of artoneutral and tendoneutral intervals via the inflection of a semisharp by half a rastma (which ends up at a full 159edo step, specifically 7 steps for the artodemisharp and 8 steps for the tendodemisharp).

The syntonic and rastmic accidentals represent their just mappings when generalized to other EDOs.{{Navbox EDO}}
{{Cat|Edos}}

159edo

2026-04-07T06:29:07Z

Aura: Added a small section for ups and downs notation

'''159edo''', or 159 equal divisions of the octave, is the equal tuning featuring steps of (1200/159) ~= 7.55 cents, 159 of which stack to the perfect octave [[2/1]]. Like [[53edo]], 159edo is an excellent approximation to Pythagorean tuning (stacking pure 3/2 fifths), however it includes near-just approximations of the 11th and 17th harmonics, and a slightly more accurate 7th harmonic, resulting in consistency up to the 17-odd-limit. The step-size, being slightly above the melodic just noticeable difference (JND) of 5 cents as well as more than twice the harmonic JND of the average trained musician at 3.5 cents, enables one to perform fluid modulations by means of [[comma pump]]s as well as by step substitutions. Furthermore, 159edo, like a number of higher edos, is characterized by being capable of imitating the intervals of smaller tuning systems - in this case, you get detemperings of [[10edo]], [[12edo]], [[13edo]], [[14edo]], [[17edo]], [[19edo]], [[22edo]], [[24edo]] and [[31edo]] among others with errors smaller than the melodic JND.

The interval qualities supported by 159edo are many, however, while every step of 159edo can be interpreted harmonically or subharmonically as being a 17-limit interval or simpler, some of the intervals you get have rather complex interpretations in terms of odd-limit. While the [[perfect fifth]] is really only divisible by three due to 159edo being the three-fold multiple of 53edo- resulting in [[slendric]] temperament and hence 159edo's distinction from 53 in the 7-limit- the [[perfect fourth]] has a little more to offer in terms of divisions. For starters, the perfect fourth can be divided into two instances of 15/13, giving us [[Semifourth-generated scales|island]] temperament. Dividing the perfect fourth into three instances of 11/10 gives us [[pine]] temperament. Dividing the perfect fourth into six instances of an interval which can be interpreted as 21/20 and 22/21 tempered together gives us [[sextilifourths]] temperament. The perfect fourth can also be cut into eleven intervals which, individually, are half of a Pythagorean limma, giving us a number of temperaments based on the exact interpretation of the semilimma.

On top of all that, there are a number of microtemperament-based structures also supported by 159edo, each of which provides some decent, unexpected melodic possibilities. For instance, one can split the Pythagorean minor third into three instances of 128/121 which gives us [[nexus]] temperament- a temperament which also happens to split the Pythagorean diatonic semitone into two, and the octave into three. One can also split the Ptolemaic minor third into three instances of 17/16, giving us [[archagall]] temperament, which is named for certain tunings found in other temperaments producing fractal-like acoustics. There's also the ability to split the Ptolemaic major sixth into six instances of 11/9, leading to [[parimic]] temperament. In addition, there's also the ability to split the septimal supermajor third into two instances of 17/15, leading to [[fidesmic]] temperament, which acts like a more accurate rendition of [[archy]] temperament in a different subgroup, and this can be exploited for modulation purposes. As if that weren't enough, there's the possibility of splitting the septimal subminor third into five instances of 33/32, leading to [[quartismic]] temperament. Furthermore, there's the possibility of splitting the greater tridecimal neutral tenth into three instances of 27/20, producing [[phaotismic]] temperament, and the list goes on.

== Theory ==
159edo was first used for maqams by Ozan Yarman. It was later put to use by Aura for its ability to handle near-just quartertones derived from the 2.3.11 subgroup on top of the 5-limit foundation provided by 53edo.

==== Edostep interpretations ====
159edo's edostep has the following interpretations in the 2.3.5.11.17 subgroup:

* 243/242, the difference between the 11-limit artoneutral third 11/9, and the 11-limit tendoneutral third 27/22
* 256/255, the difference between 16/15 and 17/16
* 289/288, the difference between 17/16 and 18/17

159edo tempers out the following commas in the 17-limit:

* The schisma (the difference between 5/4 and the Pythagorean diminished fourth)
* The vulture comma (the difference between four 320/243 intervals and the tritave)
* The amiton (the difference between a stack of five 10/9 intervals and 27/16)
* The kleisma (the difference between a stack of three 25/24 intervals and 9/8)
* The semicomma (the difference between a stack of three 75/64 intervals and 8/5)
* The gamelisma (the difference between a stack of three 8/7 intervals and 3/2)
* 385/384 (the difference between 77/64 and 6/5)
* The pine comma (the difference between a stack of three 11/10 intervals and 4/3)
* 625/624 (the difference between 25/24 and 26/25)
* 676/675 (the difference between a stack of two 15/13 intervals and the perfect fourth)
* The twosquare comma (the difference between a stack of two 33/32 intervals and 17/16)

==== JI approximation ====
Although 159edo inherits its approximations of the 5-limit from 53edo, the 5th harmonic can nonetheless be stacked twice without accumulating too much error, rendering it sufficient for Western Classical usage. While the 7th harmonic is technically more accurate in terms of absolute error than in 53edo, the relative error doesn't allow one to stack more than one instance of 7/4 without excessive error accumulation, and the same is true with 13/8. Regardless, the inconsistency remains less than 10 cents even when either interval is stacked three times, and 13/7 or 14/13 is tuned almost perfectly. As a whole, 159edo is characterized by its combination of accuracy in the 2.3.5.11.17 subgroup, and a series of compromises in the 7.13.19.23.29 subgroup- among the compromises are the slendric, [[marveltwin]], [[nestoria]], [[minor semivicemic]] and [[brunisimic]] temperaments.
{{Harmonics in ED|159|31|0}}

Currently, the [[ADIN]] system used for interval naming elsewhere on this site fails for 159edo, so another set of interval names will be used here, though the ADIN names will be referenced in places. Furthermore, because there are so many thirds, what is usually a single chart will be split into five.
{| class="wikitable"
|+Inframinor and Subminor Thirds in 159edo
!Quality
|Narrow Inframinor
|Inframinor
|Wide Inframinor
|Narrow Subminor
|Lesser Subminor (Septiminor)
|Greater Subminor
|Wide Subminor
|-
!Cents
|234
|242
|249
|257
|264
|272
|279
|-
!Just interpretation
|340/297
|1024/891
|15/13
|51/44
|7/6
|117/100
|20/17
|-
!Steps
|31
|32
|33
|34
|35
|36
|37
|}
{| class="wikitable"
|+Minor Thirds in 159edo
!Quality
|Narrow Minor (Gothminor)
|'''Pythagorean Minor (Triminor)'''
|Artomean Minor
|Tendomean Minor
|Ptolemaic Minor (Pentaminor)
|Wide Minor
|-
!Cents
|287
|'''294'''
|302
|309
|317
|325
|-
!Just interpretation
|33/28, 13/11
|'''32/27'''
|25/21
|153/128
|6/5
|135/112
|-
!Steps
|38
|'''39'''
|40
|41
|42
|43
|}
{| class="wikitable"
|+Supraminor, Neutral and Submajor Thirds in 159edo
!Quality
|Lesser Supraminor
|Greater Supraminor
|Artoneutral
|Tendoneutral
|Lesser Submajor
|Greater Submajor
|-
!Cents
|332
|340
|347
|355
|362
|370
|-
!Just interpretation
|40/33, 63/52
|39/32, 17/14
|11/9
|27/22
|16/13, 21/17
|99/80, 26/21
|-
!Steps
|44
|45
|46
|47
|48
|49
|}
{| class="wikitable"
|+Major Thirds in 159edo
!Quality
|Narrow Major
|Ptolemaic Major (Pentamajor)
|Artomean Major
|Tendomean Major
|'''Pythagorean Major (Trimajor)'''
|Wide Major (Gothmajor)
|-
!Cents
|377
|385
|392
|400
|'''408'''
|415
|-
!Just interpretation
|56/45
|5/4
|64/51
|63/50
|'''81/64'''
|14/11, 33/26
|-
!Steps
|50
|51
|52
|53
|'''54'''
|55
|}
{| class="wikitable"
|+Supermajor and Ultramajor Thirds in 159edo
!Quality
|Narrow Supermajor
|Lesser Supermajor
|Greater Supermajor (Septimajor)
|Wide Supermajor
|Narrow Ultramajor
|Ultramajor
|Wide Ultramajor
|-
!Cents
|423
|430
|438
|445
|453
|460
|468
|-
!Just interpretation
|51/40
|50/39
|9/7
|22/17
|13/10
|2673/2048
|891/680
|-
!Steps
|56
|57
|58
|59
|60
|61
|62
|}
Regular diatonic thirds are bolded.

=== Chords ===
159edo has a vast array of triads at its disposal, both fifth-bounded and fourth-bounded. However, the JI interpretation will tend to inform the usage of the various triads offered. The main exceptions to this rule involve chords that serve as detempers of other, smaller tuning systems.

== Notation ==
There are multiple different tuning systems available for 159edo.

=== Ups and downs ===
Ups and downs notation represents a single step of 159edo by means of lifts and drops while using the original ups and downs to represent single steps of 53edo.

=== Sagittal notation ===
<todo>

=== Syntonic-rastmic subchroma notation ===
Syntonic-rastmic subchroma notation, or SRS for short, is an alternative to Sagittal with fewer accidentals- one which was originally designed for 159edo. In 159edo, the synsharp (81/80) and the rasharp (243/242) are 3\159 (1\53) and 1\159 respectively, treating 159edo as a subdivision of 53edo wherein three rastmas make up a syntonic comma. So far, this is functionally identical to ups and downs notation (assuming the up and down are treated as 1\53), but syntonic-rastmic subchroma notation also supports halving of all of its accidentals. While 159edo divides neither the Pythagorean, syntonic, or rastmic accidental pairs in half, this does allow for the notation of artoneutral and tendoneutral intervals via the inflection of a semisharp by half a rastma (which ends up at a full 159edo step, specifically 7 steps for the artodemisharp and 8 steps for the tendodemisharp).

The syntonic and rastmic accidentals represent their just mappings when generalized to other EDOs.{{Navbox EDO}}
{{Cat|Edos}}

159edo

2026-04-07T06:13:43Z

Aura: /* Sagittal */

'''159edo''', or 159 equal divisions of the octave, is the equal tuning featuring steps of (1200/159) ~= 7.55 cents, 159 of which stack to the perfect octave [[2/1]]. Like [[53edo]], 159edo is an excellent approximation to Pythagorean tuning (stacking pure 3/2 fifths), however it includes near-just approximations of the 11th and 17th harmonics, and a slightly more accurate 7th harmonic, resulting in consistency up to the 17-odd-limit. The step-size, being slightly above the melodic just noticeable difference (JND) of 5 cents as well as more than twice the harmonic JND of the average trained musician at 3.5 cents, enables one to perform fluid modulations by means of [[comma pump]]s as well as by step substitutions. Furthermore, 159edo, like a number of higher edos, is characterized by being capable of imitating the intervals of smaller tuning systems - in this case, you get detemperings of [[10edo]], [[12edo]], [[13edo]], [[14edo]], [[17edo]], [[19edo]], [[22edo]], [[24edo]] and [[31edo]] among others with errors smaller than the melodic JND.

The interval qualities supported by 159edo are many, however, while every step of 159edo can be interpreted harmonically or subharmonically as being a 17-limit interval or simpler, some of the intervals you get have rather complex interpretations in terms of odd-limit. While the [[perfect fifth]] is really only divisible by three due to 159edo being the three-fold multiple of 53edo- resulting in [[slendric]] temperament and hence 159edo's distinction from 53 in the 7-limit- the [[perfect fourth]] has a little more to offer in terms of divisions. For starters, the perfect fourth can be divided into two instances of 15/13, giving us [[Semifourth-generated scales|island]] temperament. Dividing the perfect fourth into three instances of 11/10 gives us [[pine]] temperament. Dividing the perfect fourth into six instances of an interval which can be interpreted as 21/20 and 22/21 tempered together gives us [[sextilifourths]] temperament. The perfect fourth can also be cut into eleven intervals which, individually, are half of a Pythagorean limma, giving us a number of temperaments based on the exact interpretation of the semilimma.

On top of all that, there are a number of microtemperament-based structures also supported by 159edo, each of which provides some decent, unexpected melodic possibilities. For instance, one can split the Pythagorean minor third into three instances of 128/121 which gives us [[nexus]] temperament- a temperament which also happens to split the Pythagorean diatonic semitone into two, and the octave into three. One can also split the Ptolemaic minor third into three instances of 17/16, giving us [[archagall]] temperament, which is named for certain tunings found in other temperaments producing fractal-like acoustics. There's also the ability to split the Ptolemaic major sixth into six instances of 11/9, leading to [[parimic]] temperament. In addition, there's also the ability to split the septimal supermajor third into two instances of 17/15, leading to [[fidesmic]] temperament, which acts like a more accurate rendition of [[archy]] temperament in a different subgroup, and this can be exploited for modulation purposes. As if that weren't enough, there's the possibility of splitting the septimal subminor third into five instances of 33/32, leading to [[quartismic]] temperament. Furthermore, there's the possibility of splitting the greater tridecimal neutral tenth into three instances of 27/20, producing [[phaotismic]] temperament, and the list goes on.

== Theory ==
159edo was first used for maqams by Ozan Yarman. It was later put to use by Aura for its ability to handle near-just quartertones derived from the 2.3.11 subgroup on top of the 5-limit foundation provided by 53edo.

==== Edostep interpretations ====
159edo's edostep has the following interpretations in the 2.3.5.11.17 subgroup:

* 243/242, the difference between the 11-limit artoneutral third 11/9, and the 11-limit tendoneutral third 27/22
* 256/255, the difference between 16/15 and 17/16
* 289/288, the difference between 17/16 and 18/17

159edo tempers out the following commas in the 17-limit:

* The schisma (the difference between 5/4 and the Pythagorean diminished fourth)
* The vulture comma (the difference between four 320/243 intervals and the tritave)
* The amiton (the difference between a stack of five 10/9 intervals and 27/16)
* The kleisma (the difference between a stack of three 25/24 intervals and 9/8)
* The semicomma (the difference between a stack of three 75/64 intervals and 8/5)
* The gamelisma (the difference between a stack of three 8/7 intervals and 3/2)
* 385/384 (the difference between 77/64 and 6/5)
* The pine comma (the difference between a stack of three 11/10 intervals and 4/3)
* 625/624 (the difference between 25/24 and 26/25)
* 676/675 (the difference between a stack of two 15/13 intervals and the perfect fourth)
* The twosquare comma (the difference between a stack of two 33/32 intervals and 17/16)

==== JI approximation ====
Although 159edo inherits its approximations of the 5-limit from 53edo, the 5th harmonic can nonetheless be stacked twice without accumulating too much error, rendering it sufficient for Western Classical usage. While the 7th harmonic is technically more accurate in terms of absolute error than in 53edo, the relative error doesn't allow one to stack more than one instance of 7/4 without excessive error accumulation, and the same is true with 13/8. Regardless, the inconsistency remains less than 10 cents even when either interval is stacked three times, and 13/7 or 14/13 is tuned almost perfectly. As a whole, 159edo is characterized by its combination of accuracy in the 2.3.5.11.17 subgroup, and a series of compromises in the 7.13.19.23.29 subgroup- among the compromises are the slendric, [[marveltwin]], [[nestoria]], [[minor semivicemic]] and [[brunisimic]] temperaments.
{{Harmonics in ED|159|31|0}}

Currently, the [[ADIN]] system used for interval naming elsewhere on this site fails for 159edo, so another set of interval names will be used here, though the ADIN names will be referenced in places. Furthermore, because there are so many thirds, what is usually a single chart will be split into five.
{| class="wikitable"
|+Inframinor and Subminor Thirds in 159edo
!Quality
|Narrow Inframinor
|Inframinor
|Wide Inframinor
|Narrow Subminor
|Lesser Subminor (Septiminor)
|Greater Subminor
|Wide Subminor
|-
!Cents
|234
|242
|249
|257
|264
|272
|279
|-
!Just interpretation
|340/297
|1024/891
|15/13
|51/44
|7/6
|117/100
|20/17
|-
!Steps
|31
|32
|33
|34
|35
|36
|37
|}
{| class="wikitable"
|+Minor Thirds in 159edo
!Quality
|Narrow Minor (Gothminor)
|'''Pythagorean Minor (Triminor)'''
|Artomean Minor
|Tendomean Minor
|Ptolemaic Minor (Pentaminor)
|Wide Minor
|-
!Cents
|287
|'''294'''
|302
|309
|317
|325
|-
!Just interpretation
|33/28, 13/11
|'''32/27'''
|25/21
|153/128
|6/5
|135/112
|-
!Steps
|38
|'''39'''
|40
|41
|42
|43
|}
{| class="wikitable"
|+Supraminor, Neutral and Submajor Thirds in 159edo
!Quality
|Lesser Supraminor
|Greater Supraminor
|Artoneutral
|Tendoneutral
|Lesser Submajor
|Greater Submajor
|-
!Cents
|332
|340
|347
|355
|362
|370
|-
!Just interpretation
|40/33, 63/52
|39/32, 17/14
|11/9
|27/22
|16/13, 21/17
|99/80, 26/21
|-
!Steps
|44
|45
|46
|47
|48
|49
|}
{| class="wikitable"
|+Major Thirds in 159edo
!Quality
|Narrow Major
|Ptolemaic Major (Pentamajor)
|Artomean Major
|Tendomean Major
|'''Pythagorean Major (Trimajor)'''
|Wide Major (Gothmajor)
|-
!Cents
|377
|385
|392
|400
|'''408'''
|415
|-
!Just interpretation
|56/45
|5/4
|64/51
|63/50
|'''81/64'''
|14/11, 33/26
|-
!Steps
|50
|51
|52
|53
|'''54'''
|55
|}
{| class="wikitable"
|+Supermajor and Ultramajor Thirds in 159edo
!Quality
|Narrow Supermajor
|Lesser Supermajor
|Greater Supermajor (Septimajor)
|Wide Supermajor
|Narrow Ultramajor
|Ultramajor
|Wide Ultramajor
|-
!Cents
|423
|430
|438
|445
|453
|460
|468
|-
!Just interpretation
|51/40
|50/39
|9/7
|22/17
|13/10
|2673/2048
|891/680
|-
!Steps
|56
|57
|58
|59
|60
|61
|62
|}
Regular diatonic thirds are bolded.

=== Chords ===
159edo has a vast array of triads at its disposal, both fifth-bounded and fourth-bounded. However, the JI interpretation will tend to inform the usage of the various triads offered. The main exceptions to this rule involve chords that serve as detempers of other, smaller tuning systems.

== Notation ==
There are multiple different tuning systems available for 159edo.

=== Sagittal notation ===
<todo>

=== Syntonic-Rastmic Subchroma notation ===
Syntonic-rastmic subchroma notation, or SRS for short, is an alternative to Sagittal with fewer accidentals- one which was originally designed for 159edo. In 159edo, the synsharp (81/80) and the rasharp (243/242) are 3\159 (1\53) and 1\159 respectively, treating 159edo as a subdivision of 53edo wherein three rastmas make up a syntonic comma. So far, this is functionally identical to ups and downs notation (assuming the up and down are treated as 1\53), but syntonic-rastmic subchroma notation also supports halving of all of its accidentals. While 159edo divides neither the Pythagorean, syntonic, or rastmic accidental pairs in half, this does allow for the notation of artoneutral and tendoneutral intervals via the inflection of a semisharp by half a rastma (which ends up at a full 159edo step, specifically 7 steps for the artodemisharp and 8 steps for the tendodemisharp).

The syntonic and rastmic accidentals represent their just mappings when generalized to other EDOs.{{Navbox EDO}}
{{Cat|Edos}}

159edo

2026-04-07T06:12:27Z

Aura: /* Notation = */

'''159edo''', or 159 equal divisions of the octave, is the equal tuning featuring steps of (1200/159) ~= 7.55 cents, 159 of which stack to the perfect octave [[2/1]]. Like [[53edo]], 159edo is an excellent approximation to Pythagorean tuning (stacking pure 3/2 fifths), however it includes near-just approximations of the 11th and 17th harmonics, and a slightly more accurate 7th harmonic, resulting in consistency up to the 17-odd-limit. The step-size, being slightly above the melodic just noticeable difference (JND) of 5 cents as well as more than twice the harmonic JND of the average trained musician at 3.5 cents, enables one to perform fluid modulations by means of [[comma pump]]s as well as by step substitutions. Furthermore, 159edo, like a number of higher edos, is characterized by being capable of imitating the intervals of smaller tuning systems - in this case, you get detemperings of [[10edo]], [[12edo]], [[13edo]], [[14edo]], [[17edo]], [[19edo]], [[22edo]], [[24edo]] and [[31edo]] among others with errors smaller than the melodic JND.

The interval qualities supported by 159edo are many, however, while every step of 159edo can be interpreted harmonically or subharmonically as being a 17-limit interval or simpler, some of the intervals you get have rather complex interpretations in terms of odd-limit. While the [[perfect fifth]] is really only divisible by three due to 159edo being the three-fold multiple of 53edo- resulting in [[slendric]] temperament and hence 159edo's distinction from 53 in the 7-limit- the [[perfect fourth]] has a little more to offer in terms of divisions. For starters, the perfect fourth can be divided into two instances of 15/13, giving us [[Semifourth-generated scales|island]] temperament. Dividing the perfect fourth into three instances of 11/10 gives us [[pine]] temperament. Dividing the perfect fourth into six instances of an interval which can be interpreted as 21/20 and 22/21 tempered together gives us [[sextilifourths]] temperament. The perfect fourth can also be cut into eleven intervals which, individually, are half of a Pythagorean limma, giving us a number of temperaments based on the exact interpretation of the semilimma.

On top of all that, there are a number of microtemperament-based structures also supported by 159edo, each of which provides some decent, unexpected melodic possibilities. For instance, one can split the Pythagorean minor third into three instances of 128/121 which gives us [[nexus]] temperament- a temperament which also happens to split the Pythagorean diatonic semitone into two, and the octave into three. One can also split the Ptolemaic minor third into three instances of 17/16, giving us [[archagall]] temperament, which is named for certain tunings found in other temperaments producing fractal-like acoustics. There's also the ability to split the Ptolemaic major sixth into six instances of 11/9, leading to [[parimic]] temperament. In addition, there's also the ability to split the septimal supermajor third into two instances of 17/15, leading to [[fidesmic]] temperament, which acts like a more accurate rendition of [[archy]] temperament in a different subgroup, and this can be exploited for modulation purposes. As if that weren't enough, there's the possibility of splitting the septimal subminor third into five instances of 33/32, leading to [[quartismic]] temperament. Furthermore, there's the possibility of splitting the greater tridecimal neutral tenth into three instances of 27/20, producing [[phaotismic]] temperament, and the list goes on.

== Theory ==
159edo was first used for maqams by Ozan Yarman. It was later put to use by Aura for its ability to handle near-just quartertones derived from the 2.3.11 subgroup on top of the 5-limit foundation provided by 53edo.

==== Edostep interpretations ====
159edo's edostep has the following interpretations in the 2.3.5.11.17 subgroup:

* 243/242, the difference between the 11-limit artoneutral third 11/9, and the 11-limit tendoneutral third 27/22
* 256/255, the difference between 16/15 and 17/16
* 289/288, the difference between 17/16 and 18/17

159edo tempers out the following commas in the 17-limit:

* The schisma (the difference between 5/4 and the Pythagorean diminished fourth)
* The vulture comma (the difference between four 320/243 intervals and the tritave)
* The amiton (the difference between a stack of five 10/9 intervals and 27/16)
* The kleisma (the difference between a stack of three 25/24 intervals and 9/8)
* The semicomma (the difference between a stack of three 75/64 intervals and 8/5)
* The gamelisma (the difference between a stack of three 8/7 intervals and 3/2)
* 385/384 (the difference between 77/64 and 6/5)
* The pine comma (the difference between a stack of three 11/10 intervals and 4/3)
* 625/624 (the difference between 25/24 and 26/25)
* 676/675 (the difference between a stack of two 15/13 intervals and the perfect fourth)
* The twosquare comma (the difference between a stack of two 33/32 intervals and 17/16)

==== JI approximation ====
Although 159edo inherits its approximations of the 5-limit from 53edo, the 5th harmonic can nonetheless be stacked twice without accumulating too much error, rendering it sufficient for Western Classical usage. While the 7th harmonic is technically more accurate in terms of absolute error than in 53edo, the relative error doesn't allow one to stack more than one instance of 7/4 without excessive error accumulation, and the same is true with 13/8. Regardless, the inconsistency remains less than 10 cents even when either interval is stacked three times, and 13/7 or 14/13 is tuned almost perfectly. As a whole, 159edo is characterized by its combination of accuracy in the 2.3.5.11.17 subgroup, and a series of compromises in the 7.13.19.23.29 subgroup- among the compromises are the slendric, [[marveltwin]], [[nestoria]], [[minor semivicemic]] and [[brunisimic]] temperaments.
{{Harmonics in ED|159|31|0}}

Currently, the [[ADIN]] system used for interval naming elsewhere on this site fails for 159edo, so another set of interval names will be used here, though the ADIN names will be referenced in places. Furthermore, because there are so many thirds, what is usually a single chart will be split into five.
{| class="wikitable"
|+Inframinor and Subminor Thirds in 159edo
!Quality
|Narrow Inframinor
|Inframinor
|Wide Inframinor
|Narrow Subminor
|Lesser Subminor (Septiminor)
|Greater Subminor
|Wide Subminor
|-
!Cents
|234
|242
|249
|257
|264
|272
|279
|-
!Just interpretation
|340/297
|1024/891
|15/13
|51/44
|7/6
|117/100
|20/17
|-
!Steps
|31
|32
|33
|34
|35
|36
|37
|}
{| class="wikitable"
|+Minor Thirds in 159edo
!Quality
|Narrow Minor (Gothminor)
|'''Pythagorean Minor (Triminor)'''
|Artomean Minor
|Tendomean Minor
|Ptolemaic Minor (Pentaminor)
|Wide Minor
|-
!Cents
|287
|'''294'''
|302
|309
|317
|325
|-
!Just interpretation
|33/28, 13/11
|'''32/27'''
|25/21
|153/128
|6/5
|135/112
|-
!Steps
|38
|'''39'''
|40
|41
|42
|43
|}
{| class="wikitable"
|+Supraminor, Neutral and Submajor Thirds in 159edo
!Quality
|Lesser Supraminor
|Greater Supraminor
|Artoneutral
|Tendoneutral
|Lesser Submajor
|Greater Submajor
|-
!Cents
|332
|340
|347
|355
|362
|370
|-
!Just interpretation
|40/33, 63/52
|39/32, 17/14
|11/9
|27/22
|16/13, 21/17
|99/80, 26/21
|-
!Steps
|44
|45
|46
|47
|48
|49
|}
{| class="wikitable"
|+Major Thirds in 159edo
!Quality
|Narrow Major
|Ptolemaic Major (Pentamajor)
|Artomean Major
|Tendomean Major
|'''Pythagorean Major (Trimajor)'''
|Wide Major (Gothmajor)
|-
!Cents
|377
|385
|392
|400
|'''408'''
|415
|-
!Just interpretation
|56/45
|5/4
|64/51
|63/50
|'''81/64'''
|14/11, 33/26
|-
!Steps
|50
|51
|52
|53
|'''54'''
|55
|}
{| class="wikitable"
|+Supermajor and Ultramajor Thirds in 159edo
!Quality
|Narrow Supermajor
|Lesser Supermajor
|Greater Supermajor (Septimajor)
|Wide Supermajor
|Narrow Ultramajor
|Ultramajor
|Wide Ultramajor
|-
!Cents
|423
|430
|438
|445
|453
|460
|468
|-
!Just interpretation
|51/40
|50/39
|9/7
|22/17
|13/10
|2673/2048
|891/680
|-
!Steps
|56
|57
|58
|59
|60
|61
|62
|}
Regular diatonic thirds are bolded.

=== Chords ===
159edo has a vast array of triads at its disposal, both fifth-bounded and fourth-bounded. However, the JI interpretation will tend to inform the usage of the various triads offered. The main exceptions to this rule involve chords that serve as detempers of other, smaller tuning systems.

== Notation ==
There are multiple different tuning systems available for 159edo.

=== Sagittal ===
<todo>

=== Syntonic-Rastmic Subchroma notation ===
Syntonic-rastmic subchroma notation, or SRS for short, is an alternative to Sagittal with fewer accidentals- one which was originally designed for 159edo. In 159edo, the synsharp (81/80) and the rasharp (243/242) are 3\159 (1\53) and 1\159 respectively, treating 159edo as a subdivision of 53edo wherein three rastmas make up a syntonic comma. So far, this is functionally identical to ups and downs notation (assuming the up and down are treated as 1\53), but syntonic-rastmic subchroma notation also supports halving of all of its accidentals. While 159edo divides neither the Pythagorean, syntonic, or rastmic accidental pairs in half, this does allow for the notation of artoneutral and tendoneutral intervals via the inflection of a semisharp by half a rastma (which ends up at a full 159edo step, specifically 7 steps for the artodemisharp and 8 steps for the tendodemisharp).

The syntonic and rastmic accidentals represent their just mappings when generalized to other EDOs.{{Navbox EDO}}
{{Cat|Edos}}

159edo

2026-04-07T06:11:12Z

Aura: Beginning the process of making a larger Notation section- we will have to flesh this out further

'''159edo''', or 159 equal divisions of the octave, is the equal tuning featuring steps of (1200/159) ~= 7.55 cents, 159 of which stack to the perfect octave [[2/1]]. Like [[53edo]], 159edo is an excellent approximation to Pythagorean tuning (stacking pure 3/2 fifths), however it includes near-just approximations of the 11th and 17th harmonics, and a slightly more accurate 7th harmonic, resulting in consistency up to the 17-odd-limit. The step-size, being slightly above the melodic just noticeable difference (JND) of 5 cents as well as more than twice the harmonic JND of the average trained musician at 3.5 cents, enables one to perform fluid modulations by means of [[comma pump]]s as well as by step substitutions. Furthermore, 159edo, like a number of higher edos, is characterized by being capable of imitating the intervals of smaller tuning systems - in this case, you get detemperings of [[10edo]], [[12edo]], [[13edo]], [[14edo]], [[17edo]], [[19edo]], [[22edo]], [[24edo]] and [[31edo]] among others with errors smaller than the melodic JND.

The interval qualities supported by 159edo are many, however, while every step of 159edo can be interpreted harmonically or subharmonically as being a 17-limit interval or simpler, some of the intervals you get have rather complex interpretations in terms of odd-limit. While the [[perfect fifth]] is really only divisible by three due to 159edo being the three-fold multiple of 53edo- resulting in [[slendric]] temperament and hence 159edo's distinction from 53 in the 7-limit- the [[perfect fourth]] has a little more to offer in terms of divisions. For starters, the perfect fourth can be divided into two instances of 15/13, giving us [[Semifourth-generated scales|island]] temperament. Dividing the perfect fourth into three instances of 11/10 gives us [[pine]] temperament. Dividing the perfect fourth into six instances of an interval which can be interpreted as 21/20 and 22/21 tempered together gives us [[sextilifourths]] temperament. The perfect fourth can also be cut into eleven intervals which, individually, are half of a Pythagorean limma, giving us a number of temperaments based on the exact interpretation of the semilimma.

On top of all that, there are a number of microtemperament-based structures also supported by 159edo, each of which provides some decent, unexpected melodic possibilities. For instance, one can split the Pythagorean minor third into three instances of 128/121 which gives us [[nexus]] temperament- a temperament which also happens to split the Pythagorean diatonic semitone into two, and the octave into three. One can also split the Ptolemaic minor third into three instances of 17/16, giving us [[archagall]] temperament, which is named for certain tunings found in other temperaments producing fractal-like acoustics. There's also the ability to split the Ptolemaic major sixth into six instances of 11/9, leading to [[parimic]] temperament. In addition, there's also the ability to split the septimal supermajor third into two instances of 17/15, leading to [[fidesmic]] temperament, which acts like a more accurate rendition of [[archy]] temperament in a different subgroup, and this can be exploited for modulation purposes. As if that weren't enough, there's the possibility of splitting the septimal subminor third into five instances of 33/32, leading to [[quartismic]] temperament. Furthermore, there's the possibility of splitting the greater tridecimal neutral tenth into three instances of 27/20, producing [[phaotismic]] temperament, and the list goes on.

== Theory ==
159edo was first used for maqams by Ozan Yarman. It was later put to use by Aura for its ability to handle near-just quartertones derived from the 2.3.11 subgroup on top of the 5-limit foundation provided by 53edo.

==== Edostep interpretations ====
159edo's edostep has the following interpretations in the 2.3.5.11.17 subgroup:

* 243/242, the difference between the 11-limit artoneutral third 11/9, and the 11-limit tendoneutral third 27/22
* 256/255, the difference between 16/15 and 17/16
* 289/288, the difference between 17/16 and 18/17

159edo tempers out the following commas in the 17-limit:

* The schisma (the difference between 5/4 and the Pythagorean diminished fourth)
* The vulture comma (the difference between four 320/243 intervals and the tritave)
* The amiton (the difference between a stack of five 10/9 intervals and 27/16)
* The kleisma (the difference between a stack of three 25/24 intervals and 9/8)
* The semicomma (the difference between a stack of three 75/64 intervals and 8/5)
* The gamelisma (the difference between a stack of three 8/7 intervals and 3/2)
* 385/384 (the difference between 77/64 and 6/5)
* The pine comma (the difference between a stack of three 11/10 intervals and 4/3)
* 625/624 (the difference between 25/24 and 26/25)
* 676/675 (the difference between a stack of two 15/13 intervals and the perfect fourth)
* The twosquare comma (the difference between a stack of two 33/32 intervals and 17/16)

==== JI approximation ====
Although 159edo inherits its approximations of the 5-limit from 53edo, the 5th harmonic can nonetheless be stacked twice without accumulating too much error, rendering it sufficient for Western Classical usage. While the 7th harmonic is technically more accurate in terms of absolute error than in 53edo, the relative error doesn't allow one to stack more than one instance of 7/4 without excessive error accumulation, and the same is true with 13/8. Regardless, the inconsistency remains less than 10 cents even when either interval is stacked three times, and 13/7 or 14/13 is tuned almost perfectly. As a whole, 159edo is characterized by its combination of accuracy in the 2.3.5.11.17 subgroup, and a series of compromises in the 7.13.19.23.29 subgroup- among the compromises are the slendric, [[marveltwin]], [[nestoria]], [[minor semivicemic]] and [[brunisimic]] temperaments.
{{Harmonics in ED|159|31|0}}

Currently, the [[ADIN]] system used for interval naming elsewhere on this site fails for 159edo, so another set of interval names will be used here, though the ADIN names will be referenced in places. Furthermore, because there are so many thirds, what is usually a single chart will be split into five.
{| class="wikitable"
|+Inframinor and Subminor Thirds in 159edo
!Quality
|Narrow Inframinor
|Inframinor
|Wide Inframinor
|Narrow Subminor
|Lesser Subminor (Septiminor)
|Greater Subminor
|Wide Subminor
|-
!Cents
|234
|242
|249
|257
|264
|272
|279
|-
!Just interpretation
|340/297
|1024/891
|15/13
|51/44
|7/6
|117/100
|20/17
|-
!Steps
|31
|32
|33
|34
|35
|36
|37
|}
{| class="wikitable"
|+Minor Thirds in 159edo
!Quality
|Narrow Minor (Gothminor)
|'''Pythagorean Minor (Triminor)'''
|Artomean Minor
|Tendomean Minor
|Ptolemaic Minor (Pentaminor)
|Wide Minor
|-
!Cents
|287
|'''294'''
|302
|309
|317
|325
|-
!Just interpretation
|33/28, 13/11
|'''32/27'''
|25/21
|153/128
|6/5
|135/112
|-
!Steps
|38
|'''39'''
|40
|41
|42
|43
|}
{| class="wikitable"
|+Supraminor, Neutral and Submajor Thirds in 159edo
!Quality
|Lesser Supraminor
|Greater Supraminor
|Artoneutral
|Tendoneutral
|Lesser Submajor
|Greater Submajor
|-
!Cents
|332
|340
|347
|355
|362
|370
|-
!Just interpretation
|40/33, 63/52
|39/32, 17/14
|11/9
|27/22
|16/13, 21/17
|99/80, 26/21
|-
!Steps
|44
|45
|46
|47
|48
|49
|}
{| class="wikitable"
|+Major Thirds in 159edo
!Quality
|Narrow Major
|Ptolemaic Major (Pentamajor)
|Artomean Major
|Tendomean Major
|'''Pythagorean Major (Trimajor)'''
|Wide Major (Gothmajor)
|-
!Cents
|377
|385
|392
|400
|'''408'''
|415
|-
!Just interpretation
|56/45
|5/4
|64/51
|63/50
|'''81/64'''
|14/11, 33/26
|-
!Steps
|50
|51
|52
|53
|'''54'''
|55
|}
{| class="wikitable"
|+Supermajor and Ultramajor Thirds in 159edo
!Quality
|Narrow Supermajor
|Lesser Supermajor
|Greater Supermajor (Septimajor)
|Wide Supermajor
|Narrow Ultramajor
|Ultramajor
|Wide Ultramajor
|-
!Cents
|423
|430
|438
|445
|453
|460
|468
|-
!Just interpretation
|51/40
|50/39
|9/7
|22/17
|13/10
|2673/2048
|891/680
|-
!Steps
|56
|57
|58
|59
|60
|61
|62
|}
Regular diatonic thirds are bolded.

=== Chords ===
159edo has a vast array of triads at its disposal, both fifth-bounded and fourth-bounded. However, the JI interpretation will tend to inform the usage of the various triads offered. The main exceptions to this rule involve chords that serve as detempers of other, smaller tuning systems.

== Notation ===
There are multiple different tuning systems available for 159edo.

=== Sagittal ===
<todo>

=== Syntonic-Rastmic Subchroma notation ===
Syntonic-rastmic subchroma notation, or SRS for short, is an alternative to Sagittal with fewer accidentals- one which was originally designed for 159edo. In 159edo, the synsharp (81/80) and the rasharp (243/242) are 3\159 (1\53) and 1\159 respectively, treating 159edo as a subdivision of 53edo wherein three rastmas make up a syntonic comma. So far, this is functionally identical to ups and downs notation (assuming the up and down are treated as 1\53), but syntonic-rastmic subchroma notation also supports halving of all of its accidentals. While 159edo divides neither the Pythagorean, syntonic, or rastmic accidental pairs in half, this does allow for the notation of artoneutral and tendoneutral intervals via the inflection of a semisharp by half a rastma (which ends up at a full 159edo step, specifically 7 steps for the artodemisharp and 8 steps for the tendodemisharp).

The syntonic and rastmic accidentals represent their just mappings when generalized to other EDOs.{{Navbox EDO}}
{{Cat|Edos}}

106edo

2026-04-06T10:05:59Z

Aura: Added redirect to 53edo page

#redirect [[53edo#106edo]]

Template:Navbox EDO

2026-04-06T10:03:09Z

Aura: Figured out how to add EDOs to the navbox- added 106edo and will add a redirect soon

{{Navbox
| name = Navbox EDO
| Title = Equal temperaments
| Is Collapsible = true
| Header 1 = EDOs
| Header 2 = Macrotonal
| Data 2 = [[5edo|5]] • [[7edo|7]] • [[8edo|8]] • [[9edo|9]] • [[10edo|10]] • [[11edo|11]]
| Header 3 = 12-23
| Data 3 = [[12edo|12]] • [[13edo|13]] • [[14edo|14]] • [[15edo|15]] • [[16edo|16]] • [[17edo|17]] • [[18edo|18]] • [[19edo|19]] • [[20edo|20]] • [[21edo|21]] • [[22edo|22]] • [[23edo|23]]
| Header 4 = 24-35
| Data 4 = [[24edo|24]] • [[25edo|25]] • [[26edo|26]] • [[27edo|27]] • [[29edo|29]] • [[31edo|31]] • [[32edo|32]] • [[34edo|34]] • [[35edo|35]]
| Header 5 = 36-47
| Data 5 = [[36edo|36]] • [[37edo|37]] • [[39edo|39]] • [[40edo|40]] • [[41edo|41]] • [[43edo|43]] • [[44edo|44]] • [[45edo|45]] • [[46edo|46]] • [[47edo|47]]
| Header 6 = 48-59
| Data 6 = [[48edo|48]] • [[50edo|50]] • [[51edo|51]] • [[53edo|53]] • [[54edo|54]] • [[56edo|56]] • [[57edo|57]] • [[58edo|58]]
| Header 7 = 60-71
| Data 7 = [[60edo|60]] • [[63edo|63]] • [[64edo|64]] • [[65edo|65]] • [[67edo|67]] • [[68edo|68]] • [[70edo|70]]
| Header 8 = 72-83
| Data 8 = [[72edo|72]] • [[77edo|77]] • [[80edo|80]] • [[81edo|81]]
| Header 9 = 84-95
| Data 9 = [[84edo|84]] • [[87edo|87]] • [[89edo|89]] • [[90edo|90]] • [[93edo|93]] • [[94edo|94]]
| Header 10 = Large EDOs
| Data 10 = [[99edo|99]] • [[104edo|104]] • [[106edo|106]] • [[111edo|111]] • [[118edo|118]] • [[130edo|130]] • [[140edo|140]] • [[152edo|152]] • [[159edo|159]] • [[171edo|171]] • [[217edo|217]] • [[224edo|224]] • [[239edo|239]] • [[270edo|270]] • [[306edo|306]] • [[311edo|311]] • [[612edo|612]] • [[665edo|665]]
| Header 11 = Nonoctave equal temperaments
| Header 12 = Tritave
| Data 12 = [[4edt|4]] • [[9edt|9]] • [[Bohlen-Pierce|13]] • [[17edt|17]] • [[26edt|26]] • [[39edt|39]]
| Header 13 = Fifth
| Data 13 = [[8edf|8]] • [[Carlos Alpha|9]] • [[Carlos Beta|11]] • [[Carlos Gamma|20]]
| Header 14 = Other
| Data 14 =
}}

<noinclude>
Only add tritave temperaments lower than 149edt, fifth temperaments lower than 55edf, etc.

159edo

2026-04-06T09:54:50Z

Aura: Added category tag

'''159edo''', or 159 equal divisions of the octave, is the equal tuning featuring steps of (1200/159) ~= 7.55 cents, 159 of which stack to the perfect octave [[2/1]]. Like [[53edo]], 159edo is an excellent approximation to Pythagorean tuning (stacking pure 3/2 fifths), however it includes near-just approximations of the 11th and 17th harmonics, and a slightly more accurate 7th harmonic, resulting in consistency up to the 17-odd-limit. The step-size, being slightly above the melodic just noticeable difference (JND) of 5 cents as well as more than twice the harmonic JND of the average trained musician at 3.5 cents, enables one to perform fluid modulations by means of [[comma pump]]s as well as by step substitutions. Furthermore, 159edo, like a number of higher edos, is characterized by being capable of imitating the intervals of smaller tuning systems - in this case, you get detemperings of [[10edo]], [[12edo]], [[13edo]], [[14edo]], [[17edo]], [[19edo]], [[22edo]], [[24edo]] and [[31edo]] among others with errors smaller than the melodic JND.

The interval qualities supported by 159edo are many, however, while every step of 159edo can be interpreted harmonically or subharmonically as being a 17-limit interval or simpler, some of the intervals you get have rather complex interpretations in terms of odd-limit. While the [[perfect fifth]] is really only divisible by three due to 159edo being the three-fold multiple of 53edo- resulting in [[slendric]] temperament and hence 159edo's distinction from 53 in the 7-limit- the [[perfect fourth]] has a little more to offer in terms of divisions. For starters, the perfect fourth can be divided into two instances of 15/13, giving us [[Semifourth-generated scales|island]] temperament. Dividing the perfect fourth into three instances of 11/10 gives us [[pine]] temperament. Dividing the perfect fourth into six instances of an interval which can be interpreted as 21/20 and 22/21 tempered together gives us [[sextilifourths]] temperament. The perfect fourth can also be cut into eleven intervals which, individually, are half of a Pythagorean limma, giving us a number of temperaments based on the exact interpretation of the semilimma.

On top of all that, there are a number of microtemperament-based structures also supported by 159edo, each of which provides some decent, unexpected melodic possibilities. For instance, one can split the Pythagorean minor third into three instances of 128/121 which gives us [[nexus]] temperament- a temperament which also happens to split the Pythagorean diatonic semitone into two, and the octave into three. One can also split the Ptolemaic minor third into three instances of 17/16, giving us [[archagall]] temperament, which is named for certain tunings found in other temperaments producing fractal-like acoustics. There's also the ability to split the Ptolemaic major sixth into six instances of 11/9, leading to [[parimic]] temperament. In addition, there's also the ability to split the septimal supermajor third into two instances of 17/15, leading to [[fidesmic]] temperament, which acts like a more accurate rendition of [[archy]] temperament in a different subgroup, and this can be exploited for modulation purposes. As if that weren't enough, there's the possibility of splitting the septimal subminor third into five instances of 33/32, leading to [[quartismic]] temperament. Furthermore, there's the possibility of splitting the greater tridecimal neutral tenth into three instances of 27/20, producing [[phaotismic]] temperament, and the list goes on.

== Theory ==
159edo was first used for maqams by Ozan Yarman. It was later put to use by Aura for its ability to handle near-just quartertones derived from the 2.3.11 subgroup on top of the 5-limit foundation provided by 53edo.

==== Edostep interpretations ====
159edo's edostep has the following interpretations in the 2.3.5.11.17 subgroup:

* 243/242, the difference between the 11-limit artoneutral third 11/9, and the 11-limit tendoneutral third 27/22
* 256/255, the difference between 16/15 and 17/16
* 289/288, the difference between 17/16 and 18/17

159edo tempers out the following commas in the 17-limit:

* The schisma (the difference between 5/4 and the Pythagorean diminished fourth)
* The vulture comma (the difference between four 320/243 intervals and the tritave)
* The amiton (the difference between a stack of five 10/9 intervals and 27/16)
* The kleisma (the difference between a stack of three 25/24 intervals and 9/8)
* The semicomma (the difference between a stack of three 75/64 intervals and 8/5)
* The gamelisma (the difference between a stack of three 8/7 intervals and 3/2)
* 385/384 (the difference between 77/64 and 6/5)
* The pine comma (the difference between a stack of three 11/10 intervals and 4/3)
* 625/624 (the difference between 25/24 and 26/25)
* 676/675 (the difference between a stack of two 15/13 intervals and the perfect fourth)
* The twosquare comma (the difference between a stack of two 33/32 intervals and 17/16)

==== JI approximation ====
Although 159edo inherits its approximations of the 5-limit from 53edo, the 5th harmonic can nonetheless be stacked twice without accumulating too much error, rendering it sufficient for Western Classical usage. While the 7th harmonic is technically more accurate in terms of absolute error than in 53edo, the relative error doesn't allow one to stack more than one instance of 7/4 without excessive error accumulation, and the same is true with 13/8. Regardless, the inconsistency remains less than 10 cents even when either interval is stacked three times, and 13/7 or 14/13 is tuned almost perfectly. As a whole, 159edo is characterized by its combination of accuracy in the 2.3.5.11.17 subgroup, and a series of compromises in the 7.13.19.23.29 subgroup- among the compromises are the slendric, [[marveltwin]], [[nestoria]], [[minor semivicemic]] and [[brunisimic]] temperaments.
{{Harmonics in ED|159|31|0}}

Currently, the [[ADIN]] system used for interval naming elsewhere on this site fails for 159edo, so another set of interval names will be used here, though the ADIN names will be referenced in places. Furthermore, because there are so many thirds, what is usually a single chart will be split into five.
{| class="wikitable"
|+Inframinor and Subminor Thirds in 159edo
!Quality
|Narrow Inframinor
|Inframinor
|Wide Inframinor
|Narrow Subminor
|Lesser Subminor (Septiminor)
|Greater Subminor
|Wide Subminor
|-
!Cents
|234
|242
|249
|257
|264
|272
|279
|-
!Just interpretation
|340/297
|1024/891
|15/13
|51/44
|7/6
|117/100
|20/17
|-
!Steps
|31
|32
|33
|34
|35
|36
|37
|}
{| class="wikitable"
|+Minor Thirds in 159edo
!Quality
|Narrow Minor (Gothminor)
|'''Pythagorean Minor (Triminor)'''
|Artomean Minor
|Tendomean Minor
|Ptolemaic Minor (Pentaminor)
|Wide Minor
|-
!Cents
|287
|'''294'''
|302
|309
|317
|325
|-
!Just interpretation
|33/28, 13/11
|'''32/27'''
|25/21
|153/128
|6/5
|135/112
|-
!Steps
|38
|'''39'''
|40
|41
|42
|43
|}
{| class="wikitable"
|+Supraminor, Neutral and Submajor Thirds in 159edo
!Quality
|Lesser Supraminor
|Greater Supraminor
|Artoneutral
|Tendoneutral
|Lesser Submajor
|Greater Submajor
|-
!Cents
|332
|340
|347
|355
|362
|370
|-
!Just interpretation
|40/33, 63/52
|39/32, 17/14
|11/9
|27/22
|16/13, 21/17
|99/80, 26/21
|-
!Steps
|44
|45
|46
|47
|48
|49
|}
{| class="wikitable"
|+Major Thirds in 159edo
!Quality
|Narrow Major
|Ptolemaic Major (Pentamajor)
|Artomean Major
|Tendomean Major
|'''Pythagorean Major (Trimajor)'''
|Wide Major (Gothmajor)
|-
!Cents
|377
|385
|392
|400
|'''408'''
|415
|-
!Just interpretation
|56/45
|5/4
|64/51
|63/50
|'''81/64'''
|14/11, 33/26
|-
!Steps
|50
|51
|52
|53
|'''54'''
|55
|}
{| class="wikitable"
|+Supermajor and Ultramajor Thirds in 159edo
!Quality
|Narrow Supermajor
|Lesser Supermajor
|Greater Supermajor (Septimajor)
|Wide Supermajor
|Narrow Ultramajor
|Ultramajor
|Wide Ultramajor
|-
!Cents
|423
|430
|438
|445
|453
|460
|468
|-
!Just interpretation
|51/40
|50/39
|9/7
|22/17
|13/10
|2673/2048
|891/680
|-
!Steps
|56
|57
|58
|59
|60
|61
|62
|}
Regular diatonic thirds are bolded.

=== Chords ===
159edo has a vast array of triads at its disposal, both fifth-bounded and fourth-bounded. However, the JI interpretation will tend to inform the usage of the various triads offered. The main exceptions to this rule involve chords that serve as detemperings of other, smaller tuning systems.

{{Navbox EDO}}
{{Cat|Edos}}

53edo

2026-04-06T09:53:37Z

Aura: Moved the Navbox and category tag

'''53edo''', or 53 equal divisions of the octave, is the equal tuning featuring steps of (1200/53) ~= 22.64 cents, 53 of which stack to the perfect octave [[2/1]]. 53edo is an excellent approximation to Pythagorean tuning (stacking pure 3/2 fifths). Theoretical interest in this tuning system goes back to antiquity.

== Theory ==
Unless one has a set of accidentals for the syntonic comma (see the Notation section) one is left in the unenviable position of having to label a Ptolemaic major third the same way as the Pythagorean diminished fourth, for example. Apart from that issue, 53edo is very useful for 5-limit music.

==== Edostep interpretations ====
53edo's edostep has the following interpretations in the 2.3.5.7.13 subgroup:

* 65/64, the difference between the 13-limit tendoneutral third 16/13 and the classical major third 5/4
* 81/80 (the syntonic comma), the difference between 5/4 and the diatonic major third
* The Pythagorean comma, the difference between the Pythagorean diatonic and chromatic semitones
* 91/90, the difference between the 13-limit ultramajor third 13/10 and the septimal supermajor third 9/7
* 64/63, the difference between the diatonic major third and 9/7
* 512/507, the difference between the 13-limit neutral thirds

53edo tempers out the following commas:

* The schisma (the difference between 5/4 and the Pythagorean diminished fourth)
* The vulture comma (the difference between four 320/243 intervals and the tritave)
* The amiton (the difference between a stack of five 10/9 intervals and 27/16)
* The kleisma (the difference between a stack of three 25/24 intervals and 9/8)
* The semicomma (the difference between a stack of three 75/64 intervals and 8/5)
* 225/224 (the difference between 15/14 and 16/15)
* 385/384 (the difference between 77/64 and 6/5)
* 121/120 (the difference between 12/11 and 11/10)
* 625/624 (the difference between 25/24 and 26/25)
* 676/675 (the difference between a stack of two 15/13 intervals and the perfect fourth)

==== JI approximation ====
53edo is most usefully seen as a 2.3.5.7.13 tuning, but the 2.3.5.13 restriction is more accurate and shared with a number of its multiples, such as [[159edo]]. Because it is not a Meantone system, there are actually multiple potential diatonic scales to use for 5-limit harmony, one of which is the Zarlino diatonic scale (LMsLMLs), tuned in 53edo as 9-8-5-9-8-9-5, though this particular scale is arguably best used for Lydian or Locrian modes. There's also the Didymic diatonic scale, tuned in 53edo as 9-8-5-9-9-8-5, which is better suited for Ionian mode and Major tonality in general. However, 53edo also features a MOS diatonic of 9-9-4-9-9-9-4, which is basically the Pythagorean diatonic scale.

{{Harmonics in ED|53|31|0}}
{| class="wikitable"
|+Thirds in 53edo
!Quality
|Inframinor
|Subminor
|'''Farminor'''
|Nearminor
|Supraminor
|Submajor
|Nearmajor
|'''Farmajor'''
|Supermajor
|Ultramajor
|-
!Cents
|249
|272
|'''294'''
|317
|340
|362
|385
|'''408'''
|430
|453
|-
!Just interpretation
|15/13
|7/6, 75/64
|'''32/27'''
|6/5
|39/32
|16/13
|5/4
|'''81/64'''
|9/7, 32/25
|13/10
|-
!Steps
|11
|12
|'''13'''
|14
|15
|16
|17
|'''18'''
|19
|20
|}
Diatonic thirds are bolded.

=== Chords ===
53edo has four different flavors of minor and major intervals as well as supraminor and submajor intervals. Its inframinor and ultramajor thirds approximate 15/13 and 13/10 respectively. At the same time, 53edo's subminor and supermajor intervals approximate 7/6 and 9/7. Then there's the novaminor and novamajor thirds, which are extremely close approximations of Pythagorean minor and major thirds and can be referred to as such. There are also the pentaminor and pentamajor thirds, which are very close approximations of the Ptolemaic minor and major thirds and can also be referred to as such. Finally, the supraminor and submajor thirds approximate 39/32 and 16/13. For fourth-bounded triads, there's only really five options. The first two, which involve the approximations of 9/8 and 32/27, have a marked propensity to cause crowding, and thus are dissonant. Then there's the next two, the latal triads, which involve the approximations of 8/7 and 7/6, and which, due to their tuning are markedly less dissonant, but still dissonant. Finally, the last option, which splits the perfect fourth cleanly in half, is an ambisonance- that is, an interval that is halfway between the extremes of consonance and dissonance.

== Notation ==
This section provides some of the options for notating 53edo.

=== Pythagorean notation ===
In 53edo, the space between each of the notes that is separated by 2 steps in 12edo is instead 9 steps; notes separated by a single step in 12edo have to be distinguished from each other as the Pythagorean diatonic semitone is 4 steps while the Pythagorean chromatic semitone is 5 steps. Furthermore, the Pythagorean comma is a single step in 53edo, unlike in 12edo where it's tempered out. It is important to understand the usage of enharmonic equivalence here; unlike in systems such as 31edo where each note has an easily derivable "canonical" notation, it is important to understand the multiple faces of each of 53edo's pitches (which some might consider as a downside of using the Pythagorean system).

{| class="wikitable"
|+
!
!
|-
| colspan="2" |D
|-
|^^Ebb
|^D
|-
|vvEb
|^^D
|-
|vEb
|vvD#
|-
|Eb
|vD#
|-
|^Eb
|D#
|-
|^^Eb
|^D#
|-
|vvE
|^^D#
|-
|vE
|vvDx
|-
| colspan="2" |E
|}

==== Ups and Downs ====
Ups and downs naturally reflect 53edo's structure, as 5/4 is downmajor, 81/64 is major, and 9/7 is upmajor.

==== Syntonic-Rastmic Subchroma notation ====
Syntonic-Rastmic Subchroma notation, or SRS notation for short, uses '''synsharp''' and '''synflat''' as accidentals to cover the syntonic comma. However, while SRS notation is a 2.3.5.11 notation, only the 2.3.5 portion of the notation for 53edo is shared with multiples like 159edo.

==== Accidentals ====
53edo's accidentals, as mentioned and demonstrated previously, consist of sharps and flats, as well as either up and down accidentals, or, alternatively, synsharps and synflats and their derivatives.

== Multiples ==

===106edo===
106edo has inconsistent 11th and 17th harmonics, and also loses some ability to be consistent that 53edo has due to inconsistencies in the 7-odd-limit.

{{Harmonics in ED|106|31|0}}

===159edo===
''Main article: [[159edo]]''

159edo corrects 53edo's approximate 11th and 17th harmonics to near-just qualities, and also improves the 7th harmonic to a lesser extent, resulting in it being consistent to the 17-odd-limit. If you go further than that, you're forced to choose between the 17the harmonic on one hand and both the 19th and 29th harmonics on the other, but you do get a good 23rd harmonic regardless. In addition, you also gain access to a set of intervals that approximates those of simpler systems such as [[10edo]], [[12edo]], [[13edo]], [[14edo]], [[17edo]], [[19edo]], [[22edo]], [[24edo]] and [[31edo]] among others, opening up additional compositional techniques.

{{Harmonics in ED|159|31|0}}
{{Navbox EDO}}
{{Cat|Edos}}

159edo

2026-04-06T09:48:37Z

Aura: Added the EDO navbox

'''159edo''', or 159 equal divisions of the octave, is the equal tuning featuring steps of (1200/159) ~= 7.55 cents, 159 of which stack to the perfect octave [[2/1]]. Like [[53edo]], 159edo is an excellent approximation to Pythagorean tuning (stacking pure 3/2 fifths), however it includes near-just approximations of the 11th and 17th harmonics, and a slightly more accurate 7th harmonic, resulting in consistency up to the 17-odd-limit. The step-size, being slightly above the melodic just noticeable difference (JND) of 5 cents as well as more than twice the harmonic JND of the average trained musician at 3.5 cents, enables one to perform fluid modulations by means of [[comma pump]]s as well as by step substitutions. Furthermore, 159edo, like a number of higher edos, is characterized by being capable of imitating the intervals of smaller tuning systems - in this case, you get detemperings of [[10edo]], [[12edo]], [[13edo]], [[14edo]], [[17edo]], [[19edo]], [[22edo]], [[24edo]] and [[31edo]] among others with errors smaller than the melodic JND.

The interval qualities supported by 159edo are many, however, while every step of 159edo can be interpreted harmonically or subharmonically as being a 17-limit interval or simpler, some of the intervals you get have rather complex interpretations in terms of odd-limit. While the [[perfect fifth]] is really only divisible by three due to 159edo being the three-fold multiple of 53edo- resulting in [[slendric]] temperament and hence 159edo's distinction from 53 in the 7-limit- the [[perfect fourth]] has a little more to offer in terms of divisions. For starters, the perfect fourth can be divided into two instances of 15/13, giving us [[Semifourth-generated scales|island]] temperament. Dividing the perfect fourth into three instances of 11/10 gives us [[pine]] temperament. Dividing the perfect fourth into six instances of an interval which can be interpreted as 21/20 and 22/21 tempered together gives us [[sextilifourths]] temperament. The perfect fourth can also be cut into eleven intervals which, individually, are half of a Pythagorean limma, giving us a number of temperaments based on the exact interpretation of the semilimma.

On top of all that, there are a number of microtemperament-based structures also supported by 159edo, each of which provides some decent, unexpected melodic possibilities. For instance, one can split the Pythagorean minor third into three instances of 128/121 which gives us [[nexus]] temperament- a temperament which also happens to split the Pythagorean diatonic semitone into two, and the octave into three. One can also split the Ptolemaic minor third into three instances of 17/16, giving us [[archagall]] temperament, which is named for certain tunings found in other temperaments producing fractal-like acoustics. There's also the ability to split the Ptolemaic major sixth into six instances of 11/9, leading to [[parimic]] temperament. In addition, there's also the ability to split the septimal supermajor third into two instances of 17/15, leading to [[fidesmic]] temperament, which acts like a more accurate rendition of [[archy]] temperament in a different subgroup, and this can be exploited for modulation purposes. As if that weren't enough, there's the possibility of splitting the septimal subminor third into five instances of 33/32, leading to [[quartismic]] temperament. Furthermore, there's the possibility of splitting the greater tridecimal neutral tenth into three instances of 27/20, producing [[phaotismic]] temperament, and the list goes on.

== Theory ==
159edo was first used for maqams by Ozan Yarman. It was later put to use by Aura for its ability to handle near-just quartertones derived from the 2.3.11 subgroup on top of the 5-limit foundation provided by 53edo.

==== Edostep interpretations ====
159edo's edostep has the following interpretations in the 2.3.5.11.17 subgroup:

* 243/242, the difference between the 11-limit artoneutral third 11/9, and the 11-limit tendoneutral third 27/22
* 256/255, the difference between 16/15 and 17/16
* 289/288, the difference between 17/16 and 18/17

159edo tempers out the following commas in the 17-limit:

* The schisma (the difference between 5/4 and the Pythagorean diminished fourth)
* The vulture comma (the difference between four 320/243 intervals and the tritave)
* The amiton (the difference between a stack of five 10/9 intervals and 27/16)
* The kleisma (the difference between a stack of three 25/24 intervals and 9/8)
* The semicomma (the difference between a stack of three 75/64 intervals and 8/5)
* The gamelisma (the difference between a stack of three 8/7 intervals and 3/2)
* 385/384 (the difference between 77/64 and 6/5)
* The pine comma (the difference between a stack of three 11/10 intervals and 4/3)
* 625/624 (the difference between 25/24 and 26/25)
* 676/675 (the difference between a stack of two 15/13 intervals and the perfect fourth)
* The twosquare comma (the difference between a stack of two 33/32 intervals and 17/16)

==== JI approximation ====
Although 159edo inherits its approximations of the 5-limit from 53edo, the 5th harmonic can nonetheless be stacked twice without accumulating too much error, rendering it sufficient for Western Classical usage. While the 7th harmonic is technically more accurate in terms of absolute error than in 53edo, the relative error doesn't allow one to stack more than one instance of 7/4 without excessive error accumulation, and the same is true with 13/8. Regardless, the inconsistency remains less than 10 cents even when either interval is stacked three times, and 13/7 or 14/13 is tuned almost perfectly. As a whole, 159edo is characterized by its combination of accuracy in the 2.3.5.11.17 subgroup, and a series of compromises in the 7.13.19.23.29 subgroup- among the compromises are the slendric, [[marveltwin]], [[nestoria]], [[minor semivicemic]] and [[brunisimic]] temperaments.
{{Harmonics in ED|159|31|0}}

Currently, the [[ADIN]] system used for interval naming elsewhere on this site fails for 159edo, so another set of interval names will be used here, though the ADIN names will be referenced in places. Furthermore, because there are so many thirds, what is usually a single chart will be split into five.
{| class="wikitable"
|+Inframinor and Subminor Thirds in 159edo
!Quality
|Narrow Inframinor
|Inframinor
|Wide Inframinor
|Narrow Subminor
|Lesser Subminor (Septiminor)
|Greater Subminor
|Wide Subminor
|-
!Cents
|234
|242
|249
|257
|264
|272
|279
|-
!Just interpretation
|340/297
|1024/891
|15/13
|51/44
|7/6
|117/100
|20/17
|-
!Steps
|31
|32
|33
|34
|35
|36
|37
|}
{| class="wikitable"
|+Minor Thirds in 159edo
!Quality
|Narrow Minor (Gothminor)
|'''Pythagorean Minor (Triminor)'''
|Artomean Minor
|Tendomean Minor
|Ptolemaic Minor (Pentaminor)
|Wide Minor
|-
!Cents
|287
|'''294'''
|302
|309
|317
|325
|-
!Just interpretation
|33/28, 13/11
|'''32/27'''
|25/21
|153/128
|6/5
|135/112
|-
!Steps
|38
|'''39'''
|40
|41
|42
|43
|}
{| class="wikitable"
|+Supraminor, Neutral and Submajor Thirds in 159edo
!Quality
|Lesser Supraminor
|Greater Supraminor
|Artoneutral
|Tendoneutral
|Lesser Submajor
|Greater Submajor
|-
!Cents
|332
|340
|347
|355
|362
|370
|-
!Just interpretation
|40/33, 63/52
|39/32, 17/14
|11/9
|27/22
|16/13, 21/17
|99/80, 26/21
|-
!Steps
|44
|45
|46
|47
|48
|49
|}
{| class="wikitable"
|+Major Thirds in 159edo
!Quality
|Narrow Major
|Ptolemaic Major (Pentamajor)
|Artomean Major
|Tendomean Major
|'''Pythagorean Major (Trimajor)'''
|Wide Major (Gothmajor)
|-
!Cents
|377
|385
|392
|400
|'''408'''
|415
|-
!Just interpretation
|56/45
|5/4
|64/51
|63/50
|'''81/64'''
|14/11, 33/26
|-
!Steps
|50
|51
|52
|53
|'''54'''
|55
|}
{| class="wikitable"
|+Supermajor and Ultramajor Thirds in 159edo
!Quality
|Narrow Supermajor
|Lesser Supermajor
|Greater Supermajor (Septimajor)
|Wide Supermajor
|Narrow Ultramajor
|Ultramajor
|Wide Ultramajor
|-
!Cents
|423
|430
|438
|445
|453
|460
|468
|-
!Just interpretation
|51/40
|50/39
|9/7
|22/17
|13/10
|2673/2048
|891/680
|-
!Steps
|56
|57
|58
|59
|60
|61
|62
|}
Regular diatonic thirds are bolded.

=== Chords ===
159edo has a vast array of triads at its disposal, both fifth-bounded and fourth-bounded. However, the JI interpretation will tend to inform the usage of the various triads offered. The main exceptions to this rule involve chords that serve as detemperings of other, smaller tuning systems.

{{Navbox EDO}}

159edo

2026-04-04T11:16:15Z

Aura: connected the link to the page that mentions Island temperament

User:Aura/On 159edo Music Theory (Part 1)

2026-03-31T19:57:53Z

Aura:

Of all the multiples of [[53edo]], [[159edo]] is the lowest multiple that is noteworthy for being accurate in the 2.3.5.11.17 subgroup while having structural compromises in the 7.13.19.23.29 subgroup. Despite the number of pitches in this tuning system making it perhaps best fit for digital instruments of various kinds in actual performance, it is nevertheless also useful as an interval classification scheme.

== Intervals and Notation ==
159edo contains all the intervals of 53edo and can be thought of as having three fields of 53edo each separated by a third of 53edo's step. However, as some of the interpretations differ due 159edo having different mappings for certain primes, those differences show up in how harmonies are constructed. However, there's more.

Of all the intervals in 159edo, 5\159 is the first interval to be larger than the '''fission boundary''', which is where going back and forth between notes on either end of a given interval no longer sounds like a simple vibrato, but more like a dirty trill of sorts. The fission boundary further serves as the line separating melodic notes that can only be simple ornaments or quick passing tones from main melodic intervals, and since 5\159 is larger than this boundary it is the smallest interval that can serve as a main melodic interval.

The next landmark interval is 8\159, as this is the first interval to be larger than the '''gradient threshold''', which is where going back and forth between notes on either end of a given interval no longer sounds like a dirty trill, but rather a clean trill. The gradient threshold doubles as the point beyond which microtonal intervals can begin to serve as proper leading-tones.

Finally, 33\159 is the first interval to be larger than the '''trill threshold''', which is where going back and forth between notes on either end of a given interval no longer sounds like any kind of trill, and instead sounds like an arpeggio fragment. The trill threshold doubles as the boundary between intervals that are classified as steps, and those that are classified as leaps. As a consequence of this, the trill threshold marks the boundary where intervals cease to cause crowding in chords.

As if all that weren't enough, 159edo has its own variation on the [[dinner party rules]]— represented here by the Harmonic Compatibility Rating and Melodic Compatibility Rating columns in the following chart, where 10 is a full-blown friend relative to the root and −10 if a full-blown enemy relative to the root. Note that the Harmonic Compatibility and Melodic Compatibility ratings are based on octave-equivalence, and that some of the ratings are still speculative.

{| class="wikitable"
|+159edo Interval Names and Compatibility Ratings
! rowspan="2" | Step
! rowspan="2" | Cents
! colspan="3" | Interval and Note names
! colspan="2" | Compatibility rating
|-
! SKULO-based interval names
! Pythagorean-commatic-based interval names
! SRS notation
! Harmonic
! Melodic
|-
| 0
| 0
| P1
| Perfect Unison
| D
| 10
| 10
|-
| 1
| 7.5471698
| R1
| Wide Unison
| D/
| 0
| 0
|-
| 2
| 15.0943396
| rK1
| Narrow Superunison
| D↑\
| -10
| -10
|-
| 3
| 22.6415094
| K1
| Lesser Superunison
| D↑
| -10
| -3
|-
| 4
| 30.1886792
| S1, kU1
| Greater Superunison, Narrow Inframinor Second
| Edb<, Dt<↓
| -10
| 3
|-
| 5
| 37.7358491
| um2, RkU1
| Inframinor Second, Wide Superunison
| Edb>, Dt>↓
| -9
| 10
|-
| 6
| 45.2830189
| kkm2, Rum2, rU1
| Wide Inframinor Second, Narrow Ultraunison
| Eb↓↓, Dt<\
| -9
| 10
|-
| 7
| 52.8301887
| U1, rKum2
| Ultraunison, Narrow Subminor Second
| Dt<, Edb<↑
| -9
| 10
|-
| 8
| 60.3773585
| sm2, Kum2, uA1
| Lesser Subminor Second, Wide Ultraunison, Infra-Augmented Unison
| Dt>, Eb↓\
| -8
| 10
|-
| 9
| 67.9245283
| km2, RuA1, kkA1
| Greater Subminor Second, Diptolemaic Augmented Unison
| Eb↓, D#↓↓
| -8
| 9
|-
| 10
| 75.4716981
| Rkm2, rKuA1
| Wide Subminor Second, Lesser Sub-Augmented Unison
| Eb↓/, Dt<↑
| -7
| 9
|-
| 11
| 83.0188679
| rm2, KuA1
| Narrow Minor Second, Greater Sub-Augmented Unison
| Eb\, Dt>↑
| -7
| 9
|-
| 12
| 90.5660377
| m2, kA1
| Pythagorean Minor Second, Ptolemaic Augmented Unison
| Eb, D#↓
| -6
| 10
|-
| 13
| 98.1132075
| Rm2, RkA1
| Artomean Minor Second, Artomean Augmented Unison
| Eb/, D#↓/
| -6
| 10
|-
| 14
| 105.6603774
| rKm2, rA1
| Tendomean Minor Second, Tendomean Augmented Unison
| D#\, Eb↑\
| -5
| 10
|-
| 15
| 113.2075472
| Km2, A1
| Ptolemaic Minor Second, Pythagorean Augmented Unison
| D#, Eb↑
| -5
| 10
|-
| 16
| 120.7547170
| RKm2, kn2, RA1
| Wide Minor Second, Artoretromean Augmented Unison
| Ed<↓, Eb↑/, D#/
| -5
| 9
|-
| 17
| 128.3018868
| kN2, rKA1
| Lesser Supraminor Second, Tendoretromean Augmented Unison
| Ed>↓, D#↑\
| -6
| 8
|-
| 18
| 135.8490566
| KKm2, rn2, KA1
| Greater Supraminor Second, Diptolemaic Limma, Retroptolemaic Augmented Unison
| Ed<\, Eb↑↑, D#↑
| -7
| 6
|-
| 19
| 143.3962264
| n2, SA1
| Artoneutral Second, Lesser Super-Augmented Unison
| Ed<, Dt#<↓
| -8
| 5
|-
| 20
| 150.9433962
| N2, RkUA1
| Tendoneutral Second, Greater Super-Augmented Unison
| Ed>, Dt#>↓
| -7
| 6
|-
| 21
| 158.4905660
| kkM2, RN2, rUA1
| Lesser Submajor Second, Retrodiptolemaic Augmented Unison
| Ed>/, E↓↓, Dt#>↓/, D#↑↑
| -6
| 8
|-
| 22
| 166.0377358
| Kn2, UA1
| Greater Submajor Second, Ultra-Augmented Unison
| Ed<↑, Dt#<, Fb↓/
| -5
| 9
|-
| 23
| 173.5849057
| rkM2, KN2
| Narrow Major Second
| Ed>↑, E↓\, Dt#>, Fb\
| -4
| 10
|-
| 24
| 181.1320755
| kM2
| Ptolemaic Major Second
| E↓, Fb
| -3
| 10
|-
| 25
| 188.6792458
| RkM2
| Artomean Major Second
| E↓/, Fb/
| -3
| 10
|-
| 26
| 196.2264151
| rM2
| Tendomean Major Second
| E\, Fb↑\
| -2
| 10
|-
| 27
| 203.7735849
| M2
| Pythagorean Major Second
| E, Fb↑
| -2
| 10
|-
| 28
| 211.3207547
| RM2
| Wide Major Second
| E/, Fd<↓
| -1
| 10
|-
| 29
| 218.8679245
| rKM2
| Narrow Supermajor Second
| E↑\, Fd>↓
| -1
| 10
|-
| 30
| 226.4150943
| KM2
| Lesser Supermajor Second
| E↑, Fd<\, Fb↑↑, Dx
| -1
| 9
|-
| 31
| 233.9622642
| SM2, kUM2
| Greater Supermajor Second, Narrow Inframinor Third
| Fd<, Et<↓, E↑/
| 0
| 9
|-
| 32
| 241.5094340
| um3, RkUM2
| Inframinor Third, Wide Supermajor Second
| Fd>, Et>↓
| -1
| 8
|-
| 33
| 249.0566038
| kkm3, KKM2, Rum3, rUM2
| Wide Inframinor Third, Narrow Ultramajor Second, Semifourth
| Fd>/, Et<\, F↓↓, E↑↑
| 0
| 8
|-
| 34
| 256.6037736
| UM2, rKum3
| Ultramajor Second, Narrow Subminor Third
| Et<, Fd<↑
| -1
| 7
|-
| 35
| 264.1509434
| sm3, Kum3
| Lesser Subminor Third, Wide Ultramajor Second
| Et>, Fd>↑, F↓\
| 0
| 7
|-
| 36
| 271.6981132
| km3
| Greater Subminor Third
| F↓, Et>/, E#↓↓, Gbb
| -1
| 7
|-
| 37
| 279.2452830
| Rkm3
| Wide Subminor Third
| F↓/, Et<↑
| -1
| 8
|-
| 38
| 286.7924528
| rm3
| Narrow Minor Third
| F\, Et>↑
| 0
| 8
|-
| 39
| 294.3396226
| m3
| Pythagorean Minor Third
| F
| -1
| 9
|-
| 40
| 301.8867925
| Rm3
| Artomean Minor Third
| F/
| 1
| 9
|-
| 41
| 309.4339622
| rKm3
| Tendomean Minor Third
| F↑\
| 4
| 10
|-
| 42
| 316.9811321
| Km3
| Ptolemaic Minor Third
| F↑, E#
| 7
| 10
|-
| 43
| 324.5283019
| RKm3, kn3
| Wide Minor Third
| Ft<↓, F↑/, Gdb<
| 4
| 9
|-
| 44
| 332.0754717
| kN3, ud4
| Lesser Supraminor Third, Infra-Diminished Fourth
| Ft>↓, Gdb>
| 1
| 9
|-
| 45
| 339.6226415
| KKm3, rn3, Rud4
| Greater Supraminor Third, Retrodiptolemaic Diminished Fourth
| Ft<\, F↑↑, Gdb<↑\, Gb↓↓
| -1
| 8
|-
| 46
| 347.1698113
| n3, rKud4
| Artoneutral Third, Lesser Sub-Diminished Fourth
| Ft<, Gdb<↑
| 0
| 7
|-
| 47
| 354.7169811
| N3, sd4, Kud4
| Tendoneutral Third, Greater Sub-Diminished Fourth
| Ft>, Gdb>↑
| -1
| 7
|-
| 48
| 362.2641509
| kkM3, RN3, kd4
| Lesser Submajor Third, Retroptolemaic Diminished Fourth
| Ft>/, F#↓↓, Gb↓
| 0
| 8
|-
| 49
| 369.8113208
| Kn3, Rkd4
| Greater Submajor Third, Artoretromean Diminished Fourth
| Ft<↑, Gb↓/
| -1
| 9
|-
| 50
| 377.3584906
| rkM3, KN3, rd4
| Narrow Major Third, Tendoretromean Diminished Fourth
| Ft>↑, F#↓\, Gb\
| 3
| 9
|-
| 51
| 384.9056604
| kM3, d4
| Ptolemaic Major Third, Pythagorean Diminished Fourth
| Gb, F#↓
| 8
| 10
|-
| 52
| 392.4528302
| RkM3, Rd4
| Artomean Major Third, Artomean Diminished Fourth
| Gb/, F#↓/
| 4
| 10
|-
| 53
| 400
| rM3, rKd4
| Tendomean Major Third, Tendomean Diminished Fourth
| F#\, Gb↑\
| 1
| 9
|-
| 54
| 407.5471698
| M3, Kd4
| Pythagorean Major Third, Ptolemaic Diminished Fourth
| F#, Gb↑
| -1
| 9
|-
| 55
| 415.0943396
| RM3, kUd4
| Wide Major Third, Lesser Super-Diminished Fourth
| F#/, Gd<↓, Gb↑/
| 0
| 8
|-
| 56
| 422.6415094
| rKM3, RkUd4
| Narrow Supermajor Third, Greater Super-Diminished Fourth
| F#↑\, Gd>↓
| -1
| 7
|-
| 57
| 430.1886792
| KM3, rUd4, KKd4
| Lesser Supermajor Third, Diptolemaic Diminished Fourth
| F#↑, Gd<\, Gb↑↑
| -1
| 6
|-
| 58
| 437.7358491
| SM3, kUM3, rm4, Ud4
| Greater Supermajor Third, Ultra-Diminished Fourth
| Gd<, F#↑/
| 0
| 5
|-
| 59
| 445.2830189
| m4, RkUM3
| Paraminor Fourth, Wide Supermajor Third
| Gd>, Ft#>↓
| -1
| 3
|-
| 60
| 452.8301887
| Rm4, KKM3, rUM3
| Wide Paraminor Fourth, Narrow Ultramajor Third
| Gd>/, F#↑↑, G↓↓
| -2
| 1
|-
| 61
| 460.3773585
| UM3, rKm4
| Ultramajor Third, Narrow Grave Fourth
| Gd<↑, Ft#<
| -4
| -2
|-
| 62
| 467.9245283
| s4, Km4
| Lesser Grave Fourth, Wide Ultramajor Third
| Gd>↑, G↓\
| -7
| -4
|-
| 63
| 475.4716981
| k4
| Greater Grave Fourth
| G↓, Abb
| -6
| -5
|-
| 64
| 483.0188679
| Rk4
| Wide Grave Fourth
| G↓/
| -4
| 0
|-
| 65
| 490.5660377
| r4
| Narrow Fourth
| G\
| 1
| 5
|-
| 66
| 498.1132075
| P4
| Perfect Fourth
| G
| 9
| 10
|-
| 67
| 505.6603774
| R4
| Wide Fourth
| G/
| 1
| 8
|-
| 68
| 513.2075472
| rK4
| Narrow Acute Fourth
| G↑\
| -3
| 6
|-
| 69
| 520.7547170
| K4
| Lesser Acute Fourth
| G↑
| -5
| 5
|-
| 70
| 528.3018868
| S4, kM4
| Greater Acute Fourth
| Gt<↓, G↑/, Adb<
| -3
| 5
|-
| 71
| 535.8490566
| RkM4, ud5
| Wide Acute Fourth, Infra-Diminished Fifth
| Gt>↓, Adb>
| -2
| 5
|-
| 72
| 543.3962264
| rM4, Rud5
| Narrow Paramajor Fourth, Retrodiptolemaic Diminished Fifth
| Gt<\, G↑↑, Ab↓↓
| -1
| 6
|-
| 73
| 550.9433962
| M4, rKud5
| Paramajor Fourth, Lesser Sub-Diminished Fifth
| Gt<, Adb<↑
| 0
| 7
|-
| 74
| 558.4905660
| RM4, uA4, Kud5
| Infra-Augmented Fourth, Greater Sub-Diminished Fifth
| Gt>, Adb>↑
| -2
| 5
|-
| 75
| 566.0377358
| kkA4, RuA4, kd5
| Diptolemaic Augmented Fourth, Retroptolemaic Diminished Fifth
| Gt>/, G#↓↓, Ab↓
| -3
| 4
|-
| 76
| 573.5849057
| rKuA4, Rkd5
| Lesser Sub-Augmented Fourth, Artoretromean Diminished Fifth
| Gt<↑, Ab↓/
| -2
| 4
|-
| 77
| 581.1320755
| KuA4, rd5
| Greater Sub-Augmented Fourth, Tendoretromean Diminished Fifth
| Gt>↑, Ab\
| 0
| 5
|-
| 78
| 588.6792458
| kA4, d5
| Ptolemaic Augmented Fourth, Pythagorean Diminished Fifth
| Ab, G#↓
| -5
| 6
|-
| 79
| 596.2264151
| RkA4, Rd5
| Artomean Augmented Fourth, Artomean Diminished Fifth
| G#↓/, Ab/
| -9
| 7
|-
| 80
| 603.7735849
| rKd5, rA4
| Tendomean Diminished Fifth, Tendomean Augmented Fourth
| Ab↑\, G#\
| -9
| 7
|-
| 81
| 611.3207547
| Kd5, A4
| Ptolemaic Diminished Fifth, Pythagorean Augmented Fourth
| Ab↑, G#
| -5
| 6
|-
| 82
| 618.8679245
| kUd5, RA4
| Lesser Super-Diminished Fifth, Artoretromean Augmented Fourth
| Ad<↓, G#/
| 0
| 5
|-
| 83
| 626.4150943
| RkUd5, rKA4
| Greater Super-Diminished Fifth, Tendoretromean Augmented Fourth
| Ad>↓, G#↑\
| -2
| 4
|-
| 84
| 633.9622642
| KKd5, rUDd5, KA4
| Diptolemaic Diminished Fifth, Retroptolemaic Augmented Fourth
| Ad<\, Ab↑↑, G#↑
| -3
| 4
|-
| 85
| 641.5094340
| rm5, Ud5, kUA4
| Ultra-Diminished Fifth, Lesser Super-Augmented Fourth
| Ad<, Gt#<↓
| -2
| 5
|-
| 86
| 649.0566038
| m5, RkUA4
| Paraminor Fifth, Greater Super-Augmented Fourth
| Ad>, Gt#>↓
| 0
| 7
|-
| 87
| 656.6037736
| Rm5, rUA4
| Wide Paraminor Fifth, Retrodiptolemaic Augmented Fourth
| Ad>/, G#↑, Ab↑↑
| -1
| 6
|-
| 88
| 664.1509434
| rKm5, UA4
| Narrow Grave Fifth, Ultra-Augmented Fourth
| Ad<↑, Gt#<
| -2
| 5
|-
| 89
| 671.6981132
| s5, Km5
| Lesser Grave Fifth
| Ad>↑, A↓\, Gt#>
| -3
| 5
|-
| 90
| 679.2452830
| k5
| Greater Grave Fifth
| A↓
| -5
| 5
|-
| 91
| 686.7924528
| Rk5
| Wide Grave Fifth
| A↓/
| -3
| 6
|-
| 92
| 694.3396226
| r5
| Narrow Fifth
| A\
| 1
| 8
|-
| 93
| 701.8867925
| P5
| Perfect Fifth
| A
| 9
| 10
|-
| 94
| 709.4339622
| R5
| Wide Fifth
| A/
| 1
| 5
|-
| 95
| 716.9811321
| rK5
| Narrow Acute Fifth
| A↑\
| -4
| 0
|-
| 96
| 724.5283019
| K5
| Lesser Acute Fifth
| A↑, Gx
| -6
| -5
|-
| 97
| 732.0754717
| S5, kM5
| Greater Acute Fifth, Narrow Inframinor Sixth
| At<↓, A↑/
| -7
| -4
|-
| 98
| 739.6226415
| um6, RkM5
| Inframinor Sixth, Wide Acute Fifth
| At>↓, Bdb>
| -4
| -2
|-
| 99
| 747.1698113
| Rm4, KKM3, rUM3
| Narrow Paramajor Fifth, Wide Inframinor Sixth
| At<\, Bb↓↓, A↑↑
| -2
| 1
|-
| 100
| 754.7169811
| M5, rKum6
| Paramajor Fifth, Narrow Subminor Sixth
| At<, Bdb<↑
| -1
| 3
|-
| 101
| 762.2641509
| sm6, Kum6, RM5, uA5
| Lesser Subminor Sixth, Infra-Augmented Fifth
| At>, Bb↓\
| 0
| 5
|-
| 102
| 769.8113208
| km6, RuA5, kkA5
| Greater Subminor Sixth, Diptolemaic Augmented Fifth
| Bb↓, At>/, A#↓↓
| -1
| 6
|-
| 103
| 777.3584906
| Rkm6, rKuA5
| Wide Subminor Sixth, Lesser Sub-Augmented Fifth
| Bb↓/, At<↑
| -1
| 7
|-
| 104
| 784.9056604
| rm6, KuA5
| Narrow Minor Sixth, Greater Sub-Augmented Fifth
| Bb\, At>↑, A#↓\
| 0
| 8
|-
| 105
| 792.4528302
| m6, kA5
| Pythagorean Minor Sixth, Ptolemaic Augmented Fifth
| Bb, A#↓
| -1
| 9
|-
| 106
| 800
| Rm6, RkA5
| Artomean Minor Sixth, Artomean Augmented Fifth
| Bb/, A#↓/
| 1
| 9
|-
| 107
| 807.5471698
| rKm6, rA5
| Tendomean Minor Sixth, Tendomean Augmented Fifth
| A#\, Bb↑\
| 4
| 10
|-
| 108
| 815.0943396
| Km6, A5
| Ptolemaic Minor Sixth, Pythagorean Augmented Fifth
| A#, Bb↑
| 8
| 10
|-
| 109
| 822.6415094
| RKm6, kn6, RA5
|Wide Minor Sixth, Artoretromean Augmented Fifth
| Bd<↓, Bb↑/, A#/
| 3
| 9
|-
| 110
| 830.1886792
| kN6, rKA5
| Lesser Supraminor Sixth, Tendoretromean Augmented Fifth
| Bd>↓, A#↑\
| -1
| 9
|-
| 111
| 837.7358491
| KKm6, rn6, KA5
| Greater Supraminor Sixth, Retroptolemaic Augmented Fifth
| Bd<\, Bb↑↑, A#↑
| 0
| 8
|-
| 112
| 845.2830189
| n6, SA5, kUA5
| Artoneutral Sixth, Lesser Super-Augmented Fifth
| Bd<, At#<↓
| -1
| 7
|-
| 113
| 852.8301887
| N6, RkUA5
| Tendoneutral Sixth, Greater Super-Augmented Fifth
| Bd>, At#>↓
| 0
| 7
|-
| 114
| 860.3773585
| kkM6, RN6, rUA5
| Lesser Submajor Sixth, Retrodiptolemaic Augmented Fifth
| Bd>/, B↓↓, At#>↓/, A#↑↑
| -1
| 8
|-
| 115
| 867.9245283
| Kn6, UA5
| Greater Submajor Sixth, Ultra-Augmented Fifth
| Bd<↑, At#<
| 1
| 9
|-
| 116
| 875.4716981
| rkM6, KN6
| Narrow Major Sixth
| Bd>↑, B↓\, At#>
| 4
| 9
|-
| 117
| 883.0188679
| kM6
| Ptolemaic Major Sixth
| B↓, Cb
| 7
| 10
|-
| 118
| 890.5660377
| RkM6
| Artomean Major Sixth
| B↓/
| 4
| 10
|-
| 119
| 898.1132075
| rM6
| Tendomean Major Sixth
| B\
| 1
| 9
|-
| 120
| 905.6603774
| M6
| Pythagorean Major Sixth
| B
| -1
| 9
|-
| 121
| 913.2075472
| RM6
| Wide Major Sixth
| B/, Cd<↓
| 0
| 8
|-
| 122
| 920.7547170
| rKM6
| Narrow Supermajor Sixth
| B↑\, Cd>↓
| -1
| 8
|-
| 123
| 928.3018868
| KM6
| Lesser Supermajor Sixth
| B↑, Cd<\, Cb↑↑, Ax
| -1
| 7
|-
| 124
| 935.8490566
| SM6, kUM6
| Greater Supermajor Second, Narrow Inframinor Seventh
| Cd<, Bt<↓, B↑/
| 0
| 7
|-
| 125
| 943.3962264
| um7, RkUM6
| Inframinor Seventh, Wide Supermajor Sixth
| Cd>, Bt>↓
| -1
| 7
|-
| 126
| 950.9433962
| KKM6, kkm7, rUM6, Rum7
| Narrow Ultramajor Sixth, Wide Inframinor Seventh, Semitwelfth
| Bt<\, Cd>/, B↑↑, C↓↓
| 0
| 8
|-
| 127
| 958.4905660
| UM6, rKum7
| Ultramajor Sixth, Narrow Subminor Seventh
| Bt<, Cd<↑
| -1
| 8
|-
| 128
| 966.0377358
| sm7, Kum7
| Lesser Subminor Seventh, Wide Ultramajor Sixth
| Bt>, Cd>↑, C↓\
| 0
| 9
|-
| 129
| 973.5849057
| km7
| Greater Subminor Seventh
| C↓, Bt>/, B#↓↓, Dbb
| -1
| 9
|-
| 130
| 981.1320755
| Rkm7
| Wide Subminor Seventh
| C↓/, Bt<↑
| -1
| 10
|-
| 131
| 988.6792458
| rm7
| Narrow Minor Seventh
| C\, Bt>↑
| -1
| 10
|-
| 132
| 996.2264151
| m7
| Pythagorean Minor Seventh
| C, B#↓
| -2
| 10
|-
| 133
| 1003.7735849
| Rm7
| Artomean Minor Seventh
| C/, B#↓/
| -2
| 10
|-
| 134
| 1011.3207547
| rKm7
| Tendomean Minor Seventh
| C↑\, B#\
| -3
| 10
|-
| 135
| 1018.8679245
| kM2
| Ptolemaic Minor Seventh
| C↑, B#
| -3
| 10
|-
| 136
| 1026.4150943
| RKm7, kn7
| Wide Minor Seventh
| Ct<↓, C↑/, Ddb<, B#/
| -4
| 10
|-
| 137
| 1033.9622642
| kN7, ud8
| Lesser Supraminor Seventh, Infra-Diminished Octave
| Ct>↓, Ddb>, B#↑\
| -5
| 9
|-
| 138
| 1041.5094340
| KKm7, rn7, Rud8
| Greater Supraminor Seventh, Retrodiptolemaic Diminished Octave
| Ct<\, C↑↑, Ddb<↑\, Db↓↓
| -6
| 8
|-
| 139
| 1049.0566038
| n7, rKud8
| Artoneutral Seventh, Lesser Sub-Diminished Octave
| Ct<, Ddb<↑
| -7
| 6
|-
| 140
| 1056.6037736
| N7, sd8
| Tendoneutral Seventh, Greater Sub-Diminished Octave
| Ct>, Ddb>↑
| -8
| 5
|-
| 141
| 1064.1509434
| kkM7, RN7, kd8
| Lesser Submajor Seventh, Diptolemaic Major Seventh, Retroptolemaic Diminished Octave
| Ct>/, C#↓↓, Db↓
| -7
| 6
|-
| 142
| 1071.6981132
| Kn7, Rkd8
| Greater Submajor Seventh, Artoretromean Diminished Octave
| Ct<↑, Db↓/
| -6
| 8
|-
| 143
| 1079.2452830
| rkM7, KN7, rd8
| Narrow Major Seventh, Tendoretromean Diminished Octave
| Ct>↑, C#↓\, Db\
| -5
| 9
|-
| 144
| 1086.7924528
| kM7, d8
| Ptolemaic Major Seventh, Pythagorean Diminished Octave
| Db, C#↓
| -5
| 10
|-
| 145
| 1094.3396226
| RkM7, Rd8
| Artomean Major Seventh, Artomean Diminished Octave
| Db/, C#↓/
| -5
| 10
|-
| 146
| 1101.8867925
| rM7, rKd8
| Tendomean Major Seventh, Tendomean Diminished Octave
| C#\, Db↑\
| -6
| 10
|-
| 147
| 1109.4339622
| M7, Kd8
| Pythagorean Major Seventh, Ptolemaic Diminished Octave
| C#, Db↑
| -6
| 10
|-
| 148
| 1116.9811321
| RM7, kUd8
| Wide Major Seventh, Lesser Super-Diminished Octave
| C#/, Dd<↓
| -7
| 9
|-
| 149
| 1124.5283019
| rKM7, RkUd8
| Narrow Supermajor Seventh, Greater Super-Diminished Octave
| C#↑\, Dd>↓
| -7
| 9
|-
| 150
| 1132.0754717
| km2, RuA1, kkA1
| Lesser Supermajor Seventh, Diptolemaic Diminished Octave
| C#↑, Db↑↑
| -8
| 9
|-
| 151
| 1139.6226415
| SM7, kUM7, Ud8
| Greater Supermajor Seventh, Narrow Infraoctave, Ultra-Diminished Octave
| Dd<, C#↑/
| -8
| 10
|-
| 152
| 1147.1698113
| u8, RkUM7
| Infraoctave, Wide Supermajor Seventh
| Dd>, Ct#>↓
| -9
| 10
|-
| 153
| 1154.7169811
| KKM7, rUM7, Ru8
| Narrow Ultramajor Seventh, Wide Infraoctave
| C#↑↑, Dd>/
| -9
| 10
|-
| 154
| 1162.2641509
| UM7, rKu8
| Ultramajor Seventh, Wide Superprime
| Ct#<, Dd<↑
| -9
| 10
|-
| 155
| 1169.8113208
| s8, Ku8
| Lesser Suboctave, Wide Ultramajor Seventh
| Ct#>, Dd>↑
| -10
| 3
|-
| 156
| 1177.3584906
| k8
| Greater Suboctave
| D↓
| -10
| -3
|-
| 157
| 1184.9056604
| Rk8
| Wide Suboctave
| D↓/
| -10
| -10
|-
| 158
| 1192.4528302
| r8
| Narrow Octave
| D\
| 0
| 0
|-
| 159
| 1200
| P8
| Perfect Octave
| D
| 10
| 10
|}

== Trines ==
159edo has multiple types of trine. Trines are important in the aspects of 159edo music theory derived from Medieval and Neo-Medieval music theory- specifically they're the result of two notes forming an octave, along with a third note for stark contrast, being played simultaneously, which is how 3-limit harmony naturally works. That said, there are such things as dissonant trines, in which the third note is something other than a perfect fourth or perfect fifth away from the doubled root- in fact the third note can be anything from a paraminor fourth to a paramajor fifth relative to the root.

The individual intervals that constitute trines serve as the backbone of not only the triads of harmony, but the tetrachords of melody as well. Both triads and tetrachords will be covered in later installments of this series. For now, it pays to go over which three-note structures can serve as trines as well as their names.

{| class="wikitable"
|+Table of 159edo Trines
|-
! Name
! Notation (from D)
! Steps
! Approximate JI
! Notes
|-
| Otonal Perfect
| D, A, D
| 0, 93, 0
| 2:3:4
| This is the first of two trines that can be considered fully-resolved in Medieval and Neo-Medieval harmony
|-
| Utonal Perfect
| D, G, D
| 0, 66, 0
| 1/(2:3:4)
| This is the second of two trines that can be considered fully-resolved in Medieval and Neo-Medieval harmony
|-
| Otonal Archagall
| D, G\, D
| 0, 65, 0
| 64:85:128
| This trine is the first of two that are often used in the extended harmony of t<IV chords
|-
| Utonal Archagall
| D, A/, D
| 0, 94, 0
| 1/(64:85:128)
| This trine is the second of two that are often used in the extended harmony of t<IV chords
|-
| Bass-Up Marvelous
| D, A\, D
| 0, 92, 0
| 75:112:150
| This trine is the first of two that are formed from stacking identical approximations of the LCJI neutral third
|-
| Treble-Down Marvelous
| D, G/, D
| 0, 67, 0
| 1/(75:112:150)
| This trine is the second of two that are formed from stacking identical approximations of the LCJI neutral third
|-
| Narrow Supernaiadic
| D, G↓\, D
| 0, 62, 0
| 16:21:32
| This dissonant trine is common in essentially tempered chords
|-
| Wide Subcocytic
| D, A↑/, D
| 0, 97, 0
| 1/(16:21:32)
| This dissonant trine is common in essentially tempered chords
|-
| Subcocytic
| D, A↑, D
| 0, 96, 0
| 160:243:320
| This dissonant trine is common in essentially tempered chords
|-
| Supernaiadic
| D, G↓, D
| 0, 63, 0
| 1/(160:243:320)
| This dissonant trine is common in essentially tempered chords
|-
| Wide Supernaiadic
| D, G↓/, D
| 0, 64, 0
| 25:33:50
| This dissonant trine is on the outer edge of the diatonic range and is common in essentially tempered chords
|-
| Narrow Subcocytic
| D, A↑\, D
| 0, 95, 0
| 1/(25:33:50)
| This dissonant trine is on the outer edge of the diatonic range and is common in essentially tempered chords
|-
| Wide Naiadic
| D, Gd<↑, D
| 0, 61, 0
| 135:176:270
| This dissonant trine is among the more consistently complex
|-
| Narrow Cocytic
| D, At>↓, D
| 0, 98, 0
| 1/(135:176:270)
| This dissonant trine is among the more consistently complex
|-
| Naiadic
| D, Gd>/, D
| 0, 60, 0
| 10:13:20
| This dissonant trine is relatively simple and thus expected to be rather common
|-
| Cocytic
| D, At<\, D
| 0, 99, 0
| 1/(10:13:20)
| This dissonant trine is relatively simple and thus expected to be rather common
|-
| Wide Cocytic
| D, At<, D
| 0, 100, 0
| 11:17:22
| This essentially tempered trine is very likely to be used as a basis for cocytic triads
|-
| Narrow Niadic
| D, Gd>, D
| 0, 59, 0
| 1/(11:17:22)
| This essentially tempered trine is very likely to be used as a partial basis for suspended chords
|-
| Narrow Superdusthumic
| D, Ad<↑, D
| 0, 89, 0
| 128:189:256
| This dissonant trine is common in essentially tempered chords
|-
| Wide Subagallic
| D, Gt>↓, D
| 0, 70, 0
| 1/(128:189:256)
| This dissonant trine is common in essentially tempered chords
|-
| Subagallic
| D, G↑, D
| 0, 69, 0
| 20:27:40
| This dissonant trine is very likely to show up in non-meantone diatonic contexts
|-
| Superdusthumic
| D, A↓, D
| 0, 90, 0
| 1/(20:27:40)
| This dissonant trine is very likely to show up in non-meantone diatonic contexts
|-
| Narrow Subagallic
| D, G↑\, D
| 0, 68, 0
| 90:121:180
| This dissonant trine is on the outer edge of the diatonic range
|-
| Wide Superdusthumic
| D, A↓/, D
| 0, 91, 0
| 1/(90:121:180)
| This dissonant trine is on the outer edge of the diatonic range
|-
| Wide Agallic
| D, Gt<, D
| 0, 73, 0
| 8:11:16
| This ambisonant trine is very likely to be used as a partial basis for suspended chords
|-
| Narrow Dusthumic
| D, Ad>, D
| 0, 86, 0
| 1/(8:11:16)
| This ambisonant trine is very likely to be used as a basis for dusthumic triads
|-
| Dusthumic
| D, Ad<\, D
| 0, 87, 0
| 128:187:256
| This dissonant trine is common in essentially tempered chords
|-
| Agallic
| D, Gt<\, D
| 0, 72, 0
| 1/(128:187:256)
| This dissonant trine is common in essentially tempered chords
|-
| Narrow Agallic
| D, Gt>↓, D
| 0, 71, 0
| 11:15:22
| This trine is very likely to be used as a partial basis for suspended chords
|-
| Wide Dusthumic
| D, Ad<↑, D
| 0, 88, 0
| 1/(11:15:22)
| This trine is very likely to be used as a basis for dusthumic triads
|-
| Wide Subdusthumic
| D, Ad<, D
| 0, 85, 0
| 56:81:112
| This essentially tempered trine is likely to be used as a basis for subdusthumic triads
|-
| Narrow Superagallic
| D, Gt>, D
| 0, 74, 0
| 1/(56:81:112)
| This essentially tempered trine is likely to be used as a partial basis for suspended chords
|-
| Subdusthumic
| D, Ab↑↑, D
| 0, 84, 0
| 9:13:18
| This essentially tempered trine is very likely to be used as a basis for subdusthumic triads
|-
| Superagallic
| D, G#↓↓, D
| 0, 75, 0
| 1/(9:13:18)
| This essentially tempered trine is very likely to be used as a partial basis for suspended chords
|-
| Wide Superagallic
| D, Gt<↑, D
| 0, 76, 0
| 256:357:512
| This essentially tempered trine is very likely to be used as a partial basis for suspended chords
|-
| Narrow Subdusthumic
| D, Ad>↓, D
| 0, 83, 0
| 1/(256:357:512)
| This essentially tempered trine is very likely to be used as a basis for subdusthumic triads
|-
| Narrow Hyperquartal
| D, Gt>↑, D
| 0, 77, 0
| 5:7:10
| This ambisonant trine is very common as a basis for diminished chords, and is very likely to be used as a partial basis for suspended chords
|-
| Wide Hypoquintal
| D, Ad<↓, D
| 0, 82, 0
| 1/(5:7:10)
| This ambisonant trine is very common as a basis for diminished chords, and is very likely to be used as a partial basis for suspended chords
|-
| Hyperquartal
| D, G#↓, D
| 0, 78, 0
| 32:45:64
| This trine is very likely to be used as a partial basis for suspended chords
|-
| Hypoquintal
| D, Ab↑, D
| 0, 81, 0
| 1/(32:45:64)
| This trine is very common as a basis for diminished chords
|-
| Narrow Hypoquintal
| D, Ab↑\, D
| 0, 80, 0
| 12:17:24
| This trine is very common as a basis for diminished chords
|-
| Wide Hyperquartal
| D, G#↓/, D
| 0, 79, 0
| 1/(12:17:24)
| This trine is very likely to be used as a partial basis for suspended chords
|}

User:Aura/On 159edo Music Theory (Part 1)

2026-03-31T19:55:14Z

Aura: Included a temporary definition of "trine", I need to check with Margo Schulter in order to acquire a proper definition

Of all the multiples of [[53edo]], [[159edo]] is the lowest multiple that is noteworthy for being accurate in the 2.3.5.11.17 subgroup while having structural compromises in the 7.13.19.23.29 subgroup. Despite the number of pitches in this tuning system making it perhaps best fit for digital instruments of various kinds in actual performance, it is nevertheless also useful as an interval classification scheme.

== Intervals and Notation ==
159edo contains all the intervals of 53edo and can be thought of as having three fields of 53edo each separated by a third of 53edo's step. However, as some of the interpretations differ due 159edo having different mappings for certain primes, those differences show up in how harmonies are constructed. However, there's more.

Of all the intervals in 159edo, 5\159 is the first interval to be larger than the '''fission boundary''', which is where going back and forth between notes on either end of a given interval no longer sounds like a simple vibrato, but more like a dirty trill of sorts. The fission boundary further serves as the line separating melodic notes that can only be simple ornaments or quick passing tones from main melodic intervals, and since 5\159 is larger than this boundary it is the smallest interval that can serve as a main melodic interval.

The next landmark interval is 8\159, as this is the first interval to be larger than the '''gradient threshold''', which is where going back and forth between notes on either end of a given interval no longer sounds like a dirty trill, but rather a clean trill. The gradient threshold doubles as the point beyond which microtonal intervals can begin to serve as proper leading-tones.

Finally, 33\159 is the first interval to be larger than the '''trill threshold''', which is where going back and forth between notes on either end of a given interval no longer sounds like any kind of trill, and instead sounds like an arpeggio fragment. The trill threshold doubles as the boundary between intervals that are classified as steps, and those that are classified as leaps. As a consequence of this, the trill threshold marks the boundary where intervals cease to cause crowding in chords.

As if all that weren't enough, 159edo has its own variation on the [[dinner party rules]]— represented here by the Harmonic Compatibility Rating and Melodic Compatibility Rating columns in the following chart, where 10 is a full-blown friend relative to the root and −10 if a full-blown enemy relative to the root. Note that the Harmonic Compatibility and Melodic Compatibility ratings are based on octave-equivalence, and that some of the ratings are still speculative.

{| class="wikitable"
|+159edo Interval Names and Compatibility Ratings
! rowspan="2" | Step
! rowspan="2" | Cents
! colspan="3" | Interval and Note names
! colspan="2" | Compatibility rating
|-
! SKULO-based interval names
! Pythagorean-commatic-based interval names
! SRS notation
! Harmonic
! Melodic
|-
| 0
| 0
| P1
| Perfect Unison
| D
| 10
| 10
|-
| 1
| 7.5471698
| R1
| Wide Unison
| D/
| 0
| 0
|-
| 2
| 15.0943396
| rK1
| Narrow Superunison
| D↑\
| -10
| -10
|-
| 3
| 22.6415094
| K1
| Lesser Superunison
| D↑
| -10
| -3
|-
| 4
| 30.1886792
| S1, kU1
| Greater Superunison, Narrow Inframinor Second
| Edb<, Dt<↓
| -10
| 3
|-
| 5
| 37.7358491
| um2, RkU1
| Inframinor Second, Wide Superunison
| Edb>, Dt>↓
| -9
| 10
|-
| 6
| 45.2830189
| kkm2, Rum2, rU1
| Wide Inframinor Second, Narrow Ultraunison
| Eb↓↓, Dt<\
| -9
| 10
|-
| 7
| 52.8301887
| U1, rKum2
| Ultraunison, Narrow Subminor Second
| Dt<, Edb<↑
| -9
| 10
|-
| 8
| 60.3773585
| sm2, Kum2, uA1
| Lesser Subminor Second, Wide Ultraunison, Infra-Augmented Unison
| Dt>, Eb↓\
| -8
| 10
|-
| 9
| 67.9245283
| km2, RuA1, kkA1
| Greater Subminor Second, Diptolemaic Augmented Unison
| Eb↓, D#↓↓
| -8
| 9
|-
| 10
| 75.4716981
| Rkm2, rKuA1
| Wide Subminor Second, Lesser Sub-Augmented Unison
| Eb↓/, Dt<↑
| -7
| 9
|-
| 11
| 83.0188679
| rm2, KuA1
| Narrow Minor Second, Greater Sub-Augmented Unison
| Eb\, Dt>↑
| -7
| 9
|-
| 12
| 90.5660377
| m2, kA1
| Pythagorean Minor Second, Ptolemaic Augmented Unison
| Eb, D#↓
| -6
| 10
|-
| 13
| 98.1132075
| Rm2, RkA1
| Artomean Minor Second, Artomean Augmented Unison
| Eb/, D#↓/
| -6
| 10
|-
| 14
| 105.6603774
| rKm2, rA1
| Tendomean Minor Second, Tendomean Augmented Unison
| D#\, Eb↑\
| -5
| 10
|-
| 15
| 113.2075472
| Km2, A1
| Ptolemaic Minor Second, Pythagorean Augmented Unison
| D#, Eb↑
| -5
| 10
|-
| 16
| 120.7547170
| RKm2, kn2, RA1
| Wide Minor Second, Artoretromean Augmented Unison
| Ed<↓, Eb↑/, D#/
| -5
| 9
|-
| 17
| 128.3018868
| kN2, rKA1
| Lesser Supraminor Second, Tendoretromean Augmented Unison
| Ed>↓, D#↑\
| -6
| 8
|-
| 18
| 135.8490566
| KKm2, rn2, KA1
| Greater Supraminor Second, Diptolemaic Limma, Retroptolemaic Augmented Unison
| Ed<\, Eb↑↑, D#↑
| -7
| 6
|-
| 19
| 143.3962264
| n2, SA1
| Artoneutral Second, Lesser Super-Augmented Unison
| Ed<, Dt#<↓
| -8
| 5
|-
| 20
| 150.9433962
| N2, RkUA1
| Tendoneutral Second, Greater Super-Augmented Unison
| Ed>, Dt#>↓
| -7
| 6
|-
| 21
| 158.4905660
| kkM2, RN2, rUA1
| Lesser Submajor Second, Retrodiptolemaic Augmented Unison
| Ed>/, E↓↓, Dt#>↓/, D#↑↑
| -6
| 8
|-
| 22
| 166.0377358
| Kn2, UA1
| Greater Submajor Second, Ultra-Augmented Unison
| Ed<↑, Dt#<, Fb↓/
| -5
| 9
|-
| 23
| 173.5849057
| rkM2, KN2
| Narrow Major Second
| Ed>↑, E↓\, Dt#>, Fb\
| -4
| 10
|-
| 24
| 181.1320755
| kM2
| Ptolemaic Major Second
| E↓, Fb
| -3
| 10
|-
| 25
| 188.6792458
| RkM2
| Artomean Major Second
| E↓/, Fb/
| -3
| 10
|-
| 26
| 196.2264151
| rM2
| Tendomean Major Second
| E\, Fb↑\
| -2
| 10
|-
| 27
| 203.7735849
| M2
| Pythagorean Major Second
| E, Fb↑
| -2
| 10
|-
| 28
| 211.3207547
| RM2
| Wide Major Second
| E/, Fd<↓
| -1
| 10
|-
| 29
| 218.8679245
| rKM2
| Narrow Supermajor Second
| E↑\, Fd>↓
| -1
| 10
|-
| 30
| 226.4150943
| KM2
| Lesser Supermajor Second
| E↑, Fd<\, Fb↑↑, Dx
| -1
| 9
|-
| 31
| 233.9622642
| SM2, kUM2
| Greater Supermajor Second, Narrow Inframinor Third
| Fd<, Et<↓, E↑/
| 0
| 9
|-
| 32
| 241.5094340
| um3, RkUM2
| Inframinor Third, Wide Supermajor Second
| Fd>, Et>↓
| -1
| 8
|-
| 33
| 249.0566038
| kkm3, KKM2, Rum3, rUM2
| Wide Inframinor Third, Narrow Ultramajor Second, Semifourth
| Fd>/, Et<\, F↓↓, E↑↑
| 0
| 8
|-
| 34
| 256.6037736
| UM2, rKum3
| Ultramajor Second, Narrow Subminor Third
| Et<, Fd<↑
| -1
| 7
|-
| 35
| 264.1509434
| sm3, Kum3
| Lesser Subminor Third, Wide Ultramajor Second
| Et>, Fd>↑, F↓\
| 0
| 7
|-
| 36
| 271.6981132
| km3
| Greater Subminor Third
| F↓, Et>/, E#↓↓, Gbb
| -1
| 7
|-
| 37
| 279.2452830
| Rkm3
| Wide Subminor Third
| F↓/, Et<↑
| -1
| 8
|-
| 38
| 286.7924528
| rm3
| Narrow Minor Third
| F\, Et>↑
| 0
| 8
|-
| 39
| 294.3396226
| m3
| Pythagorean Minor Third
| F
| -1
| 9
|-
| 40
| 301.8867925
| Rm3
| Artomean Minor Third
| F/
| 1
| 9
|-
| 41
| 309.4339622
| rKm3
| Tendomean Minor Third
| F↑\
| 4
| 10
|-
| 42
| 316.9811321
| Km3
| Ptolemaic Minor Third
| F↑, E#
| 7
| 10
|-
| 43
| 324.5283019
| RKm3, kn3
| Wide Minor Third
| Ft<↓, F↑/, Gdb<
| 4
| 9
|-
| 44
| 332.0754717
| kN3, ud4
| Lesser Supraminor Third, Infra-Diminished Fourth
| Ft>↓, Gdb>
| 1
| 9
|-
| 45
| 339.6226415
| KKm3, rn3, Rud4
| Greater Supraminor Third, Retrodiptolemaic Diminished Fourth
| Ft<\, F↑↑, Gdb<↑\, Gb↓↓
| -1
| 8
|-
| 46
| 347.1698113
| n3, rKud4
| Artoneutral Third, Lesser Sub-Diminished Fourth
| Ft<, Gdb<↑
| 0
| 7
|-
| 47
| 354.7169811
| N3, sd4, Kud4
| Tendoneutral Third, Greater Sub-Diminished Fourth
| Ft>, Gdb>↑
| -1
| 7
|-
| 48
| 362.2641509
| kkM3, RN3, kd4
| Lesser Submajor Third, Retroptolemaic Diminished Fourth
| Ft>/, F#↓↓, Gb↓
| 0
| 8
|-
| 49
| 369.8113208
| Kn3, Rkd4
| Greater Submajor Third, Artoretromean Diminished Fourth
| Ft<↑, Gb↓/
| -1
| 9
|-
| 50
| 377.3584906
| rkM3, KN3, rd4
| Narrow Major Third, Tendoretromean Diminished Fourth
| Ft>↑, F#↓\, Gb\
| 3
| 9
|-
| 51
| 384.9056604
| kM3, d4
| Ptolemaic Major Third, Pythagorean Diminished Fourth
| Gb, F#↓
| 8
| 10
|-
| 52
| 392.4528302
| RkM3, Rd4
| Artomean Major Third, Artomean Diminished Fourth
| Gb/, F#↓/
| 4
| 10
|-
| 53
| 400
| rM3, rKd4
| Tendomean Major Third, Tendomean Diminished Fourth
| F#\, Gb↑\
| 1
| 9
|-
| 54
| 407.5471698
| M3, Kd4
| Pythagorean Major Third, Ptolemaic Diminished Fourth
| F#, Gb↑
| -1
| 9
|-
| 55
| 415.0943396
| RM3, kUd4
| Wide Major Third, Lesser Super-Diminished Fourth
| F#/, Gd<↓, Gb↑/
| 0
| 8
|-
| 56
| 422.6415094
| rKM3, RkUd4
| Narrow Supermajor Third, Greater Super-Diminished Fourth
| F#↑\, Gd>↓
| -1
| 7
|-
| 57
| 430.1886792
| KM3, rUd4, KKd4
| Lesser Supermajor Third, Diptolemaic Diminished Fourth
| F#↑, Gd<\, Gb↑↑
| -1
| 6
|-
| 58
| 437.7358491
| SM3, kUM3, rm4, Ud4
| Greater Supermajor Third, Ultra-Diminished Fourth
| Gd<, F#↑/
| 0
| 5
|-
| 59
| 445.2830189
| m4, RkUM3
| Paraminor Fourth, Wide Supermajor Third
| Gd>, Ft#>↓
| -1
| 3
|-
| 60
| 452.8301887
| Rm4, KKM3, rUM3
| Wide Paraminor Fourth, Narrow Ultramajor Third
| Gd>/, F#↑↑, G↓↓
| -2
| 1
|-
| 61
| 460.3773585
| UM3, rKm4
| Ultramajor Third, Narrow Grave Fourth
| Gd<↑, Ft#<
| -4
| -2
|-
| 62
| 467.9245283
| s4, Km4
| Lesser Grave Fourth, Wide Ultramajor Third
| Gd>↑, G↓\
| -7
| -4
|-
| 63
| 475.4716981
| k4
| Greater Grave Fourth
| G↓, Abb
| -6
| -5
|-
| 64
| 483.0188679
| Rk4
| Wide Grave Fourth
| G↓/
| -4
| 0
|-
| 65
| 490.5660377
| r4
| Narrow Fourth
| G\
| 1
| 5
|-
| 66
| 498.1132075
| P4
| Perfect Fourth
| G
| 9
| 10
|-
| 67
| 505.6603774
| R4
| Wide Fourth
| G/
| 1
| 8
|-
| 68
| 513.2075472
| rK4
| Narrow Acute Fourth
| G↑\
| -3
| 6
|-
| 69
| 520.7547170
| K4
| Lesser Acute Fourth
| G↑
| -5
| 5
|-
| 70
| 528.3018868
| S4, kM4
| Greater Acute Fourth
| Gt<↓, G↑/, Adb<
| -3
| 5
|-
| 71
| 535.8490566
| RkM4, ud5
| Wide Acute Fourth, Infra-Diminished Fifth
| Gt>↓, Adb>
| -2
| 5
|-
| 72
| 543.3962264
| rM4, Rud5
| Narrow Paramajor Fourth, Retrodiptolemaic Diminished Fifth
| Gt<\, G↑↑, Ab↓↓
| -1
| 6
|-
| 73
| 550.9433962
| M4, rKud5
| Paramajor Fourth, Lesser Sub-Diminished Fifth
| Gt<, Adb<↑
| 0
| 7
|-
| 74
| 558.4905660
| RM4, uA4, Kud5
| Infra-Augmented Fourth, Greater Sub-Diminished Fifth
| Gt>, Adb>↑
| -2
| 5
|-
| 75
| 566.0377358
| kkA4, RuA4, kd5
| Diptolemaic Augmented Fourth, Retroptolemaic Diminished Fifth
| Gt>/, G#↓↓, Ab↓
| -3
| 4
|-
| 76
| 573.5849057
| rKuA4, Rkd5
| Lesser Sub-Augmented Fourth, Artoretromean Diminished Fifth
| Gt<↑, Ab↓/
| -2
| 4
|-
| 77
| 581.1320755
| KuA4, rd5
| Greater Sub-Augmented Fourth, Tendoretromean Diminished Fifth
| Gt>↑, Ab\
| 0
| 5
|-
| 78
| 588.6792458
| kA4, d5
| Ptolemaic Augmented Fourth, Pythagorean Diminished Fifth
| Ab, G#↓
| -5
| 6
|-
| 79
| 596.2264151
| RkA4, Rd5
| Artomean Augmented Fourth, Artomean Diminished Fifth
| G#↓/, Ab/
| -9
| 7
|-
| 80
| 603.7735849
| rKd5, rA4
| Tendomean Diminished Fifth, Tendomean Augmented Fourth
| Ab↑\, G#\
| -9
| 7
|-
| 81
| 611.3207547
| Kd5, A4
| Ptolemaic Diminished Fifth, Pythagorean Augmented Fourth
| Ab↑, G#
| -5
| 6
|-
| 82
| 618.8679245
| kUd5, RA4
| Lesser Super-Diminished Fifth, Artoretromean Augmented Fourth
| Ad<↓, G#/
| 0
| 5
|-
| 83
| 626.4150943
| RkUd5, rKA4
| Greater Super-Diminished Fifth, Tendoretromean Augmented Fourth
| Ad>↓, G#↑\
| -2
| 4
|-
| 84
| 633.9622642
| KKd5, rUDd5, KA4
| Diptolemaic Diminished Fifth, Retroptolemaic Augmented Fourth
| Ad<\, Ab↑↑, G#↑
| -3
| 4
|-
| 85
| 641.5094340
| rm5, Ud5, kUA4
| Ultra-Diminished Fifth, Lesser Super-Augmented Fourth
| Ad<, Gt#<↓
| -2
| 5
|-
| 86
| 649.0566038
| m5, RkUA4
| Paraminor Fifth, Greater Super-Augmented Fourth
| Ad>, Gt#>↓
| 0
| 7
|-
| 87
| 656.6037736
| Rm5, rUA4
| Wide Paraminor Fifth, Retrodiptolemaic Augmented Fourth
| Ad>/, G#↑, Ab↑↑
| -1
| 6
|-
| 88
| 664.1509434
| rKm5, UA4
| Narrow Grave Fifth, Ultra-Augmented Fourth
| Ad<↑, Gt#<
| -2
| 5
|-
| 89
| 671.6981132
| s5, Km5
| Lesser Grave Fifth
| Ad>↑, A↓\, Gt#>
| -3
| 5
|-
| 90
| 679.2452830
| k5
| Greater Grave Fifth
| A↓
| -5
| 5
|-
| 91
| 686.7924528
| Rk5
| Wide Grave Fifth
| A↓/
| -3
| 6
|-
| 92
| 694.3396226
| r5
| Narrow Fifth
| A\
| 1
| 8
|-
| 93
| 701.8867925
| P5
| Perfect Fifth
| A
| 9
| 10
|-
| 94
| 709.4339622
| R5
| Wide Fifth
| A/
| 1
| 5
|-
| 95
| 716.9811321
| rK5
| Narrow Acute Fifth
| A↑\
| -4
| 0
|-
| 96
| 724.5283019
| K5
| Lesser Acute Fifth
| A↑, Gx
| -6
| -5
|-
| 97
| 732.0754717
| S5, kM5
| Greater Acute Fifth, Narrow Inframinor Sixth
| At<↓, A↑/
| -7
| -4
|-
| 98
| 739.6226415
| um6, RkM5
| Inframinor Sixth, Wide Acute Fifth
| At>↓, Bdb>
| -4
| -2
|-
| 99
| 747.1698113
| Rm4, KKM3, rUM3
| Narrow Paramajor Fifth, Wide Inframinor Sixth
| At<\, Bb↓↓, A↑↑
| -2
| 1
|-
| 100
| 754.7169811
| M5, rKum6
| Paramajor Fifth, Narrow Subminor Sixth
| At<, Bdb<↑
| -1
| 3
|-
| 101
| 762.2641509
| sm6, Kum6, RM5, uA5
| Lesser Subminor Sixth, Infra-Augmented Fifth
| At>, Bb↓\
| 0
| 5
|-
| 102
| 769.8113208
| km6, RuA5, kkA5
| Greater Subminor Sixth, Diptolemaic Augmented Fifth
| Bb↓, At>/, A#↓↓
| -1
| 6
|-
| 103
| 777.3584906
| Rkm6, rKuA5
| Wide Subminor Sixth, Lesser Sub-Augmented Fifth
| Bb↓/, At<↑
| -1
| 7
|-
| 104
| 784.9056604
| rm6, KuA5
| Narrow Minor Sixth, Greater Sub-Augmented Fifth
| Bb\, At>↑, A#↓\
| 0
| 8
|-
| 105
| 792.4528302
| m6, kA5
| Pythagorean Minor Sixth, Ptolemaic Augmented Fifth
| Bb, A#↓
| -1
| 9
|-
| 106
| 800
| Rm6, RkA5
| Artomean Minor Sixth, Artomean Augmented Fifth
| Bb/, A#↓/
| 1
| 9
|-
| 107
| 807.5471698
| rKm6, rA5
| Tendomean Minor Sixth, Tendomean Augmented Fifth
| A#\, Bb↑\
| 4
| 10
|-
| 108
| 815.0943396
| Km6, A5
| Ptolemaic Minor Sixth, Pythagorean Augmented Fifth
| A#, Bb↑
| 8
| 10
|-
| 109
| 822.6415094
| RKm6, kn6, RA5
|Wide Minor Sixth, Artoretromean Augmented Fifth
| Bd<↓, Bb↑/, A#/
| 3
| 9
|-
| 110
| 830.1886792
| kN6, rKA5
| Lesser Supraminor Sixth, Tendoretromean Augmented Fifth
| Bd>↓, A#↑\
| -1
| 9
|-
| 111
| 837.7358491
| KKm6, rn6, KA5
| Greater Supraminor Sixth, Retroptolemaic Augmented Fifth
| Bd<\, Bb↑↑, A#↑
| 0
| 8
|-
| 112
| 845.2830189
| n6, SA5, kUA5
| Artoneutral Sixth, Lesser Super-Augmented Fifth
| Bd<, At#<↓
| -1
| 7
|-
| 113
| 852.8301887
| N6, RkUA5
| Tendoneutral Sixth, Greater Super-Augmented Fifth
| Bd>, At#>↓
| 0
| 7
|-
| 114
| 860.3773585
| kkM6, RN6, rUA5
| Lesser Submajor Sixth, Retrodiptolemaic Augmented Fifth
| Bd>/, B↓↓, At#>↓/, A#↑↑
| -1
| 8
|-
| 115
| 867.9245283
| Kn6, UA5
| Greater Submajor Sixth, Ultra-Augmented Fifth
| Bd<↑, At#<
| 1
| 9
|-
| 116
| 875.4716981
| rkM6, KN6
| Narrow Major Sixth
| Bd>↑, B↓\, At#>
| 4
| 9
|-
| 117
| 883.0188679
| kM6
| Ptolemaic Major Sixth
| B↓, Cb
| 7
| 10
|-
| 118
| 890.5660377
| RkM6
| Artomean Major Sixth
| B↓/
| 4
| 10
|-
| 119
| 898.1132075
| rM6
| Tendomean Major Sixth
| B\
| 1
| 9
|-
| 120
| 905.6603774
| M6
| Pythagorean Major Sixth
| B
| -1
| 9
|-
| 121
| 913.2075472
| RM6
| Wide Major Sixth
| B/, Cd<↓
| 0
| 8
|-
| 122
| 920.7547170
| rKM6
| Narrow Supermajor Sixth
| B↑\, Cd>↓
| -1
| 8
|-
| 123
| 928.3018868
| KM6
| Lesser Supermajor Sixth
| B↑, Cd<\, Cb↑↑, Ax
| -1
| 7
|-
| 124
| 935.8490566
| SM6, kUM6
| Greater Supermajor Second, Narrow Inframinor Seventh
| Cd<, Bt<↓, B↑/
| 0
| 7
|-
| 125
| 943.3962264
| um7, RkUM6
| Inframinor Seventh, Wide Supermajor Sixth
| Cd>, Bt>↓
| -1
| 7
|-
| 126
| 950.9433962
| KKM6, kkm7, rUM6, Rum7
| Narrow Ultramajor Sixth, Wide Inframinor Seventh, Semitwelfth
| Bt<\, Cd>/, B↑↑, C↓↓
| 0
| 8
|-
| 127
| 958.4905660
| UM6, rKum7
| Ultramajor Sixth, Narrow Subminor Seventh
| Bt<, Cd<↑
| -1
| 8
|-
| 128
| 966.0377358
| sm7, Kum7
| Lesser Subminor Seventh, Wide Ultramajor Sixth
| Bt>, Cd>↑, C↓\
| 0
| 9
|-
| 129
| 973.5849057
| km7
| Greater Subminor Seventh
| C↓, Bt>/, B#↓↓, Dbb
| -1
| 9
|-
| 130
| 981.1320755
| Rkm7
| Wide Subminor Seventh
| C↓/, Bt<↑
| -1
| 10
|-
| 131
| 988.6792458
| rm7
| Narrow Minor Seventh
| C\, Bt>↑
| -1
| 10
|-
| 132
| 996.2264151
| m7
| Pythagorean Minor Seventh
| C, B#↓
| -2
| 10
|-
| 133
| 1003.7735849
| Rm7
| Artomean Minor Seventh
| C/, B#↓/
| -2
| 10
|-
| 134
| 1011.3207547
| rKm7
| Tendomean Minor Seventh
| C↑\, B#\
| -3
| 10
|-
| 135
| 1018.8679245
| kM2
| Ptolemaic Minor Seventh
| C↑, B#
| -3
| 10
|-
| 136
| 1026.4150943
| RKm7, kn7
| Wide Minor Seventh
| Ct<↓, C↑/, Ddb<, B#/
| -4
| 10
|-
| 137
| 1033.9622642
| kN7, ud8
| Lesser Supraminor Seventh, Infra-Diminished Octave
| Ct>↓, Ddb>, B#↑\
| -5
| 9
|-
| 138
| 1041.5094340
| KKm7, rn7, Rud8
| Greater Supraminor Seventh, Retrodiptolemaic Diminished Octave
| Ct<\, C↑↑, Ddb<↑\, Db↓↓
| -6
| 8
|-
| 139
| 1049.0566038
| n7, rKud8
| Artoneutral Seventh, Lesser Sub-Diminished Octave
| Ct<, Ddb<↑
| -7
| 6
|-
| 140
| 1056.6037736
| N7, sd8
| Tendoneutral Seventh, Greater Sub-Diminished Octave
| Ct>, Ddb>↑
| -8
| 5
|-
| 141
| 1064.1509434
| kkM7, RN7, kd8
| Lesser Submajor Seventh, Diptolemaic Major Seventh, Retroptolemaic Diminished Octave
| Ct>/, C#↓↓, Db↓
| -7
| 6
|-
| 142
| 1071.6981132
| Kn7, Rkd8
| Greater Submajor Seventh, Artoretromean Diminished Octave
| Ct<↑, Db↓/
| -6
| 8
|-
| 143
| 1079.2452830
| rkM7, KN7, rd8
| Narrow Major Seventh, Tendoretromean Diminished Octave
| Ct>↑, C#↓\, Db\
| -5
| 9
|-
| 144
| 1086.7924528
| kM7, d8
| Ptolemaic Major Seventh, Pythagorean Diminished Octave
| Db, C#↓
| -5
| 10
|-
| 145
| 1094.3396226
| RkM7, Rd8
| Artomean Major Seventh, Artomean Diminished Octave
| Db/, C#↓/
| -5
| 10
|-
| 146
| 1101.8867925
| rM7, rKd8
| Tendomean Major Seventh, Tendomean Diminished Octave
| C#\, Db↑\
| -6
| 10
|-
| 147
| 1109.4339622
| M7, Kd8
| Pythagorean Major Seventh, Ptolemaic Diminished Octave
| C#, Db↑
| -6
| 10
|-
| 148
| 1116.9811321
| RM7, kUd8
| Wide Major Seventh, Lesser Super-Diminished Octave
| C#/, Dd<↓
| -7
| 9
|-
| 149
| 1124.5283019
| rKM7, RkUd8
| Narrow Supermajor Seventh, Greater Super-Diminished Octave
| C#↑\, Dd>↓
| -7
| 9
|-
| 150
| 1132.0754717
| km2, RuA1, kkA1
| Lesser Supermajor Seventh, Diptolemaic Diminished Octave
| C#↑, Db↑↑
| -8
| 9
|-
| 151
| 1139.6226415
| SM7, kUM7, Ud8
| Greater Supermajor Seventh, Narrow Infraoctave, Ultra-Diminished Octave
| Dd<, C#↑/
| -8
| 10
|-
| 152
| 1147.1698113
| u8, RkUM7
| Infraoctave, Wide Supermajor Seventh
| Dd>, Ct#>↓
| -9
| 10
|-
| 153
| 1154.7169811
| KKM7, rUM7, Ru8
| Narrow Ultramajor Seventh, Wide Infraoctave
| C#↑↑, Dd>/
| -9
| 10
|-
| 154
| 1162.2641509
| UM7, rKu8
| Ultramajor Seventh, Wide Superprime
| Ct#<, Dd<↑
| -9
| 10
|-
| 155
| 1169.8113208
| s8, Ku8
| Lesser Suboctave, Wide Ultramajor Seventh
| Ct#>, Dd>↑
| -10
| 3
|-
| 156
| 1177.3584906
| k8
| Greater Suboctave
| D↓
| -10
| -3
|-
| 157
| 1184.9056604
| Rk8
| Wide Suboctave
| D↓/
| -10
| -10
|-
| 158
| 1192.4528302
| r8
| Narrow Octave
| D\
| 0
| 0
|-
| 159
| 1200
| P8
| Perfect Octave
| D
| 10
| 10
|}

== Trines ==
159edo has multiple types of trine. Trines are important in the aspects of 159edo music theory derived from Medieval and Neo-Medieval music theory- specifically they're the result of two notes forming an octave, along with a third note for stark contrast, being played simultaneously, which is how 3-limit harmony naturally works. That said, there are such things as dissonant trines, in which the third note is something other than a perfect fourth or perfect fifth away from the doubled root.

The individual intervals that constitute trines serve as the backbone of not only the triads of harmony, but the tetrachords of melody as well. Both triads and tetrachords will be covered in later installments of this series. For now, it pays to go over which three-note structures can serve as trines as well as their names.

{| class="wikitable"
|+Table of 159edo Trines
|-
! Name
! Notation (from D)
! Steps
! Approximate JI
! Notes
|-
| Otonal Perfect
| D, A, D
| 0, 93, 0
| 2:3:4
| This is the first of two trines that can be considered fully-resolved in Medieval and Neo-Medieval harmony
|-
| Utonal Perfect
| D, G, D
| 0, 66, 0
| 1/(2:3:4)
| This is the second of two trines that can be considered fully-resolved in Medieval and Neo-Medieval harmony
|-
| Otonal Archagall
| D, G\, D
| 0, 65, 0
| 64:85:128
| This trine is the first of two that are often used in the extended harmony of t<IV chords
|-
| Utonal Archagall
| D, A/, D
| 0, 94, 0
| 1/(64:85:128)
| This trine is the second of two that are often used in the extended harmony of t<IV chords
|-
| Bass-Up Marvelous
| D, A\, D
| 0, 92, 0
| 75:112:150
| This trine is the first of two that are formed from stacking identical approximations of the LCJI neutral third
|-
| Treble-Down Marvelous
| D, G/, D
| 0, 67, 0
| 1/(75:112:150)
| This trine is the second of two that are formed from stacking identical approximations of the LCJI neutral third
|-
| Narrow Supernaiadic
| D, G↓\, D
| 0, 62, 0
| 16:21:32
| This dissonant trine is common in essentially tempered chords
|-
| Wide Subcocytic
| D, A↑/, D
| 0, 97, 0
| 1/(16:21:32)
| This dissonant trine is common in essentially tempered chords
|-
| Subcocytic
| D, A↑, D
| 0, 96, 0
| 160:243:320
| This dissonant trine is common in essentially tempered chords
|-
| Supernaiadic
| D, G↓, D
| 0, 63, 0
| 1/(160:243:320)
| This dissonant trine is common in essentially tempered chords
|-
| Wide Supernaiadic
| D, G↓/, D
| 0, 64, 0
| 25:33:50
| This dissonant trine is on the outer edge of the diatonic range and is common in essentially tempered chords
|-
| Narrow Subcocytic
| D, A↑\, D
| 0, 95, 0
| 1/(25:33:50)
| This dissonant trine is on the outer edge of the diatonic range and is common in essentially tempered chords
|-
| Wide Naiadic
| D, Gd<↑, D
| 0, 61, 0
| 135:176:270
| This dissonant trine is among the more consistently complex
|-
| Narrow Cocytic
| D, At>↓, D
| 0, 98, 0
| 1/(135:176:270)
| This dissonant trine is among the more consistently complex
|-
| Naiadic
| D, Gd>/, D
| 0, 60, 0
| 10:13:20
| This dissonant trine is relatively simple and thus expected to be rather common
|-
| Cocytic
| D, At<\, D
| 0, 99, 0
| 1/(10:13:20)
| This dissonant trine is relatively simple and thus expected to be rather common
|-
| Wide Cocytic
| D, At<, D
| 0, 100, 0
| 11:17:22
| This essentially tempered trine is very likely to be used as a basis for cocytic triads
|-
| Narrow Niadic
| D, Gd>, D
| 0, 59, 0
| 1/(11:17:22)
| This essentially tempered trine is very likely to be used as a partial basis for suspended chords
|-
| Narrow Superdusthumic
| D, Ad<↑, D
| 0, 89, 0
| 128:189:256
| This dissonant trine is common in essentially tempered chords
|-
| Wide Subagallic
| D, Gt>↓, D
| 0, 70, 0
| 1/(128:189:256)
| This dissonant trine is common in essentially tempered chords
|-
| Subagallic
| D, G↑, D
| 0, 69, 0
| 20:27:40
| This dissonant trine is very likely to show up in non-meantone diatonic contexts
|-
| Superdusthumic
| D, A↓, D
| 0, 90, 0
| 1/(20:27:40)
| This dissonant trine is very likely to show up in non-meantone diatonic contexts
|-
| Narrow Subagallic
| D, G↑\, D
| 0, 68, 0
| 90:121:180
| This dissonant trine is on the outer edge of the diatonic range
|-
| Wide Superdusthumic
| D, A↓/, D
| 0, 91, 0
| 1/(90:121:180)
| This dissonant trine is on the outer edge of the diatonic range
|-
| Wide Agallic
| D, Gt<, D
| 0, 73, 0
| 8:11:16
| This ambisonant trine is very likely to be used as a partial basis for suspended chords
|-
| Narrow Dusthumic
| D, Ad>, D
| 0, 86, 0
| 1/(8:11:16)
| This ambisonant trine is very likely to be used as a basis for dusthumic triads
|-
| Dusthumic
| D, Ad<\, D
| 0, 87, 0
| 128:187:256
| This dissonant trine is common in essentially tempered chords
|-
| Agallic
| D, Gt<\, D
| 0, 72, 0
| 1/(128:187:256)
| This dissonant trine is common in essentially tempered chords
|-
| Narrow Agallic
| D, Gt>↓, D
| 0, 71, 0
| 11:15:22
| This trine is very likely to be used as a partial basis for suspended chords
|-
| Wide Dusthumic
| D, Ad<↑, D
| 0, 88, 0
| 1/(11:15:22)
| This trine is very likely to be used as a basis for dusthumic triads
|-
| Wide Subdusthumic
| D, Ad<, D
| 0, 85, 0
| 56:81:112
| This essentially tempered trine is likely to be used as a basis for subdusthumic triads
|-
| Narrow Superagallic
| D, Gt>, D
| 0, 74, 0
| 1/(56:81:112)
| This essentially tempered trine is likely to be used as a partial basis for suspended chords
|-
| Subdusthumic
| D, Ab↑↑, D
| 0, 84, 0
| 9:13:18
| This essentially tempered trine is very likely to be used as a basis for subdusthumic triads
|-
| Superagallic
| D, G#↓↓, D
| 0, 75, 0
| 1/(9:13:18)
| This essentially tempered trine is very likely to be used as a partial basis for suspended chords
|-
| Wide Superagallic
| D, Gt<↑, D
| 0, 76, 0
| 256:357:512
| This essentially tempered trine is very likely to be used as a partial basis for suspended chords
|-
| Narrow Subdusthumic
| D, Ad>↓, D
| 0, 83, 0
| 1/(256:357:512)
| This essentially tempered trine is very likely to be used as a basis for subdusthumic triads
|-
| Narrow Hyperquartal
| D, Gt>↑, D
| 0, 77, 0
| 5:7:10
| This ambisonant trine is very common as a basis for diminished chords, and is very likely to be used as a partial basis for suspended chords
|-
| Wide Hypoquintal
| D, Ad<↓, D
| 0, 82, 0
| 1/(5:7:10)
| This ambisonant trine is very common as a basis for diminished chords, and is very likely to be used as a partial basis for suspended chords
|-
| Hyperquartal
| D, G#↓, D
| 0, 78, 0
| 32:45:64
| This trine is very likely to be used as a partial basis for suspended chords
|-
| Hypoquintal
| D, Ab↑, D
| 0, 81, 0
| 1/(32:45:64)
| This trine is very common as a basis for diminished chords
|-
| Narrow Hypoquintal
| D, Ab↑\, D
| 0, 80, 0
| 12:17:24
| This trine is very common as a basis for diminished chords
|-
| Wide Hyperquartal
| D, G#↓/, D
| 0, 79, 0
| 1/(12:17:24)
| This trine is very likely to be used as a partial basis for suspended chords
|}

User:Aura/On 159edo Music Theory (Part 2)

2026-03-31T15:49:00Z

Aura: Starting a chart of the triads- this will take multiple edits to complete...

''It is recommended that one read [[User:Aura/On 159edo Music Theory (Part 1)|Part 1]] prior to reading this article''

Now that we have covered the intervals of [[159edo]] as well as the possible trines, it's time we begin looking at possible triads. To lay a few ground rules, the most consonant triads tend not only to involve the closest approximations of consonant just intervals, but the trine that forms their backbone is also consonant in as of itself. One should note that even in the best-case scenarios, fourth-bounded triads will be ambisonant, as there is not much room for full-fledged consonance in triads like these due to the location of trill thresholds relative to both the top and bottom notes of the triad.

== Perfect Fifth-Bounded Triads ==
Because 159edo has so many notes, there are a lot of triads to go over just counting those bounded by the perfect fifth- in fact, there are as many as twenty-eight of them which can be treated as something other than the enharmonics of suspensions. With that in mind, we will cover these first.

{| class="wikitable"
|+Table of 159edo Perfect Triads
|-
! Name
! Notation (from D)
! Steps
! Approximate JI
! Notes
|-
| Ptolemaic Major
| D, F#↓, A
| 0, 51, 93
| 4:5:6
| This is the first of two triads that can be considered fully-resolved in Baroque, Classical, and Romantic harmony
|-
| Ptolemaic Minor
| D, F↑, A
| 0, 42, 93
| 1/(4:5:6)
| This is the second of two triads that can be considered fully-resolved in Baroque, Classical, and Romantic harmony
|}

User:Aura/On 159edo Music Theory (Part 2)

2026-03-31T07:10:47Z

Aura:

User:Aura/On 159edo Music Theory (Part 1)

2026-03-31T06:55:33Z

Aura:

Of all the multiples of [[53edo]], [[159edo]] is the lowest multiple that is noteworthy for being accurate in the 2.3.5.11.17 subgroup while having structural compromises in the 7.13.19.23.29 subgroup. Despite the number of pitches in this tuning system making it perhaps best fit for digital instruments of various kinds in actual performance, it is nevertheless also useful as an interval classification scheme.

== Intervals and Notation ==
159edo contains all the intervals of 53edo and can be thought of as having three fields of 53edo each separated by a third of 53edo's step. However, as some of the interpretations differ due 159edo having different mappings for certain primes, those differences show up in how harmonies are constructed. However, there's more.

Of all the intervals in 159edo, 5\159 is the first interval to be larger than the '''fission boundary''', which is where going back and forth between notes on either end of a given interval no longer sounds like a simple vibrato, but more like a dirty trill of sorts. The fission boundary further serves as the line separating melodic notes that can only be simple ornaments or quick passing tones from main melodic intervals, and since 5\159 is larger than this boundary it is the smallest interval that can serve as a main melodic interval.

The next landmark interval is 8\159, as this is the first interval to be larger than the '''gradient threshold''', which is where going back and forth between notes on either end of a given interval no longer sounds like a dirty trill, but rather a clean trill. The gradient threshold doubles as the point beyond which microtonal intervals can begin to serve as proper leading-tones.

Finally, 33\159 is the first interval to be larger than the '''trill threshold''', which is where going back and forth between notes on either end of a given interval no longer sounds like any kind of trill, and instead sounds like an arpeggio fragment. The trill threshold doubles as the boundary between intervals that are classified as steps, and those that are classified as leaps. As a consequence of this, the trill threshold marks the boundary where intervals cease to cause crowding in chords.

As if all that weren't enough, 159edo has its own variation on the [[dinner party rules]]— represented here by the Harmonic Compatibility Rating and Melodic Compatibility Rating columns in the following chart, where 10 is a full-blown friend relative to the root and −10 if a full-blown enemy relative to the root. Note that the Harmonic Compatibility and Melodic Compatibility ratings are based on octave-equivalence, and that some of the ratings are still speculative.

{| class="wikitable"
|+159edo Interval Names and Compatibility Ratings
! rowspan="2" | Step
! rowspan="2" | Cents
! colspan="3" | Interval and Note names
! colspan="2" | Compatibility rating
|-
! SKULO-based interval names
! Pythagorean-commatic-based interval names
! SRS notation
! Harmonic
! Melodic
|-
| 0
| 0
| P1
| Perfect Unison
| D
| 10
| 10
|-
| 1
| 7.5471698
| R1
| Wide Unison
| D/
| 0
| 0
|-
| 2
| 15.0943396
| rK1
| Narrow Superunison
| D↑\
| -10
| -10
|-
| 3
| 22.6415094
| K1
| Lesser Superunison
| D↑
| -10
| -3
|-
| 4
| 30.1886792
| S1, kU1
| Greater Superunison, Narrow Inframinor Second
| Edb<, Dt<↓
| -10
| 3
|-
| 5
| 37.7358491
| um2, RkU1
| Inframinor Second, Wide Superunison
| Edb>, Dt>↓
| -9
| 10
|-
| 6
| 45.2830189
| kkm2, Rum2, rU1
| Wide Inframinor Second, Narrow Ultraunison
| Eb↓↓, Dt<\
| -9
| 10
|-
| 7
| 52.8301887
| U1, rKum2
| Ultraunison, Narrow Subminor Second
| Dt<, Edb<↑
| -9
| 10
|-
| 8
| 60.3773585
| sm2, Kum2, uA1
| Lesser Subminor Second, Wide Ultraunison, Infra-Augmented Unison
| Dt>, Eb↓\
| -8
| 10
|-
| 9
| 67.9245283
| km2, RuA1, kkA1
| Greater Subminor Second, Diptolemaic Augmented Unison
| Eb↓, D#↓↓
| -8
| 9
|-
| 10
| 75.4716981
| Rkm2, rKuA1
| Wide Subminor Second, Lesser Sub-Augmented Unison
| Eb↓/, Dt<↑
| -7
| 9
|-
| 11
| 83.0188679
| rm2, KuA1
| Narrow Minor Second, Greater Sub-Augmented Unison
| Eb\, Dt>↑
| -7
| 9
|-
| 12
| 90.5660377
| m2, kA1
| Pythagorean Minor Second, Ptolemaic Augmented Unison
| Eb, D#↓
| -6
| 10
|-
| 13
| 98.1132075
| Rm2, RkA1
| Artomean Minor Second, Artomean Augmented Unison
| Eb/, D#↓/
| -6
| 10
|-
| 14
| 105.6603774
| rKm2, rA1
| Tendomean Minor Second, Tendomean Augmented Unison
| D#\, Eb↑\
| -5
| 10
|-
| 15
| 113.2075472
| Km2, A1
| Ptolemaic Minor Second, Pythagorean Augmented Unison
| D#, Eb↑
| -5
| 10
|-
| 16
| 120.7547170
| RKm2, kn2, RA1
| Wide Minor Second, Artoretromean Augmented Unison
| Ed<↓, Eb↑/, D#/
| -5
| 9
|-
| 17
| 128.3018868
| kN2, rKA1
| Lesser Supraminor Second, Tendoretromean Augmented Unison
| Ed>↓, D#↑\
| -6
| 8
|-
| 18
| 135.8490566
| KKm2, rn2, KA1
| Greater Supraminor Second, Diptolemaic Limma, Retroptolemaic Augmented Unison
| Ed<\, Eb↑↑, D#↑
| -7
| 6
|-
| 19
| 143.3962264
| n2, SA1
| Artoneutral Second, Lesser Super-Augmented Unison
| Ed<, Dt#<↓
| -8
| 5
|-
| 20
| 150.9433962
| N2, RkUA1
| Tendoneutral Second, Greater Super-Augmented Unison
| Ed>, Dt#>↓
| -7
| 6
|-
| 21
| 158.4905660
| kkM2, RN2, rUA1
| Lesser Submajor Second, Retrodiptolemaic Augmented Unison
| Ed>/, E↓↓, Dt#>↓/, D#↑↑
| -6
| 8
|-
| 22
| 166.0377358
| Kn2, UA1
| Greater Submajor Second, Ultra-Augmented Unison
| Ed<↑, Dt#<, Fb↓/
| -5
| 9
|-
| 23
| 173.5849057
| rkM2, KN2
| Narrow Major Second
| Ed>↑, E↓\, Dt#>, Fb\
| -4
| 10
|-
| 24
| 181.1320755
| kM2
| Ptolemaic Major Second
| E↓, Fb
| -3
| 10
|-
| 25
| 188.6792458
| RkM2
| Artomean Major Second
| E↓/, Fb/
| -3
| 10
|-
| 26
| 196.2264151
| rM2
| Tendomean Major Second
| E\, Fb↑\
| -2
| 10
|-
| 27
| 203.7735849
| M2
| Pythagorean Major Second
| E, Fb↑
| -2
| 10
|-
| 28
| 211.3207547
| RM2
| Wide Major Second
| E/, Fd<↓
| -1
| 10
|-
| 29
| 218.8679245
| rKM2
| Narrow Supermajor Second
| E↑\, Fd>↓
| -1
| 10
|-
| 30
| 226.4150943
| KM2
| Lesser Supermajor Second
| E↑, Fd<\, Fb↑↑, Dx
| -1
| 9
|-
| 31
| 233.9622642
| SM2, kUM2
| Greater Supermajor Second, Narrow Inframinor Third
| Fd<, Et<↓, E↑/
| 0
| 9
|-
| 32
| 241.5094340
| um3, RkUM2
| Inframinor Third, Wide Supermajor Second
| Fd>, Et>↓
| -1
| 8
|-
| 33
| 249.0566038
| kkm3, KKM2, Rum3, rUM2
| Wide Inframinor Third, Narrow Ultramajor Second, Semifourth
| Fd>/, Et<\, F↓↓, E↑↑
| 0
| 8
|-
| 34
| 256.6037736
| UM2, rKum3
| Ultramajor Second, Narrow Subminor Third
| Et<, Fd<↑
| -1
| 7
|-
| 35
| 264.1509434
| sm3, Kum3
| Lesser Subminor Third, Wide Ultramajor Second
| Et>, Fd>↑, F↓\
| 0
| 7
|-
| 36
| 271.6981132
| km3
| Greater Subminor Third
| F↓, Et>/, E#↓↓, Gbb
| -1
| 7
|-
| 37
| 279.2452830
| Rkm3
| Wide Subminor Third
| F↓/, Et<↑
| -1
| 8
|-
| 38
| 286.7924528
| rm3
| Narrow Minor Third
| F\, Et>↑
| 0
| 8
|-
| 39
| 294.3396226
| m3
| Pythagorean Minor Third
| F
| -1
| 9
|-
| 40
| 301.8867925
| Rm3
| Artomean Minor Third
| F/
| 1
| 9
|-
| 41
| 309.4339622
| rKm3
| Tendomean Minor Third
| F↑\
| 4
| 10
|-
| 42
| 316.9811321
| Km3
| Ptolemaic Minor Third
| F↑, E#
| 7
| 10
|-
| 43
| 324.5283019
| RKm3, kn3
| Wide Minor Third
| Ft<↓, F↑/, Gdb<
| 4
| 9
|-
| 44
| 332.0754717
| kN3, ud4
| Lesser Supraminor Third, Infra-Diminished Fourth
| Ft>↓, Gdb>
| 1
| 9
|-
| 45
| 339.6226415
| KKm3, rn3, Rud4
| Greater Supraminor Third, Retrodiptolemaic Diminished Fourth
| Ft<\, F↑↑, Gdb<↑\, Gb↓↓
| -1
| 8
|-
| 46
| 347.1698113
| n3, rKud4
| Artoneutral Third, Lesser Sub-Diminished Fourth
| Ft<, Gdb<↑
| 0
| 7
|-
| 47
| 354.7169811
| N3, sd4, Kud4
| Tendoneutral Third, Greater Sub-Diminished Fourth
| Ft>, Gdb>↑
| -1
| 7
|-
| 48
| 362.2641509
| kkM3, RN3, kd4
| Lesser Submajor Third, Retroptolemaic Diminished Fourth
| Ft>/, F#↓↓, Gb↓
| 0
| 8
|-
| 49
| 369.8113208
| Kn3, Rkd4
| Greater Submajor Third, Artoretromean Diminished Fourth
| Ft<↑, Gb↓/
| -1
| 9
|-
| 50
| 377.3584906
| rkM3, KN3, rd4
| Narrow Major Third, Tendoretromean Diminished Fourth
| Ft>↑, F#↓\, Gb\
| 3
| 9
|-
| 51
| 384.9056604
| kM3, d4
| Ptolemaic Major Third, Pythagorean Diminished Fourth
| Gb, F#↓
| 8
| 10
|-
| 52
| 392.4528302
| RkM3, Rd4
| Artomean Major Third, Artomean Diminished Fourth
| Gb/, F#↓/
| 4
| 10
|-
| 53
| 400
| rM3, rKd4
| Tendomean Major Third, Tendomean Diminished Fourth
| F#\, Gb↑\
| 1
| 9
|-
| 54
| 407.5471698
| M3, Kd4
| Pythagorean Major Third, Ptolemaic Diminished Fourth
| F#, Gb↑
| -1
| 9
|-
| 55
| 415.0943396
| RM3, kUd4
| Wide Major Third, Lesser Super-Diminished Fourth
| F#/, Gd<↓, Gb↑/
| 0
| 8
|-
| 56
| 422.6415094
| rKM3, RkUd4
| Narrow Supermajor Third, Greater Super-Diminished Fourth
| F#↑\, Gd>↓
| -1
| 7
|-
| 57
| 430.1886792
| KM3, rUd4, KKd4
| Lesser Supermajor Third, Diptolemaic Diminished Fourth
| F#↑, Gd<\, Gb↑↑
| -1
| 6
|-
| 58
| 437.7358491
| SM3, kUM3, rm4, Ud4
| Greater Supermajor Third, Ultra-Diminished Fourth
| Gd<, F#↑/
| 0
| 5
|-
| 59
| 445.2830189
| m4, RkUM3
| Paraminor Fourth, Wide Supermajor Third
| Gd>, Ft#>↓
| -1
| 3
|-
| 60
| 452.8301887
| Rm4, KKM3, rUM3
| Wide Paraminor Fourth, Narrow Ultramajor Third
| Gd>/, F#↑↑, G↓↓
| -2
| 1
|-
| 61
| 460.3773585
| UM3, rKm4
| Ultramajor Third, Narrow Grave Fourth
| Gd<↑, Ft#<
| -4
| -2
|-
| 62
| 467.9245283
| s4, Km4
| Lesser Grave Fourth, Wide Ultramajor Third
| Gd>↑, G↓\
| -7
| -4
|-
| 63
| 475.4716981
| k4
| Greater Grave Fourth
| G↓, Abb
| -6
| -5
|-
| 64
| 483.0188679
| Rk4
| Wide Grave Fourth
| G↓/
| -4
| 0
|-
| 65
| 490.5660377
| r4
| Narrow Fourth
| G\
| 1
| 5
|-
| 66
| 498.1132075
| P4
| Perfect Fourth
| G
| 9
| 10
|-
| 67
| 505.6603774
| R4
| Wide Fourth
| G/
| 1
| 8
|-
| 68
| 513.2075472
| rK4
| Narrow Acute Fourth
| G↑\
| -3
| 6
|-
| 69
| 520.7547170
| K4
| Lesser Acute Fourth
| G↑
| -5
| 5
|-
| 70
| 528.3018868
| S4, kM4
| Greater Acute Fourth
| Gt<↓, G↑/, Adb<
| -3
| 5
|-
| 71
| 535.8490566
| RkM4, ud5
| Wide Acute Fourth, Infra-Diminished Fifth
| Gt>↓, Adb>
| -2
| 5
|-
| 72
| 543.3962264
| rM4, Rud5
| Narrow Paramajor Fourth, Retrodiptolemaic Diminished Fifth
| Gt<\, G↑↑, Ab↓↓
| -1
| 6
|-
| 73
| 550.9433962
| M4, rKud5
| Paramajor Fourth, Lesser Sub-Diminished Fifth
| Gt<, Adb<↑
| 0
| 7
|-
| 74
| 558.4905660
| RM4, uA4, Kud5
| Infra-Augmented Fourth, Greater Sub-Diminished Fifth
| Gt>, Adb>↑
| -2
| 5
|-
| 75
| 566.0377358
| kkA4, RuA4, kd5
| Diptolemaic Augmented Fourth, Retroptolemaic Diminished Fifth
| Gt>/, G#↓↓, Ab↓
| -3
| 4
|-
| 76
| 573.5849057
| rKuA4, Rkd5
| Lesser Sub-Augmented Fourth, Artoretromean Diminished Fifth
| Gt<↑, Ab↓/
| -2
| 4
|-
| 77
| 581.1320755
| KuA4, rd5
| Greater Sub-Augmented Fourth, Tendoretromean Diminished Fifth
| Gt>↑, Ab\
| 0
| 5
|-
| 78
| 588.6792458
| kA4, d5
| Ptolemaic Augmented Fourth, Pythagorean Diminished Fifth
| Ab, G#↓
| -5
| 6
|-
| 79
| 596.2264151
| RkA4, Rd5
| Artomean Augmented Fourth, Artomean Diminished Fifth
| G#↓/, Ab/
| -9
| 7
|-
| 80
| 603.7735849
| rKd5, rA4
| Tendomean Diminished Fifth, Tendomean Augmented Fourth
| Ab↑\, G#\
| -9
| 7
|-
| 81
| 611.3207547
| Kd5, A4
| Ptolemaic Diminished Fifth, Pythagorean Augmented Fourth
| Ab↑, G#
| -5
| 6
|-
| 82
| 618.8679245
| kUd5, RA4
| Lesser Super-Diminished Fifth, Artoretromean Augmented Fourth
| Ad<↓, G#/
| 0
| 5
|-
| 83
| 626.4150943
| RkUd5, rKA4
| Greater Super-Diminished Fifth, Tendoretromean Augmented Fourth
| Ad>↓, G#↑\
| -2
| 4
|-
| 84
| 633.9622642
| KKd5, rUDd5, KA4
| Diptolemaic Diminished Fifth, Retroptolemaic Augmented Fourth
| Ad<\, Ab↑↑, G#↑
| -3
| 4
|-
| 85
| 641.5094340
| rm5, Ud5, kUA4
| Ultra-Diminished Fifth, Lesser Super-Augmented Fourth
| Ad<, Gt#<↓
| -2
| 5
|-
| 86
| 649.0566038
| m5, RkUA4
| Paraminor Fifth, Greater Super-Augmented Fourth
| Ad>, Gt#>↓
| 0
| 7
|-
| 87
| 656.6037736
| Rm5, rUA4
| Wide Paraminor Fifth, Retrodiptolemaic Augmented Fourth
| Ad>/, G#↑, Ab↑↑
| -1
| 6
|-
| 88
| 664.1509434
| rKm5, UA4
| Narrow Grave Fifth, Ultra-Augmented Fourth
| Ad<↑, Gt#<
| -2
| 5
|-
| 89
| 671.6981132
| s5, Km5
| Lesser Grave Fifth
| Ad>↑, A↓\, Gt#>
| -3
| 5
|-
| 90
| 679.2452830
| k5
| Greater Grave Fifth
| A↓
| -5
| 5
|-
| 91
| 686.7924528
| Rk5
| Wide Grave Fifth
| A↓/
| -3
| 6
|-
| 92
| 694.3396226
| r5
| Narrow Fifth
| A\
| 1
| 8
|-
| 93
| 701.8867925
| P5
| Perfect Fifth
| A
| 9
| 10
|-
| 94
| 709.4339622
| R5
| Wide Fifth
| A/
| 1
| 5
|-
| 95
| 716.9811321
| rK5
| Narrow Acute Fifth
| A↑\
| -4
| 0
|-
| 96
| 724.5283019
| K5
| Lesser Acute Fifth
| A↑, Gx
| -6
| -5
|-
| 97
| 732.0754717
| S5, kM5
| Greater Acute Fifth, Narrow Inframinor Sixth
| At<↓, A↑/
| -7
| -4
|-
| 98
| 739.6226415
| um6, RkM5
| Inframinor Sixth, Wide Acute Fifth
| At>↓, Bdb>
| -4
| -2
|-
| 99
| 747.1698113
| Rm4, KKM3, rUM3
| Narrow Paramajor Fifth, Wide Inframinor Sixth
| At<\, Bb↓↓, A↑↑
| -2
| 1
|-
| 100
| 754.7169811
| M5, rKum6
| Paramajor Fifth, Narrow Subminor Sixth
| At<, Bdb<↑
| -1
| 3
|-
| 101
| 762.2641509
| sm6, Kum6, RM5, uA5
| Lesser Subminor Sixth, Infra-Augmented Fifth
| At>, Bb↓\
| 0
| 5
|-
| 102
| 769.8113208
| km6, RuA5, kkA5
| Greater Subminor Sixth, Diptolemaic Augmented Fifth
| Bb↓, At>/, A#↓↓
| -1
| 6
|-
| 103
| 777.3584906
| Rkm6, rKuA5
| Wide Subminor Sixth, Lesser Sub-Augmented Fifth
| Bb↓/, At<↑
| -1
| 7
|-
| 104
| 784.9056604
| rm6, KuA5
| Narrow Minor Sixth, Greater Sub-Augmented Fifth
| Bb\, At>↑, A#↓\
| 0
| 8
|-
| 105
| 792.4528302
| m6, kA5
| Pythagorean Minor Sixth, Ptolemaic Augmented Fifth
| Bb, A#↓
| -1
| 9
|-
| 106
| 800
| Rm6, RkA5
| Artomean Minor Sixth, Artomean Augmented Fifth
| Bb/, A#↓/
| 1
| 9
|-
| 107
| 807.5471698
| rKm6, rA5
| Tendomean Minor Sixth, Tendomean Augmented Fifth
| A#\, Bb↑\
| 4
| 10
|-
| 108
| 815.0943396
| Km6, A5
| Ptolemaic Minor Sixth, Pythagorean Augmented Fifth
| A#, Bb↑
| 8
| 10
|-
| 109
| 822.6415094
| RKm6, kn6, RA5
|Wide Minor Sixth, Artoretromean Augmented Fifth
| Bd<↓, Bb↑/, A#/
| 3
| 9
|-
| 110
| 830.1886792
| kN6, rKA5
| Lesser Supraminor Sixth, Tendoretromean Augmented Fifth
| Bd>↓, A#↑\
| -1
| 9
|-
| 111
| 837.7358491
| KKm6, rn6, KA5
| Greater Supraminor Sixth, Retroptolemaic Augmented Fifth
| Bd<\, Bb↑↑, A#↑
| 0
| 8
|-
| 112
| 845.2830189
| n6, SA5, kUA5
| Artoneutral Sixth, Lesser Super-Augmented Fifth
| Bd<, At#<↓
| -1
| 7
|-
| 113
| 852.8301887
| N6, RkUA5
| Tendoneutral Sixth, Greater Super-Augmented Fifth
| Bd>, At#>↓
| 0
| 7
|-
| 114
| 860.3773585
| kkM6, RN6, rUA5
| Lesser Submajor Sixth, Retrodiptolemaic Augmented Fifth
| Bd>/, B↓↓, At#>↓/, A#↑↑
| -1
| 8
|-
| 115
| 867.9245283
| Kn6, UA5
| Greater Submajor Sixth, Ultra-Augmented Fifth
| Bd<↑, At#<
| 1
| 9
|-
| 116
| 875.4716981
| rkM6, KN6
| Narrow Major Sixth
| Bd>↑, B↓\, At#>
| 4
| 9
|-
| 117
| 883.0188679
| kM6
| Ptolemaic Major Sixth
| B↓, Cb
| 7
| 10
|-
| 118
| 890.5660377
| RkM6
| Artomean Major Sixth
| B↓/
| 4
| 10
|-
| 119
| 898.1132075
| rM6
| Tendomean Major Sixth
| B\
| 1
| 9
|-
| 120
| 905.6603774
| M6
| Pythagorean Major Sixth
| B
| -1
| 9
|-
| 121
| 913.2075472
| RM6
| Wide Major Sixth
| B/, Cd<↓
| 0
| 8
|-
| 122
| 920.7547170
| rKM6
| Narrow Supermajor Sixth
| B↑\, Cd>↓
| -1
| 8
|-
| 123
| 928.3018868
| KM6
| Lesser Supermajor Sixth
| B↑, Cd<\, Cb↑↑, Ax
| -1
| 7
|-
| 124
| 935.8490566
| SM6, kUM6
| Greater Supermajor Second, Narrow Inframinor Seventh
| Cd<, Bt<↓, B↑/
| 0
| 7
|-
| 125
| 943.3962264
| um7, RkUM6
| Inframinor Seventh, Wide Supermajor Sixth
| Cd>, Bt>↓
| -1
| 7
|-
| 126
| 950.9433962
| KKM6, kkm7, rUM6, Rum7
| Narrow Ultramajor Sixth, Wide Inframinor Seventh, Semitwelfth
| Bt<\, Cd>/, B↑↑, C↓↓
| 0
| 8
|-
| 127
| 958.4905660
| UM6, rKum7
| Ultramajor Sixth, Narrow Subminor Seventh
| Bt<, Cd<↑
| -1
| 8
|-
| 128
| 966.0377358
| sm7, Kum7
| Lesser Subminor Seventh, Wide Ultramajor Sixth
| Bt>, Cd>↑, C↓\
| 0
| 9
|-
| 129
| 973.5849057
| km7
| Greater Subminor Seventh
| C↓, Bt>/, B#↓↓, Dbb
| -1
| 9
|-
| 130
| 981.1320755
| Rkm7
| Wide Subminor Seventh
| C↓/, Bt<↑
| -1
| 10
|-
| 131
| 988.6792458
| rm7
| Narrow Minor Seventh
| C\, Bt>↑
| -1
| 10
|-
| 132
| 996.2264151
| m7
| Pythagorean Minor Seventh
| C, B#↓
| -2
| 10
|-
| 133
| 1003.7735849
| Rm7
| Artomean Minor Seventh
| C/, B#↓/
| -2
| 10
|-
| 134
| 1011.3207547
| rKm7
| Tendomean Minor Seventh
| C↑\, B#\
| -3
| 10
|-
| 135
| 1018.8679245
| kM2
| Ptolemaic Minor Seventh
| C↑, B#
| -3
| 10
|-
| 136
| 1026.4150943
| RKm7, kn7
| Wide Minor Seventh
| Ct<↓, C↑/, Ddb<, B#/
| -4
| 10
|-
| 137
| 1033.9622642
| kN7, ud8
| Lesser Supraminor Seventh, Infra-Diminished Octave
| Ct>↓, Ddb>, B#↑\
| -5
| 9
|-
| 138
| 1041.5094340
| KKm7, rn7, Rud8
| Greater Supraminor Seventh, Retrodiptolemaic Diminished Octave
| Ct<\, C↑↑, Ddb<↑\, Db↓↓
| -6
| 8
|-
| 139
| 1049.0566038
| n7, rKud8
| Artoneutral Seventh, Lesser Sub-Diminished Octave
| Ct<, Ddb<↑
| -7
| 6
|-
| 140
| 1056.6037736
| N7, sd8
| Tendoneutral Seventh, Greater Sub-Diminished Octave
| Ct>, Ddb>↑
| -8
| 5
|-
| 141
| 1064.1509434
| kkM7, RN7, kd8
| Lesser Submajor Seventh, Diptolemaic Major Seventh, Retroptolemaic Diminished Octave
| Ct>/, C#↓↓, Db↓
| -7
| 6
|-
| 142
| 1071.6981132
| Kn7, Rkd8
| Greater Submajor Seventh, Artoretromean Diminished Octave
| Ct<↑, Db↓/
| -6
| 8
|-
| 143
| 1079.2452830
| rkM7, KN7, rd8
| Narrow Major Seventh, Tendoretromean Diminished Octave
| Ct>↑, C#↓\, Db\
| -5
| 9
|-
| 144
| 1086.7924528
| kM7, d8
| Ptolemaic Major Seventh, Pythagorean Diminished Octave
| Db, C#↓
| -5
| 10
|-
| 145
| 1094.3396226
| RkM7, Rd8
| Artomean Major Seventh, Artomean Diminished Octave
| Db/, C#↓/
| -5
| 10
|-
| 146
| 1101.8867925
| rM7, rKd8
| Tendomean Major Seventh, Tendomean Diminished Octave
| C#\, Db↑\
| -6
| 10
|-
| 147
| 1109.4339622
| M7, Kd8
| Pythagorean Major Seventh, Ptolemaic Diminished Octave
| C#, Db↑
| -6
| 10
|-
| 148
| 1116.9811321
| RM7, kUd8
| Wide Major Seventh, Lesser Super-Diminished Octave
| C#/, Dd<↓
| -7
| 9
|-
| 149
| 1124.5283019
| rKM7, RkUd8
| Narrow Supermajor Seventh, Greater Super-Diminished Octave
| C#↑\, Dd>↓
| -7
| 9
|-
| 150
| 1132.0754717
| km2, RuA1, kkA1
| Lesser Supermajor Seventh, Diptolemaic Diminished Octave
| C#↑, Db↑↑
| -8
| 9
|-
| 151
| 1139.6226415
| SM7, kUM7, Ud8
| Greater Supermajor Seventh, Narrow Infraoctave, Ultra-Diminished Octave
| Dd<, C#↑/
| -8
| 10
|-
| 152
| 1147.1698113
| u8, RkUM7
| Infraoctave, Wide Supermajor Seventh
| Dd>, Ct#>↓
| -9
| 10
|-
| 153
| 1154.7169811
| KKM7, rUM7, Ru8
| Narrow Ultramajor Seventh, Wide Infraoctave
| C#↑↑, Dd>/
| -9
| 10
|-
| 154
| 1162.2641509
| UM7, rKu8
| Ultramajor Seventh, Wide Superprime
| Ct#<, Dd<↑
| -9
| 10
|-
| 155
| 1169.8113208
| s8, Ku8
| Lesser Suboctave, Wide Ultramajor Seventh
| Ct#>, Dd>↑
| -10
| 3
|-
| 156
| 1177.3584906
| k8
| Greater Suboctave
| D↓
| -10
| -3
|-
| 157
| 1184.9056604
| Rk8
| Wide Suboctave
| D↓/
| -10
| -10
|-
| 158
| 1192.4528302
| r8
| Narrow Octave
| D\
| 0
| 0
|-
| 159
| 1200
| P8
| Perfect Octave
| D
| 10
| 10
|}

== Trines ==
159edo has multiple types of trine. Trines are important in the aspects of 159edo music theory derived from Medieval and Neo-Medieval music theory. In fact, the individual intervals that constitute trines serve as the backbone of not only the triads of harmony, but the tetrachords of melody as well. Both triads and tetrachords will be covered in later installments of this series. For now, it pays to go over which three-note structures can serve as trines as well as their names.

{| class="wikitable"
|+Table of 159edo Trines
|-
! Name
! Notation (from D)
! Steps
! Approximate JI
! Notes
|-
| Otonal Perfect
| D, A, D
| 0, 93, 0
| 2:3:4
| This is the first of two trines that can be considered fully-resolved in Medieval and Neo-Medieval harmony
|-
| Utonal Perfect
| D, G, D
| 0, 66, 0
| 1/(2:3:4)
| This is the second of two trines that can be considered fully-resolved in Medieval and Neo-Medieval harmony
|-
| Otonal Archagall
| D, G\, D
| 0, 65, 0
| 64:85:128
| This trine is the first of two that are often used in the extended harmony of t<IV chords
|-
| Utonal Archagall
| D, A/, D
| 0, 94, 0
| 1/(64:85:128)
| This trine is the second of two that are often used in the extended harmony of t<IV chords
|-
| Bass-Up Marvelous
| D, A\, D
| 0, 92, 0
| 75:112:150
| This trine is the first of two that are formed from stacking identical approximations of the LCJI neutral third
|-
| Treble-Down Marvelous
| D, G/, D
| 0, 67, 0
| 1/(75:112:150)
| This trine is the second of two that are formed from stacking identical approximations of the LCJI neutral third
|-
| Narrow Supernaiadic
| D, G↓\, D
| 0, 62, 0
| 16:21:32
| This dissonant trine is common in essentially tempered chords
|-
| Wide Subcocytic
| D, A↑/, D
| 0, 97, 0
| 1/(16:21:32)
| This dissonant trine is common in essentially tempered chords
|-
| Subcocytic
| D, A↑, D
| 0, 96, 0
| 160:243:320
| This dissonant trine is common in essentially tempered chords
|-
| Supernaiadic
| D, G↓, D
| 0, 63, 0
| 1/(160:243:320)
| This dissonant trine is common in essentially tempered chords
|-
| Wide Supernaiadic
| D, G↓/, D
| 0, 64, 0
| 25:33:50
| This dissonant trine is on the outer edge of the diatonic range and is common in essentially tempered chords
|-
| Narrow Subcocytic
| D, A↑\, D
| 0, 95, 0
| 1/(25:33:50)
| This dissonant trine is on the outer edge of the diatonic range and is common in essentially tempered chords
|-
| Wide Naiadic
| D, Gd<↑, D
| 0, 61, 0
| 135:176:270
| This dissonant trine is among the more consistently complex
|-
| Narrow Cocytic
| D, At>↓, D
| 0, 98, 0
| 1/(135:176:270)
| This dissonant trine is among the more consistently complex
|-
| Naiadic
| D, Gd>/, D
| 0, 60, 0
| 10:13:20
| This dissonant trine is relatively simple and thus expected to be rather common
|-
| Cocytic
| D, At<\, D
| 0, 99, 0
| 1/(10:13:20)
| This dissonant trine is relatively simple and thus expected to be rather common
|-
| Wide Cocytic
| D, At<, D
| 0, 100, 0
| 11:17:22
| This essentially tempered trine is very likely to be used as a basis for cocytic triads
|-
| Narrow Niadic
| D, Gd>, D
| 0, 59, 0
| 1/(11:17:22)
| This essentially tempered trine is very likely to be used as a partial basis for suspended chords
|-
| Narrow Superdusthumic
| D, Ad<↑, D
| 0, 89, 0
| 128:189:256
| This dissonant trine is common in essentially tempered chords
|-
| Wide Subagallic
| D, Gt>↓, D
| 0, 70, 0
| 1/(128:189:256)
| This dissonant trine is common in essentially tempered chords
|-
| Subagallic
| D, G↑, D
| 0, 69, 0
| 20:27:40
| This dissonant trine is very likely to show up in non-meantone diatonic contexts
|-
| Superdusthumic
| D, A↓, D
| 0, 90, 0
| 1/(20:27:40)
| This dissonant trine is very likely to show up in non-meantone diatonic contexts
|-
| Narrow Subagallic
| D, G↑\, D
| 0, 68, 0
| 90:121:180
| This dissonant trine is on the outer edge of the diatonic range
|-
| Wide Superdusthumic
| D, A↓/, D
| 0, 91, 0
| 1/(90:121:180)
| This dissonant trine is on the outer edge of the diatonic range
|-
| Wide Agallic
| D, Gt<, D
| 0, 73, 0
| 8:11:16
| This ambisonant trine is very likely to be used as a partial basis for suspended chords
|-
| Narrow Dusthumic
| D, Ad>, D
| 0, 86, 0
| 1/(8:11:16)
| This ambisonant trine is very likely to be used as a basis for dusthumic triads
|-
| Dusthumic
| D, Ad<\, D
| 0, 87, 0
| 128:187:256
| This dissonant trine is common in essentially tempered chords
|-
| Agallic
| D, Gt<\, D
| 0, 72, 0
| 1/(128:187:256)
| This dissonant trine is common in essentially tempered chords
|-
| Narrow Agallic
| D, Gt>↓, D
| 0, 71, 0
| 11:15:22
| This trine is very likely to be used as a partial basis for suspended chords
|-
| Wide Dusthumic
| D, Ad<↑, D
| 0, 88, 0
| 1/(11:15:22)
| This trine is very likely to be used as a basis for dusthumic triads
|-
| Wide Subdusthumic
| D, Ad<, D
| 0, 85, 0
| 56:81:112
| This essentially tempered trine is likely to be used as a basis for subdusthumic triads
|-
| Narrow Superagallic
| D, Gt>, D
| 0, 74, 0
| 1/(56:81:112)
| This essentially tempered trine is likely to be used as a partial basis for suspended chords
|-
| Subdusthumic
| D, Ab↑↑, D
| 0, 84, 0
| 9:13:18
| This essentially tempered trine is very likely to be used as a basis for subdusthumic triads
|-
| Superagallic
| D, G#↓↓, D
| 0, 75, 0
| 1/(9:13:18)
| This essentially tempered trine is very likely to be used as a partial basis for suspended chords
|-
| Wide Superagallic
| D, Gt<↑, D
| 0, 76, 0
| 256:357:512
| This essentially tempered trine is very likely to be used as a partial basis for suspended chords
|-
| Narrow Subdusthumic
| D, Ad>↓, D
| 0, 83, 0
| 1/(256:357:512)
| This essentially tempered trine is very likely to be used as a basis for subdusthumic triads
|-
| Narrow Hyperquartal
| D, Gt>↑, D
| 0, 77, 0
| 5:7:10
| This ambisonant trine is very common as a basis for diminished chords, and is very likely to be used as a partial basis for suspended chords
|-
| Wide Hypoquintal
| D, Ad<↓, D
| 0, 82, 0
| 1/(5:7:10)
| This ambisonant trine is very common as a basis for diminished chords, and is very likely to be used as a partial basis for suspended chords
|-
| Hyperquartal
| D, G#↓, D
| 0, 78, 0
| 32:45:64
| This trine is very likely to be used as a partial basis for suspended chords
|-
| Hypoquintal
| D, Ab↑, D
| 0, 81, 0
| 1/(32:45:64)
| This trine is very common as a basis for diminished chords
|-
| Narrow Hypoquintal
| D, Ab↑\, D
| 0, 80, 0
| 12:17:24
| This trine is very common as a basis for diminished chords
|-
| Wide Hyperquartal
| D, G#↓/, D
| 0, 79, 0
| 1/(12:17:24)
| This trine is very likely to be used as a partial basis for suspended chords
|}

User:Aura/On 159edo Music Theory (Part 2)

2026-03-31T06:52:32Z

Aura:

User:Aura/On 159edo Music Theory (Part 2)

2026-03-31T06:46:01Z

Aura: Creating page. I'll have to expand on the content of this page soon...

User:Aura/On 159edo Music Theory (Part 1)

2026-03-31T06:26:59Z

Aura: Going to have to relabel more the consonances and dissonances of the various trines eventually

Of all the multiples of [[53edo]], [[159edo]] is the lowest multiple that is noteworthy for being accurate in the 2.3.5.11.17 subgroup while having structural compromises in the 7.13.19.23.29 subgroup. Despite the number of pitches in this tuning system making it perhaps best fit for digital instruments of various kinds in actual performance, it is nevertheless also useful as an interval classification scheme.

== Intervals and Notation ==
159edo contains all the intervals of 53edo and can be thought of as having three fields of 53edo each separated by a third of 53edo's step. However, as some of the interpretations differ due 159edo having different mappings for certain primes, those differences show up in how harmonies are constructed. However, there's more.

Of all the intervals in 159edo, 5\159 is the first interval to be larger than the '''fission boundary''', which is where going back and forth between notes on either end of a given interval no longer sounds like a simple vibrato, but more like a dirty trill of sorts. The fission boundary further serves as the line separating melodic notes that can only be simple ornaments or quick passing tones from main melodic intervals, and since 5\159 is larger than this boundary it is the smallest interval that can serve as a main melodic interval.

The next landmark interval is 8\159, as this is the first interval to be larger than the '''gradient threshold''', which is where going back and forth between notes on either end of a given interval no longer sounds like a dirty trill, but rather a clean trill. The gradient threshold doubles as the point beyond which microtonal intervals can begin to serve as proper leading-tones.

Finally, 33\159 is the first interval to be larger than the '''trill threshold''', which is where going back and forth between notes on either end of a given interval no longer sounds like any kind of trill, and instead sounds like an arpeggio fragment. The trill threshold doubles as the boundary between intervals that are classified as steps, and those that are classified as leaps.

As if all that weren't enough, 159edo has its own variation on the [[dinner party rules]]— represented here by the Harmonic Compatibility Rating and Melodic Compatibility Rating columns in the following chart, where 10 is a full-blown friend relative to the root and −10 if a full-blown enemy relative to the root. Note that the Harmonic Compatibility and Melodic Compatibility ratings are based on octave-equivalence, and that some of the ratings are still speculative.

{| class="wikitable"
|+159edo Interval Names and Compatibility Ratings
! rowspan="2" | Step
! rowspan="2" | Cents
! colspan="3" | Interval and Note names
! colspan="2" | Compatibility rating
|-
! SKULO-based interval names
! Pythagorean-commatic-based interval names
! SRS notation
! Harmonic
! Melodic
|-
| 0
| 0
| P1
| Perfect Unison
| D
| 10
| 10
|-
| 1
| 7.5471698
| R1
| Wide Unison
| D/
| 0
| 0
|-
| 2
| 15.0943396
| rK1
| Narrow Superunison
| D↑\
| -10
| -10
|-
| 3
| 22.6415094
| K1
| Lesser Superunison
| D↑
| -10
| -3
|-
| 4
| 30.1886792
| S1, kU1
| Greater Superunison, Narrow Inframinor Second
| Edb<, Dt<↓
| -10
| 3
|-
| 5
| 37.7358491
| um2, RkU1
| Inframinor Second, Wide Superunison
| Edb>, Dt>↓
| -9
| 10
|-
| 6
| 45.2830189
| kkm2, Rum2, rU1
| Wide Inframinor Second, Narrow Ultraunison
| Eb↓↓, Dt<\
| -9
| 10
|-
| 7
| 52.8301887
| U1, rKum2
| Ultraunison, Narrow Subminor Second
| Dt<, Edb<↑
| -9
| 10
|-
| 8
| 60.3773585
| sm2, Kum2, uA1
| Lesser Subminor Second, Wide Ultraunison, Infra-Augmented Unison
| Dt>, Eb↓\
| -8
| 10
|-
| 9
| 67.9245283
| km2, RuA1, kkA1
| Greater Subminor Second, Diptolemaic Augmented Unison
| Eb↓, D#↓↓
| -8
| 9
|-
| 10
| 75.4716981
| Rkm2, rKuA1
| Wide Subminor Second, Lesser Sub-Augmented Unison
| Eb↓/, Dt<↑
| -7
| 9
|-
| 11
| 83.0188679
| rm2, KuA1
| Narrow Minor Second, Greater Sub-Augmented Unison
| Eb\, Dt>↑
| -7
| 9
|-
| 12
| 90.5660377
| m2, kA1
| Pythagorean Minor Second, Ptolemaic Augmented Unison
| Eb, D#↓
| -6
| 10
|-
| 13
| 98.1132075
| Rm2, RkA1
| Artomean Minor Second, Artomean Augmented Unison
| Eb/, D#↓/
| -6
| 10
|-
| 14
| 105.6603774
| rKm2, rA1
| Tendomean Minor Second, Tendomean Augmented Unison
| D#\, Eb↑\
| -5
| 10
|-
| 15
| 113.2075472
| Km2, A1
| Ptolemaic Minor Second, Pythagorean Augmented Unison
| D#, Eb↑
| -5
| 10
|-
| 16
| 120.7547170
| RKm2, kn2, RA1
| Wide Minor Second, Artoretromean Augmented Unison
| Ed<↓, Eb↑/, D#/
| -5
| 9
|-
| 17
| 128.3018868
| kN2, rKA1
| Lesser Supraminor Second, Tendoretromean Augmented Unison
| Ed>↓, D#↑\
| -6
| 8
|-
| 18
| 135.8490566
| KKm2, rn2, KA1
| Greater Supraminor Second, Diptolemaic Limma, Retroptolemaic Augmented Unison
| Ed<\, Eb↑↑, D#↑
| -7
| 6
|-
| 19
| 143.3962264
| n2, SA1
| Artoneutral Second, Lesser Super-Augmented Unison
| Ed<, Dt#<↓
| -8
| 5
|-
| 20
| 150.9433962
| N2, RkUA1
| Tendoneutral Second, Greater Super-Augmented Unison
| Ed>, Dt#>↓
| -7
| 6
|-
| 21
| 158.4905660
| kkM2, RN2, rUA1
| Lesser Submajor Second, Retrodiptolemaic Augmented Unison
| Ed>/, E↓↓, Dt#>↓/, D#↑↑
| -6
| 8
|-
| 22
| 166.0377358
| Kn2, UA1
| Greater Submajor Second, Ultra-Augmented Unison
| Ed<↑, Dt#<, Fb↓/
| -5
| 9
|-
| 23
| 173.5849057
| rkM2, KN2
| Narrow Major Second
| Ed>↑, E↓\, Dt#>, Fb\
| -4
| 10
|-
| 24
| 181.1320755
| kM2
| Ptolemaic Major Second
| E↓, Fb
| -3
| 10
|-
| 25
| 188.6792458
| RkM2
| Artomean Major Second
| E↓/, Fb/
| -3
| 10
|-
| 26
| 196.2264151
| rM2
| Tendomean Major Second
| E\, Fb↑\
| -2
| 10
|-
| 27
| 203.7735849
| M2
| Pythagorean Major Second
| E, Fb↑
| -2
| 10
|-
| 28
| 211.3207547
| RM2
| Wide Major Second
| E/, Fd<↓
| -1
| 10
|-
| 29
| 218.8679245
| rKM2
| Narrow Supermajor Second
| E↑\, Fd>↓
| -1
| 10
|-
| 30
| 226.4150943
| KM2
| Lesser Supermajor Second
| E↑, Fd<\, Fb↑↑, Dx
| -1
| 9
|-
| 31
| 233.9622642
| SM2, kUM2
| Greater Supermajor Second, Narrow Inframinor Third
| Fd<, Et<↓, E↑/
| 0
| 9
|-
| 32
| 241.5094340
| um3, RkUM2
| Inframinor Third, Wide Supermajor Second
| Fd>, Et>↓
| -1
| 8
|-
| 33
| 249.0566038
| kkm3, KKM2, Rum3, rUM2
| Wide Inframinor Third, Narrow Ultramajor Second, Semifourth
| Fd>/, Et<\, F↓↓, E↑↑
| 0
| 8
|-
| 34
| 256.6037736
| UM2, rKum3
| Ultramajor Second, Narrow Subminor Third
| Et<, Fd<↑
| -1
| 7
|-
| 35
| 264.1509434
| sm3, Kum3
| Lesser Subminor Third, Wide Ultramajor Second
| Et>, Fd>↑, F↓\
| 0
| 7
|-
| 36
| 271.6981132
| km3
| Greater Subminor Third
| F↓, Et>/, E#↓↓, Gbb
| -1
| 7
|-
| 37
| 279.2452830
| Rkm3
| Wide Subminor Third
| F↓/, Et<↑
| -1
| 8
|-
| 38
| 286.7924528
| rm3
| Narrow Minor Third
| F\, Et>↑
| 0
| 8
|-
| 39
| 294.3396226
| m3
| Pythagorean Minor Third
| F
| -1
| 9
|-
| 40
| 301.8867925
| Rm3
| Artomean Minor Third
| F/
| 1
| 9
|-
| 41
| 309.4339622
| rKm3
| Tendomean Minor Third
| F↑\
| 4
| 10
|-
| 42
| 316.9811321
| Km3
| Ptolemaic Minor Third
| F↑, E#
| 7
| 10
|-
| 43
| 324.5283019
| RKm3, kn3
| Wide Minor Third
| Ft<↓, F↑/, Gdb<
| 4
| 9
|-
| 44
| 332.0754717
| kN3, ud4
| Lesser Supraminor Third, Infra-Diminished Fourth
| Ft>↓, Gdb>
| 1
| 9
|-
| 45
| 339.6226415
| KKm3, rn3, Rud4
| Greater Supraminor Third, Retrodiptolemaic Diminished Fourth
| Ft<\, F↑↑, Gdb<↑\, Gb↓↓
| -1
| 8
|-
| 46
| 347.1698113
| n3, rKud4
| Artoneutral Third, Lesser Sub-Diminished Fourth
| Ft<, Gdb<↑
| 0
| 7
|-
| 47
| 354.7169811
| N3, sd4, Kud4
| Tendoneutral Third, Greater Sub-Diminished Fourth
| Ft>, Gdb>↑
| -1
| 7
|-
| 48
| 362.2641509
| kkM3, RN3, kd4
| Lesser Submajor Third, Retroptolemaic Diminished Fourth
| Ft>/, F#↓↓, Gb↓
| 0
| 8
|-
| 49
| 369.8113208
| Kn3, Rkd4
| Greater Submajor Third, Artoretromean Diminished Fourth
| Ft<↑, Gb↓/
| -1
| 9
|-
| 50
| 377.3584906
| rkM3, KN3, rd4
| Narrow Major Third, Tendoretromean Diminished Fourth
| Ft>↑, F#↓\, Gb\
| 3
| 9
|-
| 51
| 384.9056604
| kM3, d4
| Ptolemaic Major Third, Pythagorean Diminished Fourth
| Gb, F#↓
| 8
| 10
|-
| 52
| 392.4528302
| RkM3, Rd4
| Artomean Major Third, Artomean Diminished Fourth
| Gb/, F#↓/
| 4
| 10
|-
| 53
| 400
| rM3, rKd4
| Tendomean Major Third, Tendomean Diminished Fourth
| F#\, Gb↑\
| 1
| 9
|-
| 54
| 407.5471698
| M3, Kd4
| Pythagorean Major Third, Ptolemaic Diminished Fourth
| F#, Gb↑
| -1
| 9
|-
| 55
| 415.0943396
| RM3, kUd4
| Wide Major Third, Lesser Super-Diminished Fourth
| F#/, Gd<↓, Gb↑/
| 0
| 8
|-
| 56
| 422.6415094
| rKM3, RkUd4
| Narrow Supermajor Third, Greater Super-Diminished Fourth
| F#↑\, Gd>↓
| -1
| 7
|-
| 57
| 430.1886792
| KM3, rUd4, KKd4
| Lesser Supermajor Third, Diptolemaic Diminished Fourth
| F#↑, Gd<\, Gb↑↑
| -1
| 6
|-
| 58
| 437.7358491
| SM3, kUM3, rm4, Ud4
| Greater Supermajor Third, Ultra-Diminished Fourth
| Gd<, F#↑/
| 0
| 5
|-
| 59
| 445.2830189
| m4, RkUM3
| Paraminor Fourth, Wide Supermajor Third
| Gd>, Ft#>↓
| -1
| 3
|-
| 60
| 452.8301887
| Rm4, KKM3, rUM3
| Wide Paraminor Fourth, Narrow Ultramajor Third
| Gd>/, F#↑↑, G↓↓
| -2
| 1
|-
| 61
| 460.3773585
| UM3, rKm4
| Ultramajor Third, Narrow Grave Fourth
| Gd<↑, Ft#<
| -4
| -2
|-
| 62
| 467.9245283
| s4, Km4
| Lesser Grave Fourth, Wide Ultramajor Third
| Gd>↑, G↓\
| -7
| -4
|-
| 63
| 475.4716981
| k4
| Greater Grave Fourth
| G↓, Abb
| -6
| -5
|-
| 64
| 483.0188679
| Rk4
| Wide Grave Fourth
| G↓/
| -4
| 0
|-
| 65
| 490.5660377
| r4
| Narrow Fourth
| G\
| 1
| 5
|-
| 66
| 498.1132075
| P4
| Perfect Fourth
| G
| 9
| 10
|-
| 67
| 505.6603774
| R4
| Wide Fourth
| G/
| 1
| 8
|-
| 68
| 513.2075472
| rK4
| Narrow Acute Fourth
| G↑\
| -3
| 6
|-
| 69
| 520.7547170
| K4
| Lesser Acute Fourth
| G↑
| -5
| 5
|-
| 70
| 528.3018868
| S4, kM4
| Greater Acute Fourth
| Gt<↓, G↑/, Adb<
| -3
| 5
|-
| 71
| 535.8490566
| RkM4, ud5
| Wide Acute Fourth, Infra-Diminished Fifth
| Gt>↓, Adb>
| -2
| 5
|-
| 72
| 543.3962264
| rM4, Rud5
| Narrow Paramajor Fourth, Retrodiptolemaic Diminished Fifth
| Gt<\, G↑↑, Ab↓↓
| -1
| 6
|-
| 73
| 550.9433962
| M4, rKud5
| Paramajor Fourth, Lesser Sub-Diminished Fifth
| Gt<, Adb<↑
| 0
| 7
|-
| 74
| 558.4905660
| RM4, uA4, Kud5
| Infra-Augmented Fourth, Greater Sub-Diminished Fifth
| Gt>, Adb>↑
| -2
| 5
|-
| 75
| 566.0377358
| kkA4, RuA4, kd5
| Diptolemaic Augmented Fourth, Retroptolemaic Diminished Fifth
| Gt>/, G#↓↓, Ab↓
| -3
| 4
|-
| 76
| 573.5849057
| rKuA4, Rkd5
| Lesser Sub-Augmented Fourth, Artoretromean Diminished Fifth
| Gt<↑, Ab↓/
| -2
| 4
|-
| 77
| 581.1320755
| KuA4, rd5
| Greater Sub-Augmented Fourth, Tendoretromean Diminished Fifth
| Gt>↑, Ab\
| 0
| 5
|-
| 78
| 588.6792458
| kA4, d5
| Ptolemaic Augmented Fourth, Pythagorean Diminished Fifth
| Ab, G#↓
| -5
| 6
|-
| 79
| 596.2264151
| RkA4, Rd5
| Artomean Augmented Fourth, Artomean Diminished Fifth
| G#↓/, Ab/
| -9
| 7
|-
| 80
| 603.7735849
| rKd5, rA4
| Tendomean Diminished Fifth, Tendomean Augmented Fourth
| Ab↑\, G#\
| -9
| 7
|-
| 81
| 611.3207547
| Kd5, A4
| Ptolemaic Diminished Fifth, Pythagorean Augmented Fourth
| Ab↑, G#
| -5
| 6
|-
| 82
| 618.8679245
| kUd5, RA4
| Lesser Super-Diminished Fifth, Artoretromean Augmented Fourth
| Ad<↓, G#/
| 0
| 5
|-
| 83
| 626.4150943
| RkUd5, rKA4
| Greater Super-Diminished Fifth, Tendoretromean Augmented Fourth
| Ad>↓, G#↑\
| -2
| 4
|-
| 84
| 633.9622642
| KKd5, rUDd5, KA4
| Diptolemaic Diminished Fifth, Retroptolemaic Augmented Fourth
| Ad<\, Ab↑↑, G#↑
| -3
| 4
|-
| 85
| 641.5094340
| rm5, Ud5, kUA4
| Ultra-Diminished Fifth, Lesser Super-Augmented Fourth
| Ad<, Gt#<↓
| -2
| 5
|-
| 86
| 649.0566038
| m5, RkUA4
| Paraminor Fifth, Greater Super-Augmented Fourth
| Ad>, Gt#>↓
| 0
| 7
|-
| 87
| 656.6037736
| Rm5, rUA4
| Wide Paraminor Fifth, Retrodiptolemaic Augmented Fourth
| Ad>/, G#↑, Ab↑↑
| -1
| 6
|-
| 88
| 664.1509434
| rKm5, UA4
| Narrow Grave Fifth, Ultra-Augmented Fourth
| Ad<↑, Gt#<
| -2
| 5
|-
| 89
| 671.6981132
| s5, Km5
| Lesser Grave Fifth
| Ad>↑, A↓\, Gt#>
| -3
| 5
|-
| 90
| 679.2452830
| k5
| Greater Grave Fifth
| A↓
| -5
| 5
|-
| 91
| 686.7924528
| Rk5
| Wide Grave Fifth
| A↓/
| -3
| 6
|-
| 92
| 694.3396226
| r5
| Narrow Fifth
| A\
| 1
| 8
|-
| 93
| 701.8867925
| P5
| Perfect Fifth
| A
| 9
| 10
|-
| 94
| 709.4339622
| R5
| Wide Fifth
| A/
| 1
| 5
|-
| 95
| 716.9811321
| rK5
| Narrow Acute Fifth
| A↑\
| -4
| 0
|-
| 96
| 724.5283019
| K5
| Lesser Acute Fifth
| A↑, Gx
| -6
| -5
|-
| 97
| 732.0754717
| S5, kM5
| Greater Acute Fifth, Narrow Inframinor Sixth
| At<↓, A↑/
| -7
| -4
|-
| 98
| 739.6226415
| um6, RkM5
| Inframinor Sixth, Wide Acute Fifth
| At>↓, Bdb>
| -4
| -2
|-
| 99
| 747.1698113
| Rm4, KKM3, rUM3
| Narrow Paramajor Fifth, Wide Inframinor Sixth
| At<\, Bb↓↓, A↑↑
| -2
| 1
|-
| 100
| 754.7169811
| M5, rKum6
| Paramajor Fifth, Narrow Subminor Sixth
| At<, Bdb<↑
| -1
| 3
|-
| 101
| 762.2641509
| sm6, Kum6, RM5, uA5
| Lesser Subminor Sixth, Infra-Augmented Fifth
| At>, Bb↓\
| 0
| 5
|-
| 102
| 769.8113208
| km6, RuA5, kkA5
| Greater Subminor Sixth, Diptolemaic Augmented Fifth
| Bb↓, At>/, A#↓↓
| -1
| 6
|-
| 103
| 777.3584906
| Rkm6, rKuA5
| Wide Subminor Sixth, Lesser Sub-Augmented Fifth
| Bb↓/, At<↑
| -1
| 7
|-
| 104
| 784.9056604
| rm6, KuA5
| Narrow Minor Sixth, Greater Sub-Augmented Fifth
| Bb\, At>↑, A#↓\
| 0
| 8
|-
| 105
| 792.4528302
| m6, kA5
| Pythagorean Minor Sixth, Ptolemaic Augmented Fifth
| Bb, A#↓
| -1
| 9
|-
| 106
| 800
| Rm6, RkA5
| Artomean Minor Sixth, Artomean Augmented Fifth
| Bb/, A#↓/
| 1
| 9
|-
| 107
| 807.5471698
| rKm6, rA5
| Tendomean Minor Sixth, Tendomean Augmented Fifth
| A#\, Bb↑\
| 4
| 10
|-
| 108
| 815.0943396
| Km6, A5
| Ptolemaic Minor Sixth, Pythagorean Augmented Fifth
| A#, Bb↑
| 8
| 10
|-
| 109
| 822.6415094
| RKm6, kn6, RA5
|Wide Minor Sixth, Artoretromean Augmented Fifth
| Bd<↓, Bb↑/, A#/
| 3
| 9
|-
| 110
| 830.1886792
| kN6, rKA5
| Lesser Supraminor Sixth, Tendoretromean Augmented Fifth
| Bd>↓, A#↑\
| -1
| 9
|-
| 111
| 837.7358491
| KKm6, rn6, KA5
| Greater Supraminor Sixth, Retroptolemaic Augmented Fifth
| Bd<\, Bb↑↑, A#↑
| 0
| 8
|-
| 112
| 845.2830189
| n6, SA5, kUA5
| Artoneutral Sixth, Lesser Super-Augmented Fifth
| Bd<, At#<↓
| -1
| 7
|-
| 113
| 852.8301887
| N6, RkUA5
| Tendoneutral Sixth, Greater Super-Augmented Fifth
| Bd>, At#>↓
| 0
| 7
|-
| 114
| 860.3773585
| kkM6, RN6, rUA5
| Lesser Submajor Sixth, Retrodiptolemaic Augmented Fifth
| Bd>/, B↓↓, At#>↓/, A#↑↑
| -1
| 8
|-
| 115
| 867.9245283
| Kn6, UA5
| Greater Submajor Sixth, Ultra-Augmented Fifth
| Bd<↑, At#<
| 1
| 9
|-
| 116
| 875.4716981
| rkM6, KN6
| Narrow Major Sixth
| Bd>↑, B↓\, At#>
| 4
| 9
|-
| 117
| 883.0188679
| kM6
| Ptolemaic Major Sixth
| B↓, Cb
| 7
| 10
|-
| 118
| 890.5660377
| RkM6
| Artomean Major Sixth
| B↓/
| 4
| 10
|-
| 119
| 898.1132075
| rM6
| Tendomean Major Sixth
| B\
| 1
| 9
|-
| 120
| 905.6603774
| M6
| Pythagorean Major Sixth
| B
| -1
| 9
|-
| 121
| 913.2075472
| RM6
| Wide Major Sixth
| B/, Cd<↓
| 0
| 8
|-
| 122
| 920.7547170
| rKM6
| Narrow Supermajor Sixth
| B↑\, Cd>↓
| -1
| 8
|-
| 123
| 928.3018868
| KM6
| Lesser Supermajor Sixth
| B↑, Cd<\, Cb↑↑, Ax
| -1
| 7
|-
| 124
| 935.8490566
| SM6, kUM6
| Greater Supermajor Second, Narrow Inframinor Seventh
| Cd<, Bt<↓, B↑/
| 0
| 7
|-
| 125
| 943.3962264
| um7, RkUM6
| Inframinor Seventh, Wide Supermajor Sixth
| Cd>, Bt>↓
| -1
| 7
|-
| 126
| 950.9433962
| KKM6, kkm7, rUM6, Rum7
| Narrow Ultramajor Sixth, Wide Inframinor Seventh, Semitwelfth
| Bt<\, Cd>/, B↑↑, C↓↓
| 0
| 8
|-
| 127
| 958.4905660
| UM6, rKum7
| Ultramajor Sixth, Narrow Subminor Seventh
| Bt<, Cd<↑
| -1
| 8
|-
| 128
| 966.0377358
| sm7, Kum7
| Lesser Subminor Seventh, Wide Ultramajor Sixth
| Bt>, Cd>↑, C↓\
| 0
| 9
|-
| 129
| 973.5849057
| km7
| Greater Subminor Seventh
| C↓, Bt>/, B#↓↓, Dbb
| -1
| 9
|-
| 130
| 981.1320755
| Rkm7
| Wide Subminor Seventh
| C↓/, Bt<↑
| -1
| 10
|-
| 131
| 988.6792458
| rm7
| Narrow Minor Seventh
| C\, Bt>↑
| -1
| 10
|-
| 132
| 996.2264151
| m7
| Pythagorean Minor Seventh
| C, B#↓
| -2
| 10
|-
| 133
| 1003.7735849
| Rm7
| Artomean Minor Seventh
| C/, B#↓/
| -2
| 10
|-
| 134
| 1011.3207547
| rKm7
| Tendomean Minor Seventh
| C↑\, B#\
| -3
| 10
|-
| 135
| 1018.8679245
| kM2
| Ptolemaic Minor Seventh
| C↑, B#
| -3
| 10
|-
| 136
| 1026.4150943
| RKm7, kn7
| Wide Minor Seventh
| Ct<↓, C↑/, Ddb<, B#/
| -4
| 10
|-
| 137
| 1033.9622642
| kN7, ud8
| Lesser Supraminor Seventh, Infra-Diminished Octave
| Ct>↓, Ddb>, B#↑\
| -5
| 9
|-
| 138
| 1041.5094340
| KKm7, rn7, Rud8
| Greater Supraminor Seventh, Retrodiptolemaic Diminished Octave
| Ct<\, C↑↑, Ddb<↑\, Db↓↓
| -6
| 8
|-
| 139
| 1049.0566038
| n7, rKud8
| Artoneutral Seventh, Lesser Sub-Diminished Octave
| Ct<, Ddb<↑
| -7
| 6
|-
| 140
| 1056.6037736
| N7, sd8
| Tendoneutral Seventh, Greater Sub-Diminished Octave
| Ct>, Ddb>↑
| -8
| 5
|-
| 141
| 1064.1509434
| kkM7, RN7, kd8
| Lesser Submajor Seventh, Diptolemaic Major Seventh, Retroptolemaic Diminished Octave
| Ct>/, C#↓↓, Db↓
| -7
| 6
|-
| 142
| 1071.6981132
| Kn7, Rkd8
| Greater Submajor Seventh, Artoretromean Diminished Octave
| Ct<↑, Db↓/
| -6
| 8
|-
| 143
| 1079.2452830
| rkM7, KN7, rd8
| Narrow Major Seventh, Tendoretromean Diminished Octave
| Ct>↑, C#↓\, Db\
| -5
| 9
|-
| 144
| 1086.7924528
| kM7, d8
| Ptolemaic Major Seventh, Pythagorean Diminished Octave
| Db, C#↓
| -5
| 10
|-
| 145
| 1094.3396226
| RkM7, Rd8
| Artomean Major Seventh, Artomean Diminished Octave
| Db/, C#↓/
| -5
| 10
|-
| 146
| 1101.8867925
| rM7, rKd8
| Tendomean Major Seventh, Tendomean Diminished Octave
| C#\, Db↑\
| -6
| 10
|-
| 147
| 1109.4339622
| M7, Kd8
| Pythagorean Major Seventh, Ptolemaic Diminished Octave
| C#, Db↑
| -6
| 10
|-
| 148
| 1116.9811321
| RM7, kUd8
| Wide Major Seventh, Lesser Super-Diminished Octave
| C#/, Dd<↓
| -7
| 9
|-
| 149
| 1124.5283019
| rKM7, RkUd8
| Narrow Supermajor Seventh, Greater Super-Diminished Octave
| C#↑\, Dd>↓
| -7
| 9
|-
| 150
| 1132.0754717
| km2, RuA1, kkA1
| Lesser Supermajor Seventh, Diptolemaic Diminished Octave
| C#↑, Db↑↑
| -8
| 9
|-
| 151
| 1139.6226415
| SM7, kUM7, Ud8
| Greater Supermajor Seventh, Narrow Infraoctave, Ultra-Diminished Octave
| Dd<, C#↑/
| -8
| 10
|-
| 152
| 1147.1698113
| u8, RkUM7
| Infraoctave, Wide Supermajor Seventh
| Dd>, Ct#>↓
| -9
| 10
|-
| 153
| 1154.7169811
| KKM7, rUM7, Ru8
| Narrow Ultramajor Seventh, Wide Infraoctave
| C#↑↑, Dd>/
| -9
| 10
|-
| 154
| 1162.2641509
| UM7, rKu8
| Ultramajor Seventh, Wide Superprime
| Ct#<, Dd<↑
| -9
| 10
|-
| 155
| 1169.8113208
| s8, Ku8
| Lesser Suboctave, Wide Ultramajor Seventh
| Ct#>, Dd>↑
| -10
| 3
|-
| 156
| 1177.3584906
| k8
| Greater Suboctave
| D↓
| -10
| -3
|-
| 157
| 1184.9056604
| Rk8
| Wide Suboctave
| D↓/
| -10
| -10
|-
| 158
| 1192.4528302
| r8
| Narrow Octave
| D\
| 0
| 0
|-
| 159
| 1200
| P8
| Perfect Octave
| D
| 10
| 10
|}

== Trines ==
159edo has multiple types of trine. Trines are important in the aspects of 159edo music theory derived from Medieval and Neo-Medieval music theory. In fact, the individual intervals that constitute trines serve as the backbone of not only the triads of harmony, but the tetrachords of melody as well. Both triads and tetrachords will be covered in later installments of this series. For now, it pays to go over which three-note structures can serve as trines as well as their names.

{| class="wikitable"
|+Table of 159edo Trines
|-
! Name
! Notation (from D)
! Steps
! Approximate JI
! Notes
|-
| Otonal Perfect
| D, A, D
| 0, 93, 0
| 2:3:4
| This is the first of two trines that can be considered fully-resolved in Medieval and Neo-Medieval harmony
|-
| Utonal Perfect
| D, G, D
| 0, 66, 0
| 1/(2:3:4)
| This is the second of two trines that can be considered fully-resolved in Medieval and Neo-Medieval harmony
|-
| Otonal Archagall
| D, G\, D
| 0, 65, 0
| 64:85:128
| This trine is the first of two that are often used in the extended harmony of t<IV chords
|-
| Utonal Archagall
| D, A/, D
| 0, 94, 0
| 1/(64:85:128)
| This trine is the second of two that are often used in the extended harmony of t<IV chords
|-
| Bass-Up Marvelous
| D, A\, D
| 0, 92, 0
| 75:112:150
| This trine is the first of two that are formed from stacking identical approximations of the LCJI neutral third
|-
| Treble-Down Marvelous
| D, G/, D
| 0, 67, 0
| 1/(75:112:150)
| This trine is the second of two that are formed from stacking identical approximations of the LCJI neutral third
|-
| Narrow Supernaiadic
| D, G↓\, D
| 0, 62, 0
| 16:21:32
| This dissonant trine is common in essentially tempered chords
|-
| Wide Subcocytic
| D, A↑/, D
| 0, 97, 0
| 1/(16:21:32)
| This dissonant trine is common in essentially tempered chords
|-
| Subcocytic
| D, A↑, D
| 0, 96, 0
| 160:243:320
| This dissonant trine is common in essentially tempered chords
|-
| Supernaiadic
| D, G↓, D
| 0, 63, 0
| 1/(160:243:320)
| This dissonant trine is common in essentially tempered chords
|-
| Wide Supernaiadic
| D, G↓/, D
| 0, 64, 0
| 25:33:50
| This dissonant trine is on the outer edge of the diatonic range and is common in essentially tempered chords
|-
| Narrow Subcocytic
| D, A↑\, D
| 0, 95, 0
| 1/(25:33:50)
| This dissonant trine is on the outer edge of the diatonic range and is common in essentially tempered chords
|-
| Wide Naiadic
| D, Gd<↑, D
| 0, 61, 0
| 135:176:270
| This dissonant trine is among the more consistently complex
|-
| Narrow Cocytic
| D, At>↓, D
| 0, 98, 0
| 1/(135:176:270)
| This dissonant trine is among the more consistently complex
|-
| Naiadic
| D, Gd>/, D
| 0, 60, 0
| 10:13:20
| This dissonant trine is relatively simple and thus expected to be rather common
|-
| Cocytic
| D, At<\, D
| 0, 99, 0
| 1/(10:13:20)
| This dissonant trine is relatively simple and thus expected to be rather common
|-
| Wide Cocytic
| D, At<, D
| 0, 100, 0
| 11:17:22
| This essentially tempered trine is very likely to be used as a basis for cocytic triads
|-
| Narrow Niadic
| D, Gd>, D
| 0, 59, 0
| 1/(11:17:22)
| This essentially tempered trine is very likely to be used as a partial basis for suspended chords
|-
| Narrow Superdusthumic
| D, Ad<↑, D
| 0, 89, 0
| 128:189:256
| This dissonant trine is common in essentially tempered chords
|-
| Wide Subagallic
| D, Gt>↓, D
| 0, 70, 0
| 1/(128:189:256)
| This dissonant trine is common in essentially tempered chords
|-
| Subagallic
| D, G↑, D
| 0, 69, 0
| 20:27:40
| This dissonant trine is very likely to show up in non-meantone diatonic contexts
|-
| Superdusthumic
| D, A↓, D
| 0, 90, 0
| 1/(20:27:40)
| This dissonant trine is very likely to show up in non-meantone diatonic contexts
|-
| Narrow Subagallic
| D, G↑\, D
| 0, 68, 0
| 90:121:180
| This dissonant trine is on the outer edge of the diatonic range
|-
| Wide Superdusthumic
| D, A↓/, D
| 0, 91, 0
| 1/(90:121:180)
| This dissonant trine is on the outer edge of the diatonic range
|-
| Wide Agallic
| D, Gt<, D
| 0, 73, 0
| 8:11:16
| This ambisonant trine is very likely to be used as a partial basis for suspended chords
|-
| Narrow Dusthumic
| D, Ad>, D
| 0, 86, 0
| 1/(8:11:16)
| This ambisonant trine is very likely to be used as a basis for dusthumic triads
|-
| Dusthumic
| D, Ad<\, D
| 0, 87, 0
| 128:187:256
| This dissonant trine is common in essentially tempered chords
|-
| Agallic
| D, Gt<\, D
| 0, 72, 0
| 1/(128:187:256)
| This dissonant trine is common in essentially tempered chords
|-
| Narrow Agallic
| D, Gt>↓, D
| 0, 71, 0
| 11:15:22
| This trine is very likely to be used as a partial basis for suspended chords
|-
| Wide Dusthumic
| D, Ad<↑, D
| 0, 88, 0
| 1/(11:15:22)
| This trine is very likely to be used as a basis for dusthumic triads
|-
| Wide Subdusthumic
| D, Ad<, D
| 0, 85, 0
| 56:81:112
| This essentially tempered trine is likely to be used as a basis for subdusthumic triads
|-
| Narrow Superagallic
| D, Gt>, D
| 0, 74, 0
| 1/(56:81:112)
| This essentially tempered trine is likely to be used as a partial basis for suspended chords
|-
| Subdusthumic
| D, Ab↑↑, D
| 0, 84, 0
| 9:13:18
| This essentially tempered trine is very likely to be used as a basis for subdusthumic triads
|-
| Superagallic
| D, G#↓↓, D
| 0, 75, 0
| 1/(9:13:18)
| This essentially tempered trine is very likely to be used as a partial basis for suspended chords
|-
| Wide Superagallic
| D, Gt<↑, D
| 0, 76, 0
| 256:357:512
| This essentially tempered trine is very likely to be used as a partial basis for suspended chords
|-
| Narrow Subdusthumic
| D, Ad>↓, D
| 0, 83, 0
| 1/(256:357:512)
| This essentially tempered trine is very likely to be used as a basis for subdusthumic triads
|-
| Narrow Hyperquartal
| D, Gt>↑, D
| 0, 77, 0
| 5:7:10
| This ambisonant trine is very common as a basis for diminished chords, and is very likely to be used as a partial basis for suspended chords
|-
| Wide Hypoquintal
| D, Ad<↓, D
| 0, 82, 0
| 1/(5:7:10)
| This ambisonant trine is very common as a basis for diminished chords, and is very likely to be used as a partial basis for suspended chords
|-
| Hyperquartal
| D, G#↓, D
| 0, 78, 0
| 32:45:64
| This trine is very likely to be used as a partial basis for suspended chords
|-
| Hypoquintal
| D, Ab↑, D
| 0, 81, 0
| 1/(32:45:64)
| This trine is very common as a basis for diminished chords
|-
| Narrow Hypoquintal
| D, Ab↑\, D
| 0, 80, 0
| 12:17:24
| This trine is very common as a basis for diminished chords
|-
| Wide Hyperquartal
| D, G#↓/, D
| 0, 79, 0
| 1/(12:17:24)
| This trine is very likely to be used as a partial basis for suspended chords
|}

User:Aura/On 159edo Music Theory (Part 1)

2026-03-31T06:21:53Z

Aura: Rewrote a few things, and decided to cover triads and tetrachords in later installments

Of all the multiples of [[53edo]], [[159edo]] is the lowest multiple that is noteworthy for being accurate in the 2.3.5.11.17 subgroup while having structural compromises in the 7.13.19.23.29 subgroup. Despite the number of pitches in this tuning system making it perhaps best fit for digital instruments of various kinds in actual performance, it is nevertheless also useful as an interval classification scheme.

== Intervals and Notation ==
159edo contains all the intervals of 53edo and can be thought of as having three fields of 53edo each separated by a third of 53edo's step. However, as some of the interpretations differ due 159edo having different mappings for certain primes, those differences show up in how harmonies are constructed. However, there's more.

Of all the intervals in 159edo, 5\159 is the first interval to be larger than the '''fission boundary''', which is where going back and forth between notes on either end of a given interval no longer sounds like a simple vibrato, but more like a dirty trill of sorts. The fission boundary further serves as the line separating melodic notes that can only be simple ornaments or quick passing tones from main melodic intervals, and since 5\159 is larger than this boundary it is the smallest interval that can serve as a main melodic interval.

The next landmark interval is 8\159, as this is the first interval to be larger than the '''gradient threshold''', which is where going back and forth between notes on either end of a given interval no longer sounds like a dirty trill, but rather a clean trill. The gradient threshold doubles as the point beyond which microtonal intervals can begin to serve as proper leading-tones.

Finally, 33\159 is the first interval to be larger than the '''trill threshold''', which is where going back and forth between notes on either end of a given interval no longer sounds like any kind of trill, and instead sounds like an arpeggio fragment. The trill threshold doubles as the boundary between intervals that are classified as steps, and those that are classified as leaps.

As if all that weren't enough, 159edo has its own variation on the [[dinner party rules]]— represented here by the Harmonic Compatibility Rating and Melodic Compatibility Rating columns in the following chart, where 10 is a full-blown friend relative to the root and −10 if a full-blown enemy relative to the root. Note that the Harmonic Compatibility and Melodic Compatibility ratings are based on octave-equivalence, and that some of the ratings are still speculative.

{| class="wikitable"
|+159edo Interval Names and Compatibility Ratings
! rowspan="2" | Step
! rowspan="2" | Cents
! colspan="3" | Interval and Note names
! colspan="2" | Compatibility rating
|-
! SKULO-based interval names
! Pythagorean-commatic-based interval names
! SRS notation
! Harmonic
! Melodic
|-
| 0
| 0
| P1
| Perfect Unison
| D
| 10
| 10
|-
| 1
| 7.5471698
| R1
| Wide Unison
| D/
| 0
| 0
|-
| 2
| 15.0943396
| rK1
| Narrow Superunison
| D↑\
| -10
| -10
|-
| 3
| 22.6415094
| K1
| Lesser Superunison
| D↑
| -10
| -3
|-
| 4
| 30.1886792
| S1, kU1
| Greater Superunison, Narrow Inframinor Second
| Edb<, Dt<↓
| -10
| 3
|-
| 5
| 37.7358491
| um2, RkU1
| Inframinor Second, Wide Superunison
| Edb>, Dt>↓
| -9
| 10
|-
| 6
| 45.2830189
| kkm2, Rum2, rU1
| Wide Inframinor Second, Narrow Ultraunison
| Eb↓↓, Dt<\
| -9
| 10
|-
| 7
| 52.8301887
| U1, rKum2
| Ultraunison, Narrow Subminor Second
| Dt<, Edb<↑
| -9
| 10
|-
| 8
| 60.3773585
| sm2, Kum2, uA1
| Lesser Subminor Second, Wide Ultraunison, Infra-Augmented Unison
| Dt>, Eb↓\
| -8
| 10
|-
| 9
| 67.9245283
| km2, RuA1, kkA1
| Greater Subminor Second, Diptolemaic Augmented Unison
| Eb↓, D#↓↓
| -8
| 9
|-
| 10
| 75.4716981
| Rkm2, rKuA1
| Wide Subminor Second, Lesser Sub-Augmented Unison
| Eb↓/, Dt<↑
| -7
| 9
|-
| 11
| 83.0188679
| rm2, KuA1
| Narrow Minor Second, Greater Sub-Augmented Unison
| Eb\, Dt>↑
| -7
| 9
|-
| 12
| 90.5660377
| m2, kA1
| Pythagorean Minor Second, Ptolemaic Augmented Unison
| Eb, D#↓
| -6
| 10
|-
| 13
| 98.1132075
| Rm2, RkA1
| Artomean Minor Second, Artomean Augmented Unison
| Eb/, D#↓/
| -6
| 10
|-
| 14
| 105.6603774
| rKm2, rA1
| Tendomean Minor Second, Tendomean Augmented Unison
| D#\, Eb↑\
| -5
| 10
|-
| 15
| 113.2075472
| Km2, A1
| Ptolemaic Minor Second, Pythagorean Augmented Unison
| D#, Eb↑
| -5
| 10
|-
| 16
| 120.7547170
| RKm2, kn2, RA1
| Wide Minor Second, Artoretromean Augmented Unison
| Ed<↓, Eb↑/, D#/
| -5
| 9
|-
| 17
| 128.3018868
| kN2, rKA1
| Lesser Supraminor Second, Tendoretromean Augmented Unison
| Ed>↓, D#↑\
| -6
| 8
|-
| 18
| 135.8490566
| KKm2, rn2, KA1
| Greater Supraminor Second, Diptolemaic Limma, Retroptolemaic Augmented Unison
| Ed<\, Eb↑↑, D#↑
| -7
| 6
|-
| 19
| 143.3962264
| n2, SA1
| Artoneutral Second, Lesser Super-Augmented Unison
| Ed<, Dt#<↓
| -8
| 5
|-
| 20
| 150.9433962
| N2, RkUA1
| Tendoneutral Second, Greater Super-Augmented Unison
| Ed>, Dt#>↓
| -7
| 6
|-
| 21
| 158.4905660
| kkM2, RN2, rUA1
| Lesser Submajor Second, Retrodiptolemaic Augmented Unison
| Ed>/, E↓↓, Dt#>↓/, D#↑↑
| -6
| 8
|-
| 22
| 166.0377358
| Kn2, UA1
| Greater Submajor Second, Ultra-Augmented Unison
| Ed<↑, Dt#<, Fb↓/
| -5
| 9
|-
| 23
| 173.5849057
| rkM2, KN2
| Narrow Major Second
| Ed>↑, E↓\, Dt#>, Fb\
| -4
| 10
|-
| 24
| 181.1320755
| kM2
| Ptolemaic Major Second
| E↓, Fb
| -3
| 10
|-
| 25
| 188.6792458
| RkM2
| Artomean Major Second
| E↓/, Fb/
| -3
| 10
|-
| 26
| 196.2264151
| rM2
| Tendomean Major Second
| E\, Fb↑\
| -2
| 10
|-
| 27
| 203.7735849
| M2
| Pythagorean Major Second
| E, Fb↑
| -2
| 10
|-
| 28
| 211.3207547
| RM2
| Wide Major Second
| E/, Fd<↓
| -1
| 10
|-
| 29
| 218.8679245
| rKM2
| Narrow Supermajor Second
| E↑\, Fd>↓
| -1
| 10
|-
| 30
| 226.4150943
| KM2
| Lesser Supermajor Second
| E↑, Fd<\, Fb↑↑, Dx
| -1
| 9
|-
| 31
| 233.9622642
| SM2, kUM2
| Greater Supermajor Second, Narrow Inframinor Third
| Fd<, Et<↓, E↑/
| 0
| 9
|-
| 32
| 241.5094340
| um3, RkUM2
| Inframinor Third, Wide Supermajor Second
| Fd>, Et>↓
| -1
| 8
|-
| 33
| 249.0566038
| kkm3, KKM2, Rum3, rUM2
| Wide Inframinor Third, Narrow Ultramajor Second, Semifourth
| Fd>/, Et<\, F↓↓, E↑↑
| 0
| 8
|-
| 34
| 256.6037736
| UM2, rKum3
| Ultramajor Second, Narrow Subminor Third
| Et<, Fd<↑
| -1
| 7
|-
| 35
| 264.1509434
| sm3, Kum3
| Lesser Subminor Third, Wide Ultramajor Second
| Et>, Fd>↑, F↓\
| 0
| 7
|-
| 36
| 271.6981132
| km3
| Greater Subminor Third
| F↓, Et>/, E#↓↓, Gbb
| -1
| 7
|-
| 37
| 279.2452830
| Rkm3
| Wide Subminor Third
| F↓/, Et<↑
| -1
| 8
|-
| 38
| 286.7924528
| rm3
| Narrow Minor Third
| F\, Et>↑
| 0
| 8
|-
| 39
| 294.3396226
| m3
| Pythagorean Minor Third
| F
| -1
| 9
|-
| 40
| 301.8867925
| Rm3
| Artomean Minor Third
| F/
| 1
| 9
|-
| 41
| 309.4339622
| rKm3
| Tendomean Minor Third
| F↑\
| 4
| 10
|-
| 42
| 316.9811321
| Km3
| Ptolemaic Minor Third
| F↑, E#
| 7
| 10
|-
| 43
| 324.5283019
| RKm3, kn3
| Wide Minor Third
| Ft<↓, F↑/, Gdb<
| 4
| 9
|-
| 44
| 332.0754717
| kN3, ud4
| Lesser Supraminor Third, Infra-Diminished Fourth
| Ft>↓, Gdb>
| 1
| 9
|-
| 45
| 339.6226415
| KKm3, rn3, Rud4
| Greater Supraminor Third, Retrodiptolemaic Diminished Fourth
| Ft<\, F↑↑, Gdb<↑\, Gb↓↓
| -1
| 8
|-
| 46
| 347.1698113
| n3, rKud4
| Artoneutral Third, Lesser Sub-Diminished Fourth
| Ft<, Gdb<↑
| 0
| 7
|-
| 47
| 354.7169811
| N3, sd4, Kud4
| Tendoneutral Third, Greater Sub-Diminished Fourth
| Ft>, Gdb>↑
| -1
| 7
|-
| 48
| 362.2641509
| kkM3, RN3, kd4
| Lesser Submajor Third, Retroptolemaic Diminished Fourth
| Ft>/, F#↓↓, Gb↓
| 0
| 8
|-
| 49
| 369.8113208
| Kn3, Rkd4
| Greater Submajor Third, Artoretromean Diminished Fourth
| Ft<↑, Gb↓/
| -1
| 9
|-
| 50
| 377.3584906
| rkM3, KN3, rd4
| Narrow Major Third, Tendoretromean Diminished Fourth
| Ft>↑, F#↓\, Gb\
| 3
| 9
|-
| 51
| 384.9056604
| kM3, d4
| Ptolemaic Major Third, Pythagorean Diminished Fourth
| Gb, F#↓
| 8
| 10
|-
| 52
| 392.4528302
| RkM3, Rd4
| Artomean Major Third, Artomean Diminished Fourth
| Gb/, F#↓/
| 4
| 10
|-
| 53
| 400
| rM3, rKd4
| Tendomean Major Third, Tendomean Diminished Fourth
| F#\, Gb↑\
| 1
| 9
|-
| 54
| 407.5471698
| M3, Kd4
| Pythagorean Major Third, Ptolemaic Diminished Fourth
| F#, Gb↑
| -1
| 9
|-
| 55
| 415.0943396
| RM3, kUd4
| Wide Major Third, Lesser Super-Diminished Fourth
| F#/, Gd<↓, Gb↑/
| 0
| 8
|-
| 56
| 422.6415094
| rKM3, RkUd4
| Narrow Supermajor Third, Greater Super-Diminished Fourth
| F#↑\, Gd>↓
| -1
| 7
|-
| 57
| 430.1886792
| KM3, rUd4, KKd4
| Lesser Supermajor Third, Diptolemaic Diminished Fourth
| F#↑, Gd<\, Gb↑↑
| -1
| 6
|-
| 58
| 437.7358491
| SM3, kUM3, rm4, Ud4
| Greater Supermajor Third, Ultra-Diminished Fourth
| Gd<, F#↑/
| 0
| 5
|-
| 59
| 445.2830189
| m4, RkUM3
| Paraminor Fourth, Wide Supermajor Third
| Gd>, Ft#>↓
| -1
| 3
|-
| 60
| 452.8301887
| Rm4, KKM3, rUM3
| Wide Paraminor Fourth, Narrow Ultramajor Third
| Gd>/, F#↑↑, G↓↓
| -2
| 1
|-
| 61
| 460.3773585
| UM3, rKm4
| Ultramajor Third, Narrow Grave Fourth
| Gd<↑, Ft#<
| -4
| -2
|-
| 62
| 467.9245283
| s4, Km4
| Lesser Grave Fourth, Wide Ultramajor Third
| Gd>↑, G↓\
| -7
| -4
|-
| 63
| 475.4716981
| k4
| Greater Grave Fourth
| G↓, Abb
| -6
| -5
|-
| 64
| 483.0188679
| Rk4
| Wide Grave Fourth
| G↓/
| -4
| 0
|-
| 65
| 490.5660377
| r4
| Narrow Fourth
| G\
| 1
| 5
|-
| 66
| 498.1132075
| P4
| Perfect Fourth
| G
| 9
| 10
|-
| 67
| 505.6603774
| R4
| Wide Fourth
| G/
| 1
| 8
|-
| 68
| 513.2075472
| rK4
| Narrow Acute Fourth
| G↑\
| -3
| 6
|-
| 69
| 520.7547170
| K4
| Lesser Acute Fourth
| G↑
| -5
| 5
|-
| 70
| 528.3018868
| S4, kM4
| Greater Acute Fourth
| Gt<↓, G↑/, Adb<
| -3
| 5
|-
| 71
| 535.8490566
| RkM4, ud5
| Wide Acute Fourth, Infra-Diminished Fifth
| Gt>↓, Adb>
| -2
| 5
|-
| 72
| 543.3962264
| rM4, Rud5
| Narrow Paramajor Fourth, Retrodiptolemaic Diminished Fifth
| Gt<\, G↑↑, Ab↓↓
| -1
| 6
|-
| 73
| 550.9433962
| M4, rKud5
| Paramajor Fourth, Lesser Sub-Diminished Fifth
| Gt<, Adb<↑
| 0
| 7
|-
| 74
| 558.4905660
| RM4, uA4, Kud5
| Infra-Augmented Fourth, Greater Sub-Diminished Fifth
| Gt>, Adb>↑
| -2
| 5
|-
| 75
| 566.0377358
| kkA4, RuA4, kd5
| Diptolemaic Augmented Fourth, Retroptolemaic Diminished Fifth
| Gt>/, G#↓↓, Ab↓
| -3
| 4
|-
| 76
| 573.5849057
| rKuA4, Rkd5
| Lesser Sub-Augmented Fourth, Artoretromean Diminished Fifth
| Gt<↑, Ab↓/
| -2
| 4
|-
| 77
| 581.1320755
| KuA4, rd5
| Greater Sub-Augmented Fourth, Tendoretromean Diminished Fifth
| Gt>↑, Ab\
| 0
| 5
|-
| 78
| 588.6792458
| kA4, d5
| Ptolemaic Augmented Fourth, Pythagorean Diminished Fifth
| Ab, G#↓
| -5
| 6
|-
| 79
| 596.2264151
| RkA4, Rd5
| Artomean Augmented Fourth, Artomean Diminished Fifth
| G#↓/, Ab/
| -9
| 7
|-
| 80
| 603.7735849
| rKd5, rA4
| Tendomean Diminished Fifth, Tendomean Augmented Fourth
| Ab↑\, G#\
| -9
| 7
|-
| 81
| 611.3207547
| Kd5, A4
| Ptolemaic Diminished Fifth, Pythagorean Augmented Fourth
| Ab↑, G#
| -5
| 6
|-
| 82
| 618.8679245
| kUd5, RA4
| Lesser Super-Diminished Fifth, Artoretromean Augmented Fourth
| Ad<↓, G#/
| 0
| 5
|-
| 83
| 626.4150943
| RkUd5, rKA4
| Greater Super-Diminished Fifth, Tendoretromean Augmented Fourth
| Ad>↓, G#↑\
| -2
| 4
|-
| 84
| 633.9622642
| KKd5, rUDd5, KA4
| Diptolemaic Diminished Fifth, Retroptolemaic Augmented Fourth
| Ad<\, Ab↑↑, G#↑
| -3
| 4
|-
| 85
| 641.5094340
| rm5, Ud5, kUA4
| Ultra-Diminished Fifth, Lesser Super-Augmented Fourth
| Ad<, Gt#<↓
| -2
| 5
|-
| 86
| 649.0566038
| m5, RkUA4
| Paraminor Fifth, Greater Super-Augmented Fourth
| Ad>, Gt#>↓
| 0
| 7
|-
| 87
| 656.6037736
| Rm5, rUA4
| Wide Paraminor Fifth, Retrodiptolemaic Augmented Fourth
| Ad>/, G#↑, Ab↑↑
| -1
| 6
|-
| 88
| 664.1509434
| rKm5, UA4
| Narrow Grave Fifth, Ultra-Augmented Fourth
| Ad<↑, Gt#<
| -2
| 5
|-
| 89
| 671.6981132
| s5, Km5
| Lesser Grave Fifth
| Ad>↑, A↓\, Gt#>
| -3
| 5
|-
| 90
| 679.2452830
| k5
| Greater Grave Fifth
| A↓
| -5
| 5
|-
| 91
| 686.7924528
| Rk5
| Wide Grave Fifth
| A↓/
| -3
| 6
|-
| 92
| 694.3396226
| r5
| Narrow Fifth
| A\
| 1
| 8
|-
| 93
| 701.8867925
| P5
| Perfect Fifth
| A
| 9
| 10
|-
| 94
| 709.4339622
| R5
| Wide Fifth
| A/
| 1
| 5
|-
| 95
| 716.9811321
| rK5
| Narrow Acute Fifth
| A↑\
| -4
| 0
|-
| 96
| 724.5283019
| K5
| Lesser Acute Fifth
| A↑, Gx
| -6
| -5
|-
| 97
| 732.0754717
| S5, kM5
| Greater Acute Fifth, Narrow Inframinor Sixth
| At<↓, A↑/
| -7
| -4
|-
| 98
| 739.6226415
| um6, RkM5
| Inframinor Sixth, Wide Acute Fifth
| At>↓, Bdb>
| -4
| -2
|-
| 99
| 747.1698113
| Rm4, KKM3, rUM3
| Narrow Paramajor Fifth, Wide Inframinor Sixth
| At<\, Bb↓↓, A↑↑
| -2
| 1
|-
| 100
| 754.7169811
| M5, rKum6
| Paramajor Fifth, Narrow Subminor Sixth
| At<, Bdb<↑
| -1
| 3
|-
| 101
| 762.2641509
| sm6, Kum6, RM5, uA5
| Lesser Subminor Sixth, Infra-Augmented Fifth
| At>, Bb↓\
| 0
| 5
|-
| 102
| 769.8113208
| km6, RuA5, kkA5
| Greater Subminor Sixth, Diptolemaic Augmented Fifth
| Bb↓, At>/, A#↓↓
| -1
| 6
|-
| 103
| 777.3584906
| Rkm6, rKuA5
| Wide Subminor Sixth, Lesser Sub-Augmented Fifth
| Bb↓/, At<↑
| -1
| 7
|-
| 104
| 784.9056604
| rm6, KuA5
| Narrow Minor Sixth, Greater Sub-Augmented Fifth
| Bb\, At>↑, A#↓\
| 0
| 8
|-
| 105
| 792.4528302
| m6, kA5
| Pythagorean Minor Sixth, Ptolemaic Augmented Fifth
| Bb, A#↓
| -1
| 9
|-
| 106
| 800
| Rm6, RkA5
| Artomean Minor Sixth, Artomean Augmented Fifth
| Bb/, A#↓/
| 1
| 9
|-
| 107
| 807.5471698
| rKm6, rA5
| Tendomean Minor Sixth, Tendomean Augmented Fifth
| A#\, Bb↑\
| 4
| 10
|-
| 108
| 815.0943396
| Km6, A5
| Ptolemaic Minor Sixth, Pythagorean Augmented Fifth
| A#, Bb↑
| 8
| 10
|-
| 109
| 822.6415094
| RKm6, kn6, RA5
|Wide Minor Sixth, Artoretromean Augmented Fifth
| Bd<↓, Bb↑/, A#/
| 3
| 9
|-
| 110
| 830.1886792
| kN6, rKA5
| Lesser Supraminor Sixth, Tendoretromean Augmented Fifth
| Bd>↓, A#↑\
| -1
| 9
|-
| 111
| 837.7358491
| KKm6, rn6, KA5
| Greater Supraminor Sixth, Retroptolemaic Augmented Fifth
| Bd<\, Bb↑↑, A#↑
| 0
| 8
|-
| 112
| 845.2830189
| n6, SA5, kUA5
| Artoneutral Sixth, Lesser Super-Augmented Fifth
| Bd<, At#<↓
| -1
| 7
|-
| 113
| 852.8301887
| N6, RkUA5
| Tendoneutral Sixth, Greater Super-Augmented Fifth
| Bd>, At#>↓
| 0
| 7
|-
| 114
| 860.3773585
| kkM6, RN6, rUA5
| Lesser Submajor Sixth, Retrodiptolemaic Augmented Fifth
| Bd>/, B↓↓, At#>↓/, A#↑↑
| -1
| 8
|-
| 115
| 867.9245283
| Kn6, UA5
| Greater Submajor Sixth, Ultra-Augmented Fifth
| Bd<↑, At#<
| 1
| 9
|-
| 116
| 875.4716981
| rkM6, KN6
| Narrow Major Sixth
| Bd>↑, B↓\, At#>
| 4
| 9
|-
| 117
| 883.0188679
| kM6
| Ptolemaic Major Sixth
| B↓, Cb
| 7
| 10
|-
| 118
| 890.5660377
| RkM6
| Artomean Major Sixth
| B↓/
| 4
| 10
|-
| 119
| 898.1132075
| rM6
| Tendomean Major Sixth
| B\
| 1
| 9
|-
| 120
| 905.6603774
| M6
| Pythagorean Major Sixth
| B
| -1
| 9
|-
| 121
| 913.2075472
| RM6
| Wide Major Sixth
| B/, Cd<↓
| 0
| 8
|-
| 122
| 920.7547170
| rKM6
| Narrow Supermajor Sixth
| B↑\, Cd>↓
| -1
| 8
|-
| 123
| 928.3018868
| KM6
| Lesser Supermajor Sixth
| B↑, Cd<\, Cb↑↑, Ax
| -1
| 7
|-
| 124
| 935.8490566
| SM6, kUM6
| Greater Supermajor Second, Narrow Inframinor Seventh
| Cd<, Bt<↓, B↑/
| 0
| 7
|-
| 125
| 943.3962264
| um7, RkUM6
| Inframinor Seventh, Wide Supermajor Sixth
| Cd>, Bt>↓
| -1
| 7
|-
| 126
| 950.9433962
| KKM6, kkm7, rUM6, Rum7
| Narrow Ultramajor Sixth, Wide Inframinor Seventh, Semitwelfth
| Bt<\, Cd>/, B↑↑, C↓↓
| 0
| 8
|-
| 127
| 958.4905660
| UM6, rKum7
| Ultramajor Sixth, Narrow Subminor Seventh
| Bt<, Cd<↑
| -1
| 8
|-
| 128
| 966.0377358
| sm7, Kum7
| Lesser Subminor Seventh, Wide Ultramajor Sixth
| Bt>, Cd>↑, C↓\
| 0
| 9
|-
| 129
| 973.5849057
| km7
| Greater Subminor Seventh
| C↓, Bt>/, B#↓↓, Dbb
| -1
| 9
|-
| 130
| 981.1320755
| Rkm7
| Wide Subminor Seventh
| C↓/, Bt<↑
| -1
| 10
|-
| 131
| 988.6792458
| rm7
| Narrow Minor Seventh
| C\, Bt>↑
| -1
| 10
|-
| 132
| 996.2264151
| m7
| Pythagorean Minor Seventh
| C, B#↓
| -2
| 10
|-
| 133
| 1003.7735849
| Rm7
| Artomean Minor Seventh
| C/, B#↓/
| -2
| 10
|-
| 134
| 1011.3207547
| rKm7
| Tendomean Minor Seventh
| C↑\, B#\
| -3
| 10
|-
| 135
| 1018.8679245
| kM2
| Ptolemaic Minor Seventh
| C↑, B#
| -3
| 10
|-
| 136
| 1026.4150943
| RKm7, kn7
| Wide Minor Seventh
| Ct<↓, C↑/, Ddb<, B#/
| -4
| 10
|-
| 137
| 1033.9622642
| kN7, ud8
| Lesser Supraminor Seventh, Infra-Diminished Octave
| Ct>↓, Ddb>, B#↑\
| -5
| 9
|-
| 138
| 1041.5094340
| KKm7, rn7, Rud8
| Greater Supraminor Seventh, Retrodiptolemaic Diminished Octave
| Ct<\, C↑↑, Ddb<↑\, Db↓↓
| -6
| 8
|-
| 139
| 1049.0566038
| n7, rKud8
| Artoneutral Seventh, Lesser Sub-Diminished Octave
| Ct<, Ddb<↑
| -7
| 6
|-
| 140
| 1056.6037736
| N7, sd8
| Tendoneutral Seventh, Greater Sub-Diminished Octave
| Ct>, Ddb>↑
| -8
| 5
|-
| 141
| 1064.1509434
| kkM7, RN7, kd8
| Lesser Submajor Seventh, Diptolemaic Major Seventh, Retroptolemaic Diminished Octave
| Ct>/, C#↓↓, Db↓
| -7
| 6
|-
| 142
| 1071.6981132
| Kn7, Rkd8
| Greater Submajor Seventh, Artoretromean Diminished Octave
| Ct<↑, Db↓/
| -6
| 8
|-
| 143
| 1079.2452830
| rkM7, KN7, rd8
| Narrow Major Seventh, Tendoretromean Diminished Octave
| Ct>↑, C#↓\, Db\
| -5
| 9
|-
| 144
| 1086.7924528
| kM7, d8
| Ptolemaic Major Seventh, Pythagorean Diminished Octave
| Db, C#↓
| -5
| 10
|-
| 145
| 1094.3396226
| RkM7, Rd8
| Artomean Major Seventh, Artomean Diminished Octave
| Db/, C#↓/
| -5
| 10
|-
| 146
| 1101.8867925
| rM7, rKd8
| Tendomean Major Seventh, Tendomean Diminished Octave
| C#\, Db↑\
| -6
| 10
|-
| 147
| 1109.4339622
| M7, Kd8
| Pythagorean Major Seventh, Ptolemaic Diminished Octave
| C#, Db↑
| -6
| 10
|-
| 148
| 1116.9811321
| RM7, kUd8
| Wide Major Seventh, Lesser Super-Diminished Octave
| C#/, Dd<↓
| -7
| 9
|-
| 149
| 1124.5283019
| rKM7, RkUd8
| Narrow Supermajor Seventh, Greater Super-Diminished Octave
| C#↑\, Dd>↓
| -7
| 9
|-
| 150
| 1132.0754717
| km2, RuA1, kkA1
| Lesser Supermajor Seventh, Diptolemaic Diminished Octave
| C#↑, Db↑↑
| -8
| 9
|-
| 151
| 1139.6226415
| SM7, kUM7, Ud8
| Greater Supermajor Seventh, Narrow Infraoctave, Ultra-Diminished Octave
| Dd<, C#↑/
| -8
| 10
|-
| 152
| 1147.1698113
| u8, RkUM7
| Infraoctave, Wide Supermajor Seventh
| Dd>, Ct#>↓
| -9
| 10
|-
| 153
| 1154.7169811
| KKM7, rUM7, Ru8
| Narrow Ultramajor Seventh, Wide Infraoctave
| C#↑↑, Dd>/
| -9
| 10
|-
| 154
| 1162.2641509
| UM7, rKu8
| Ultramajor Seventh, Wide Superprime
| Ct#<, Dd<↑
| -9
| 10
|-
| 155
| 1169.8113208
| s8, Ku8
| Lesser Suboctave, Wide Ultramajor Seventh
| Ct#>, Dd>↑
| -10
| 3
|-
| 156
| 1177.3584906
| k8
| Greater Suboctave
| D↓
| -10
| -3
|-
| 157
| 1184.9056604
| Rk8
| Wide Suboctave
| D↓/
| -10
| -10
|-
| 158
| 1192.4528302
| r8
| Narrow Octave
| D\
| 0
| 0
|-
| 159
| 1200
| P8
| Perfect Octave
| D
| 10
| 10
|}

== Trines ==
159edo has multiple types of trine. Trines are important in the aspects of 159edo music theory derived from Medieval and Neo-Medieval music theory. In fact, the individual intervals that constitute trines serve as the backbone of not only the triads of harmony, but the tetrachords of melody as well. Both triads and tetrachords will be covered in later installments of this series. For now, it pays to go over which three-note structures can serve as trines as well as their names.

{| class="wikitable"
|+Table of 159edo Trines
|-
! Name
! Notation (from D)
! Steps
! Approximate JI
! Notes
|-
| Otonal Perfect
| D, A, D
| 0, 93, 0
| 2:3:4
| This is the first of two trines that can be considered fully-resolved in Medieval and Neo-Medieval harmony
|-
| Utonal Perfect
| D, G, D
| 0, 66, 0
| 1/(2:3:4)
| This is the second of two trines that can be considered fully-resolved in Medieval and Neo-Medieval harmony
|-
| Otonal Archagall
| D, G\, D
| 0, 65, 0
| 64:85:128
| This trine is the first of two that are often used in the extended harmony of t<IV chords and is considered a dissonance
|-
| Utonal Archagall
| D, A/, D
| 0, 94, 0
| 1/(64:85:128)
| This trine is the second of two that are often used in the extended harmony of t<IV chords and is considered a dissonance
|-
| Bass-Up Marvelous
| D, A\, D
| 0, 92, 0
| 75:112:150
| This dissonant trine is the first of two that are formed from stacking identical approximations of the LCJI neutral third
|-
| Treble-Down Marvelous
| D, G/, D
| 0, 67, 0
| 1/(75:112:150)
| This dissonant trine is the second of two that are formed from stacking identical approximations of the LCJI neutral third
|-
| Narrow Supernaiadic
| D, G↓\, D
| 0, 62, 0
| 16:21:32
| This dissonant trine is common in essentially tempered chords
|-
| Wide Subcocytic
| D, A↑/, D
| 0, 97, 0
| 1/(16:21:32)
| This dissonant trine is common in essentially tempered chords
|-
| Subcocytic
| D, A↑, D
| 0, 96, 0
| 160:243:320
| This dissonant trine is common in essentially tempered chords
|-
| Supernaiadic
| D, G↓, D
| 0, 63, 0
| 1/(160:243:320)
| This dissonant trine is common in essentially tempered chords
|-
| Wide Supernaiadic
| D, G↓/, D
| 0, 64, 0
| 25:33:50
| This dissonant trine is on the outer edge of the diatonic range and is common in essentially tempered chords
|-
| Narrow Subcocytic
| D, A↑\, D
| 0, 95, 0
| 1/(25:33:50)
| This dissonant trine is on the outer edge of the diatonic range and is common in essentially tempered chords
|-
| Wide Naiadic
| D, Gd<↑, D
| 0, 61, 0
| 135:176:270
| This dissonant trine is among the more consistently complex
|-
| Narrow Cocytic
| D, At>↓, D
| 0, 98, 0
| 1/(135:176:270)
| This dissonant trine is among the more consistently complex
|-
| Naiadic
| D, Gd>/, D
| 0, 60, 0
| 10:13:20
| This dissonant trine is relatively simple and thus expected to be rather common
|-
| Cocytic
| D, At<\, D
| 0, 99, 0
| 1/(10:13:20)
| This dissonant trine is relatively simple and thus expected to be rather common
|-
| Wide Cocytic
| D, At<, D
| 0, 100, 0
| 11:17:22
| This essentially tempered trine is very likely to be used as a basis for cocytic triads
|-
| Narrow Niadic
| D, Gd>, D
| 0, 59, 0
| 1/(11:17:22)
| This essentially tempered trine is very likely to be used as a partial basis for suspended chords
|-
| Narrow Superdusthumic
| D, Ad<↑, D
| 0, 89, 0
| 128:189:256
| This dissonant trine is common in essentially tempered chords
|-
| Wide Subagallic
| D, Gt>↓, D
| 0, 70, 0
| 1/(128:189:256)
| This dissonant trine is common in essentially tempered chords
|-
| Subagallic
| D, G↑, D
| 0, 69, 0
| 20:27:40
| This dissonant trine is very likely to show up in non-meantone diatonic contexts
|-
| Superdusthumic
| D, A↓, D
| 0, 90, 0
| 1/(20:27:40)
| This dissonant trine is very likely to show up in non-meantone diatonic contexts
|-
| Narrow Subagallic
| D, G↑\, D
| 0, 68, 0
| 90:121:180
| This dissonant trine is on the outer edge of the diatonic range
|-
| Wide Superdusthumic
| D, A↓/, D
| 0, 91, 0
| 1/(90:121:180)
| This dissonant trine is on the outer edge of the diatonic range
|-
| Wide Agallic
| D, Gt<, D
| 0, 73, 0
| 8:11:16
| This ambisonant trine is very likely to be used as a partial basis for suspended chords
|-
| Narrow Dusthumic
| D, Ad>, D
| 0, 86, 0
| 1/(8:11:16)
| This ambisonant trine is very likely to be used as a basis for dusthumic triads
|-
| Dusthumic
| D, Ad<\, D
| 0, 87, 0
| 128:187:256
| This dissonant trine is common in essentially tempered chords
|-
| Agallic
| D, Gt<\, D
| 0, 72, 0
| 1/(128:187:256)
| This dissonant trine is common in essentially tempered chords
|-
| Narrow Agallic
| D, Gt>↓, D
| 0, 71, 0
| 11:15:22
| This trine is very likely to be used as a partial basis for suspended chords
|-
| Wide Dusthumic
| D, Ad<↑, D
| 0, 88, 0
| 1/(11:15:22)
| This trine is very likely to be used as a basis for dusthumic triads
|-
| Wide Subdusthumic
| D, Ad<, D
| 0, 85, 0
| 56:81:112
| This essentially tempered trine is likely to be used as a basis for subdusthumic triads
|-
| Narrow Superagallic
| D, Gt>, D
| 0, 74, 0
| 1/(56:81:112)
| This essentially tempered trine is likely to be used as a partial basis for suspended chords
|-
| Subdusthumic
| D, Ab↑↑, D
| 0, 84, 0
| 9:13:18
| This essentially tempered trine is very likely to be used as a basis for subdusthumic triads
|-
| Superagallic
| D, G#↓↓, D
| 0, 75, 0
| 1/(9:13:18)
| This essentially tempered trine is very likely to be used as a partial basis for suspended chords
|-
| Wide Superagallic
| D, Gt<↑, D
| 0, 76, 0
| 256:357:512
| This essentially tempered trine is very likely to be used as a partial basis for suspended chords
|-
| Narrow Subdusthumic
| D, Ad>↓, D
| 0, 83, 0
| 1/(256:357:512)
| This essentially tempered trine is very likely to be used as a basis for subdusthumic triads
|-
| Narrow Hyperquartal
| D, Gt>↑, D
| 0, 77, 0
| 5:7:10
| This ambisonant trine is very common as a basis for diminished chords, and is very likely to be used as a partial basis for suspended chords
|-
| Wide Hypoquintal
| D, Ad<↓, D
| 0, 82, 0
| 1/(5:7:10)
| This ambisonant trine is very common as a basis for diminished chords, and is very likely to be used as a partial basis for suspended chords
|-
| Hyperquartal
| D, G#↓, D
| 0, 78, 0
| 32:45:64
| This trine is very likely to be used as a partial basis for suspended chords
|-
| Hypoquintal
| D, Ab↑, D
| 0, 81, 0
| 1/(32:45:64)
| This trine is very common as a basis for diminished chords
|-
| Narrow Hypoquintal
| D, Ab↑\, D
| 0, 80, 0
| 12:17:24
| This trine is very common as a basis for diminished chords
|-
| Wide Hyperquartal
| D, G#↓/, D
| 0, 79, 0
| 1/(12:17:24)
| This trine is very likely to be used as a partial basis for suspended chords
|}

User:Aura/On 159edo Music Theory (Part 1)

2026-03-31T00:24:59Z

Aura: Will have to remove the tetrachord chart for now until I figure out a way to cover them better...

Of all the multiples of [[53edo]], [[159edo]] is the lowest multiple that is noteworthy for being accurate in the 2.3.5.11.17 subgroup while having structural compromises in the 7.13.19.23.29 subgroup. Despite the number of pitches in this tuning system making it perhaps best fit for digital instruments of various kinds in actual performance, it is nevertheless also useful as an interval classification scheme.

== Intervals and Notation ==
159edo contains all the intervals of 53edo and can be thought of as having three fields of 53edo each separated by a third of 53edo's step. However, as some of the interpretations differ due 159edo having different mappings for certain primes, those differences show up in how harmonies are constructed.

159edo has its own variation on the [[dinner party rules]]— represented here by the Harmonic Compatibility Rating and Melodic Compatibility Rating columns in the following chart, where 10 is a full-blown friend relative to the root and −10 if a full-blown enemy relative to the root. Note that the Harmonic Compatibility and Melodic Compatibility ratings are based on octave-equivalence, and that some of the ratings are still speculative.

{| class="wikitable"
|+159edo Interval Names and Compatibility Ratings
! rowspan="2" | Step
! rowspan="2" | Cents
! colspan="3" | Interval and Note names
! colspan="2" | Compatibility rating
|-
! SKULO-based interval names
! Pythagorean-commatic-based interval names
! SRS notation
! Harmonic
! Melodic
|-
| 0
| 0
| P1
| Perfect Unison
| D
| 10
| 10
|-
| 1
| 7.5471698
| R1
| Wide Unison
| D/
| 0
| 0
|-
| 2
| 15.0943396
| rK1
| Narrow Superunison
| D↑\
| -10
| -10
|-
| 3
| 22.6415094
| K1
| Lesser Superunison
| D↑
| -10
| -3
|-
| 4
| 30.1886792
| S1, kU1
| Greater Superunison, Narrow Inframinor Second
| Edb<, Dt<↓
| -10
| 3
|-
| 5
| 37.7358491
| um2, RkU1
| Inframinor Second, Wide Superunison
| Edb>, Dt>↓
| -9
| 10
|-
| 6
| 45.2830189
| kkm2, Rum2, rU1
| Wide Inframinor Second, Narrow Ultraunison
| Eb↓↓, Dt<\
| -9
| 10
|-
| 7
| 52.8301887
| U1, rKum2
| Ultraunison, Narrow Subminor Second
| Dt<, Edb<↑
| -9
| 10
|-
| 8
| 60.3773585
| sm2, Kum2, uA1
| Lesser Subminor Second, Wide Ultraunison, Infra-Augmented Unison
| Dt>, Eb↓\
| -8
| 10
|-
| 9
| 67.9245283
| km2, RuA1, kkA1
| Greater Subminor Second, Diptolemaic Augmented Unison
| Eb↓, D#↓↓
| -8
| 9
|-
| 10
| 75.4716981
| Rkm2, rKuA1
| Wide Subminor Second, Lesser Sub-Augmented Unison
| Eb↓/, Dt<↑
| -7
| 9
|-
| 11
| 83.0188679
| rm2, KuA1
| Narrow Minor Second, Greater Sub-Augmented Unison
| Eb\, Dt>↑
| -7
| 9
|-
| 12
| 90.5660377
| m2, kA1
| Pythagorean Minor Second, Ptolemaic Augmented Unison
| Eb, D#↓
| -6
| 10
|-
| 13
| 98.1132075
| Rm2, RkA1
| Artomean Minor Second, Artomean Augmented Unison
| Eb/, D#↓/
| -6
| 10
|-
| 14
| 105.6603774
| rKm2, rA1
| Tendomean Minor Second, Tendomean Augmented Unison
| D#\, Eb↑\
| -5
| 10
|-
| 15
| 113.2075472
| Km2, A1
| Ptolemaic Minor Second, Pythagorean Augmented Unison
| D#, Eb↑
| -5
| 10
|-
| 16
| 120.7547170
| RKm2, kn2, RA1
| Wide Minor Second, Artoretromean Augmented Unison
| Ed<↓, Eb↑/, D#/
| -5
| 9
|-
| 17
| 128.3018868
| kN2, rKA1
| Lesser Supraminor Second, Tendoretromean Augmented Unison
| Ed>↓, D#↑\
| -6
| 8
|-
| 18
| 135.8490566
| KKm2, rn2, KA1
| Greater Supraminor Second, Diptolemaic Limma, Retroptolemaic Augmented Unison
| Ed<\, Eb↑↑, D#↑
| -7
| 6
|-
| 19
| 143.3962264
| n2, SA1
| Artoneutral Second, Lesser Super-Augmented Unison
| Ed<, Dt#<↓
| -8
| 5
|-
| 20
| 150.9433962
| N2, RkUA1
| Tendoneutral Second, Greater Super-Augmented Unison
| Ed>, Dt#>↓
| -7
| 6
|-
| 21
| 158.4905660
| kkM2, RN2, rUA1
| Lesser Submajor Second, Retrodiptolemaic Augmented Unison
| Ed>/, E↓↓, Dt#>↓/, D#↑↑
| -6
| 8
|-
| 22
| 166.0377358
| Kn2, UA1
| Greater Submajor Second, Ultra-Augmented Unison
| Ed<↑, Dt#<, Fb↓/
| -5
| 9
|-
| 23
| 173.5849057
| rkM2, KN2
| Narrow Major Second
| Ed>↑, E↓\, Dt#>, Fb\
| -4
| 10
|-
| 24
| 181.1320755
| kM2
| Ptolemaic Major Second
| E↓, Fb
| -3
| 10
|-
| 25
| 188.6792458
| RkM2
| Artomean Major Second
| E↓/, Fb/
| -3
| 10
|-
| 26
| 196.2264151
| rM2
| Tendomean Major Second
| E\, Fb↑\
| -2
| 10
|-
| 27
| 203.7735849
| M2
| Pythagorean Major Second
| E, Fb↑
| -2
| 10
|-
| 28
| 211.3207547
| RM2
| Wide Major Second
| E/, Fd<↓
| -1
| 10
|-
| 29
| 218.8679245
| rKM2
| Narrow Supermajor Second
| E↑\, Fd>↓
| -1
| 10
|-
| 30
| 226.4150943
| KM2
| Lesser Supermajor Second
| E↑, Fd<\, Fb↑↑, Dx
| -1
| 9
|-
| 31
| 233.9622642
| SM2, kUM2
| Greater Supermajor Second, Narrow Inframinor Third
| Fd<, Et<↓, E↑/
| 0
| 9
|-
| 32
| 241.5094340
| um3, RkUM2
| Inframinor Third, Wide Supermajor Second
| Fd>, Et>↓
| -1
| 8
|-
| 33
| 249.0566038
| kkm3, KKM2, Rum3, rUM2
| Wide Inframinor Third, Narrow Ultramajor Second, Semifourth
| Fd>/, Et<\, F↓↓, E↑↑
| 0
| 8
|-
| 34
| 256.6037736
| UM2, rKum3
| Ultramajor Second, Narrow Subminor Third
| Et<, Fd<↑
| -1
| 7
|-
| 35
| 264.1509434
| sm3, Kum3
| Lesser Subminor Third, Wide Ultramajor Second
| Et>, Fd>↑, F↓\
| 0
| 7
|-
| 36
| 271.6981132
| km3
| Greater Subminor Third
| F↓, Et>/, E#↓↓, Gbb
| -1
| 7
|-
| 37
| 279.2452830
| Rkm3
| Wide Subminor Third
| F↓/, Et<↑
| -1
| 8
|-
| 38
| 286.7924528
| rm3
| Narrow Minor Third
| F\, Et>↑
| 0
| 8
|-
| 39
| 294.3396226
| m3
| Pythagorean Minor Third
| F
| -1
| 9
|-
| 40
| 301.8867925
| Rm3
| Artomean Minor Third
| F/
| 1
| 9
|-
| 41
| 309.4339622
| rKm3
| Tendomean Minor Third
| F↑\
| 4
| 10
|-
| 42
| 316.9811321
| Km3
| Ptolemaic Minor Third
| F↑, E#
| 7
| 10
|-
| 43
| 324.5283019
| RKm3, kn3
| Wide Minor Third
| Ft<↓, F↑/, Gdb<
| 4
| 9
|-
| 44
| 332.0754717
| kN3, ud4
| Lesser Supraminor Third, Infra-Diminished Fourth
| Ft>↓, Gdb>
| 1
| 9
|-
| 45
| 339.6226415
| KKm3, rn3, Rud4
| Greater Supraminor Third, Retrodiptolemaic Diminished Fourth
| Ft<\, F↑↑, Gdb<↑\, Gb↓↓
| -1
| 8
|-
| 46
| 347.1698113
| n3, rKud4
| Artoneutral Third, Lesser Sub-Diminished Fourth
| Ft<, Gdb<↑
| 0
| 7
|-
| 47
| 354.7169811
| N3, sd4, Kud4
| Tendoneutral Third, Greater Sub-Diminished Fourth
| Ft>, Gdb>↑
| -1
| 7
|-
| 48
| 362.2641509
| kkM3, RN3, kd4
| Lesser Submajor Third, Retroptolemaic Diminished Fourth
| Ft>/, F#↓↓, Gb↓
| 0
| 8
|-
| 49
| 369.8113208
| Kn3, Rkd4
| Greater Submajor Third, Artoretromean Diminished Fourth
| Ft<↑, Gb↓/
| -1
| 9
|-
| 50
| 377.3584906
| rkM3, KN3, rd4
| Narrow Major Third, Tendoretromean Diminished Fourth
| Ft>↑, F#↓\, Gb\
| 3
| 9
|-
| 51
| 384.9056604
| kM3, d4
| Ptolemaic Major Third, Pythagorean Diminished Fourth
| Gb, F#↓
| 8
| 10
|-
| 52
| 392.4528302
| RkM3, Rd4
| Artomean Major Third, Artomean Diminished Fourth
| Gb/, F#↓/
| 4
| 10
|-
| 53
| 400
| rM3, rKd4
| Tendomean Major Third, Tendomean Diminished Fourth
| F#\, Gb↑\
| 1
| 9
|-
| 54
| 407.5471698
| M3, Kd4
| Pythagorean Major Third, Ptolemaic Diminished Fourth
| F#, Gb↑
| -1
| 9
|-
| 55
| 415.0943396
| RM3, kUd4
| Wide Major Third, Lesser Super-Diminished Fourth
| F#/, Gd<↓, Gb↑/
| 0
| 8
|-
| 56
| 422.6415094
| rKM3, RkUd4
| Narrow Supermajor Third, Greater Super-Diminished Fourth
| F#↑\, Gd>↓
| -1
| 7
|-
| 57
| 430.1886792
| KM3, rUd4, KKd4
| Lesser Supermajor Third, Diptolemaic Diminished Fourth
| F#↑, Gd<\, Gb↑↑
| -1
| 6
|-
| 58
| 437.7358491
| SM3, kUM3, rm4, Ud4
| Greater Supermajor Third, Ultra-Diminished Fourth
| Gd<, F#↑/
| 0
| 5
|-
| 59
| 445.2830189
| m4, RkUM3
| Paraminor Fourth, Wide Supermajor Third
| Gd>, Ft#>↓
| -1
| 3
|-
| 60
| 452.8301887
| Rm4, KKM3, rUM3
| Wide Paraminor Fourth, Narrow Ultramajor Third
| Gd>/, F#↑↑, G↓↓
| -2
| 1
|-
| 61
| 460.3773585
| UM3, rKm4
| Ultramajor Third, Narrow Grave Fourth
| Gd<↑, Ft#<
| -4
| -2
|-
| 62
| 467.9245283
| s4, Km4
| Lesser Grave Fourth, Wide Ultramajor Third
| Gd>↑, G↓\
| -7
| -4
|-
| 63
| 475.4716981
| k4
| Greater Grave Fourth
| G↓, Abb
| -6
| -5
|-
| 64
| 483.0188679
| Rk4
| Wide Grave Fourth
| G↓/
| -4
| 0
|-
| 65
| 490.5660377
| r4
| Narrow Fourth
| G\
| 1
| 5
|-
| 66
| 498.1132075
| P4
| Perfect Fourth
| G
| 9
| 10
|-
| 67
| 505.6603774
| R4
| Wide Fourth
| G/
| 1
| 8
|-
| 68
| 513.2075472
| rK4
| Narrow Acute Fourth
| G↑\
| -3
| 6
|-
| 69
| 520.7547170
| K4
| Lesser Acute Fourth
| G↑
| -5
| 5
|-
| 70
| 528.3018868
| S4, kM4
| Greater Acute Fourth
| Gt<↓, G↑/, Adb<
| -3
| 5
|-
| 71
| 535.8490566
| RkM4, ud5
| Wide Acute Fourth, Infra-Diminished Fifth
| Gt>↓, Adb>
| -2
| 5
|-
| 72
| 543.3962264
| rM4, Rud5
| Narrow Paramajor Fourth, Retrodiptolemaic Diminished Fifth
| Gt<\, G↑↑, Ab↓↓
| -1
| 6
|-
| 73
| 550.9433962
| M4, rKud5
| Paramajor Fourth, Lesser Sub-Diminished Fifth
| Gt<, Adb<↑
| 0
| 7
|-
| 74
| 558.4905660
| RM4, uA4, Kud5
| Infra-Augmented Fourth, Greater Sub-Diminished Fifth
| Gt>, Adb>↑
| -2
| 5
|-
| 75
| 566.0377358
| kkA4, RuA4, kd5
| Diptolemaic Augmented Fourth, Retroptolemaic Diminished Fifth
| Gt>/, G#↓↓, Ab↓
| -3
| 4
|-
| 76
| 573.5849057
| rKuA4, Rkd5
| Lesser Sub-Augmented Fourth, Artoretromean Diminished Fifth
| Gt<↑, Ab↓/
| -2
| 4
|-
| 77
| 581.1320755
| KuA4, rd5
| Greater Sub-Augmented Fourth, Tendoretromean Diminished Fifth
| Gt>↑, Ab\
| 0
| 5
|-
| 78
| 588.6792458
| kA4, d5
| Ptolemaic Augmented Fourth, Pythagorean Diminished Fifth
| Ab, G#↓
| -5
| 6
|-
| 79
| 596.2264151
| RkA4, Rd5
| Artomean Augmented Fourth, Artomean Diminished Fifth
| G#↓/, Ab/
| -9
| 7
|-
| 80
| 603.7735849
| rKd5, rA4
| Tendomean Diminished Fifth, Tendomean Augmented Fourth
| Ab↑\, G#\
| -9
| 7
|-
| 81
| 611.3207547
| Kd5, A4
| Ptolemaic Diminished Fifth, Pythagorean Augmented Fourth
| Ab↑, G#
| -5
| 6
|-
| 82
| 618.8679245
| kUd5, RA4
| Lesser Super-Diminished Fifth, Artoretromean Augmented Fourth
| Ad<↓, G#/
| 0
| 5
|-
| 83
| 626.4150943
| RkUd5, rKA4
| Greater Super-Diminished Fifth, Tendoretromean Augmented Fourth
| Ad>↓, G#↑\
| -2
| 4
|-
| 84
| 633.9622642
| KKd5, rUDd5, KA4
| Diptolemaic Diminished Fifth, Retroptolemaic Augmented Fourth
| Ad<\, Ab↑↑, G#↑
| -3
| 4
|-
| 85
| 641.5094340
| rm5, Ud5, kUA4
| Ultra-Diminished Fifth, Lesser Super-Augmented Fourth
| Ad<, Gt#<↓
| -2
| 5
|-
| 86
| 649.0566038
| m5, RkUA4
| Paraminor Fifth, Greater Super-Augmented Fourth
| Ad>, Gt#>↓
| 0
| 7
|-
| 87
| 656.6037736
| Rm5, rUA4
| Wide Paraminor Fifth, Retrodiptolemaic Augmented Fourth
| Ad>/, G#↑, Ab↑↑
| -1
| 6
|-
| 88
| 664.1509434
| rKm5, UA4
| Narrow Grave Fifth, Ultra-Augmented Fourth
| Ad<↑, Gt#<
| -2
| 5
|-
| 89
| 671.6981132
| s5, Km5
| Lesser Grave Fifth
| Ad>↑, A↓\, Gt#>
| -3
| 5
|-
| 90
| 679.2452830
| k5
| Greater Grave Fifth
| A↓
| -5
| 5
|-
| 91
| 686.7924528
| Rk5
| Wide Grave Fifth
| A↓/
| -3
| 6
|-
| 92
| 694.3396226
| r5
| Narrow Fifth
| A\
| 1
| 8
|-
| 93
| 701.8867925
| P5
| Perfect Fifth
| A
| 9
| 10
|-
| 94
| 709.4339622
| R5
| Wide Fifth
| A/
| 1
| 5
|-
| 95
| 716.9811321
| rK5
| Narrow Acute Fifth
| A↑\
| -4
| 0
|-
| 96
| 724.5283019
| K5
| Lesser Acute Fifth
| A↑, Gx
| -6
| -5
|-
| 97
| 732.0754717
| S5, kM5
| Greater Acute Fifth, Narrow Inframinor Sixth
| At<↓, A↑/
| -7
| -4
|-
| 98
| 739.6226415
| um6, RkM5
| Inframinor Sixth, Wide Acute Fifth
| At>↓, Bdb>
| -4
| -2
|-
| 99
| 747.1698113
| Rm4, KKM3, rUM3
| Narrow Paramajor Fifth, Wide Inframinor Sixth
| At<\, Bb↓↓, A↑↑
| -2
| 1
|-
| 100
| 754.7169811
| M5, rKum6
| Paramajor Fifth, Narrow Subminor Sixth
| At<, Bdb<↑
| -1
| 3
|-
| 101
| 762.2641509
| sm6, Kum6, RM5, uA5
| Lesser Subminor Sixth, Infra-Augmented Fifth
| At>, Bb↓\
| 0
| 5
|-
| 102
| 769.8113208
| km6, RuA5, kkA5
| Greater Subminor Sixth, Diptolemaic Augmented Fifth
| Bb↓, At>/, A#↓↓
| -1
| 6
|-
| 103
| 777.3584906
| Rkm6, rKuA5
| Wide Subminor Sixth, Lesser Sub-Augmented Fifth
| Bb↓/, At<↑
| -1
| 7
|-
| 104
| 784.9056604
| rm6, KuA5
| Narrow Minor Sixth, Greater Sub-Augmented Fifth
| Bb\, At>↑, A#↓\
| 0
| 8
|-
| 105
| 792.4528302
| m6, kA5
| Pythagorean Minor Sixth, Ptolemaic Augmented Fifth
| Bb, A#↓
| -1
| 9
|-
| 106
| 800
| Rm6, RkA5
| Artomean Minor Sixth, Artomean Augmented Fifth
| Bb/, A#↓/
| 1
| 9
|-
| 107
| 807.5471698
| rKm6, rA5
| Tendomean Minor Sixth, Tendomean Augmented Fifth
| A#\, Bb↑\
| 4
| 10
|-
| 108
| 815.0943396
| Km6, A5
| Ptolemaic Minor Sixth, Pythagorean Augmented Fifth
| A#, Bb↑
| 8
| 10
|-
| 109
| 822.6415094
| RKm6, kn6, RA5
|Wide Minor Sixth, Artoretromean Augmented Fifth
| Bd<↓, Bb↑/, A#/
| 3
| 9
|-
| 110
| 830.1886792
| kN6, rKA5
| Lesser Supraminor Sixth, Tendoretromean Augmented Fifth
| Bd>↓, A#↑\
| -1
| 9
|-
| 111
| 837.7358491
| KKm6, rn6, KA5
| Greater Supraminor Sixth, Retroptolemaic Augmented Fifth
| Bd<\, Bb↑↑, A#↑
| 0
| 8
|-
| 112
| 845.2830189
| n6, SA5, kUA5
| Artoneutral Sixth, Lesser Super-Augmented Fifth
| Bd<, At#<↓
| -1
| 7
|-
| 113
| 852.8301887
| N6, RkUA5
| Tendoneutral Sixth, Greater Super-Augmented Fifth
| Bd>, At#>↓
| 0
| 7
|-
| 114
| 860.3773585
| kkM6, RN6, rUA5
| Lesser Submajor Sixth, Retrodiptolemaic Augmented Fifth
| Bd>/, B↓↓, At#>↓/, A#↑↑
| -1
| 8
|-
| 115
| 867.9245283
| Kn6, UA5
| Greater Submajor Sixth, Ultra-Augmented Fifth
| Bd<↑, At#<
| 1
| 9
|-
| 116
| 875.4716981
| rkM6, KN6
| Narrow Major Sixth
| Bd>↑, B↓\, At#>
| 4
| 9
|-
| 117
| 883.0188679
| kM6
| Ptolemaic Major Sixth
| B↓, Cb
| 7
| 10
|-
| 118
| 890.5660377
| RkM6
| Artomean Major Sixth
| B↓/
| 4
| 10
|-
| 119
| 898.1132075
| rM6
| Tendomean Major Sixth
| B\
| 1
| 9
|-
| 120
| 905.6603774
| M6
| Pythagorean Major Sixth
| B
| -1
| 9
|-
| 121
| 913.2075472
| RM6
| Wide Major Sixth
| B/, Cd<↓
| 0
| 8
|-
| 122
| 920.7547170
| rKM6
| Narrow Supermajor Sixth
| B↑\, Cd>↓
| -1
| 8
|-
| 123
| 928.3018868
| KM6
| Lesser Supermajor Sixth
| B↑, Cd<\, Cb↑↑, Ax
| -1
| 7
|-
| 124
| 935.8490566
| SM6, kUM6
| Greater Supermajor Second, Narrow Inframinor Seventh
| Cd<, Bt<↓, B↑/
| 0
| 7
|-
| 125
| 943.3962264
| um7, RkUM6
| Inframinor Seventh, Wide Supermajor Sixth
| Cd>, Bt>↓
| -1
| 7
|-
| 126
| 950.9433962
| KKM6, kkm7, rUM6, Rum7
| Narrow Ultramajor Sixth, Wide Inframinor Seventh, Semitwelfth
| Bt<\, Cd>/, B↑↑, C↓↓
| 0
| 8
|-
| 127
| 958.4905660
| UM6, rKum7
| Ultramajor Sixth, Narrow Subminor Seventh
| Bt<, Cd<↑
| -1
| 8
|-
| 128
| 966.0377358
| sm7, Kum7
| Lesser Subminor Seventh, Wide Ultramajor Sixth
| Bt>, Cd>↑, C↓\
| 0
| 9
|-
| 129
| 973.5849057
| km7
| Greater Subminor Seventh
| C↓, Bt>/, B#↓↓, Dbb
| -1
| 9
|-
| 130
| 981.1320755
| Rkm7
| Wide Subminor Seventh
| C↓/, Bt<↑
| -1
| 10
|-
| 131
| 988.6792458
| rm7
| Narrow Minor Seventh
| C\, Bt>↑
| -1
| 10
|-
| 132
| 996.2264151
| m7
| Pythagorean Minor Seventh
| C, B#↓
| -2
| 10
|-
| 133
| 1003.7735849
| Rm7
| Artomean Minor Seventh
| C/, B#↓/
| -2
| 10
|-
| 134
| 1011.3207547
| rKm7
| Tendomean Minor Seventh
| C↑\, B#\
| -3
| 10
|-
| 135
| 1018.8679245
| kM2
| Ptolemaic Minor Seventh
| C↑, B#
| -3
| 10
|-
| 136
| 1026.4150943
| RKm7, kn7
| Wide Minor Seventh
| Ct<↓, C↑/, Ddb<, B#/
| -4
| 10
|-
| 137
| 1033.9622642
| kN7, ud8
| Lesser Supraminor Seventh, Infra-Diminished Octave
| Ct>↓, Ddb>, B#↑\
| -5
| 9
|-
| 138
| 1041.5094340
| KKm7, rn7, Rud8
| Greater Supraminor Seventh, Retrodiptolemaic Diminished Octave
| Ct<\, C↑↑, Ddb<↑\, Db↓↓
| -6
| 8
|-
| 139
| 1049.0566038
| n7, rKud8
| Artoneutral Seventh, Lesser Sub-Diminished Octave
| Ct<, Ddb<↑
| -7
| 6
|-
| 140
| 1056.6037736
| N7, sd8
| Tendoneutral Seventh, Greater Sub-Diminished Octave
| Ct>, Ddb>↑
| -8
| 5
|-
| 141
| 1064.1509434
| kkM7, RN7, kd8
| Lesser Submajor Seventh, Diptolemaic Major Seventh, Retroptolemaic Diminished Octave
| Ct>/, C#↓↓, Db↓
| -7
| 6
|-
| 142
| 1071.6981132
| Kn7, Rkd8
| Greater Submajor Seventh, Artoretromean Diminished Octave
| Ct<↑, Db↓/
| -6
| 8
|-
| 143
| 1079.2452830
| rkM7, KN7, rd8
| Narrow Major Seventh, Tendoretromean Diminished Octave
| Ct>↑, C#↓\, Db\
| -5
| 9
|-
| 144
| 1086.7924528
| kM7, d8
| Ptolemaic Major Seventh, Pythagorean Diminished Octave
| Db, C#↓
| -5
| 10
|-
| 145
| 1094.3396226
| RkM7, Rd8
| Artomean Major Seventh, Artomean Diminished Octave
| Db/, C#↓/
| -5
| 10
|-
| 146
| 1101.8867925
| rM7, rKd8
| Tendomean Major Seventh, Tendomean Diminished Octave
| C#\, Db↑\
| -6
| 10
|-
| 147
| 1109.4339622
| M7, Kd8
| Pythagorean Major Seventh, Ptolemaic Diminished Octave
| C#, Db↑
| -6
| 10
|-
| 148
| 1116.9811321
| RM7, kUd8
| Wide Major Seventh, Lesser Super-Diminished Octave
| C#/, Dd<↓
| -7
| 9
|-
| 149
| 1124.5283019
| rKM7, RkUd8
| Narrow Supermajor Seventh, Greater Super-Diminished Octave
| C#↑\, Dd>↓
| -7
| 9
|-
| 150
| 1132.0754717
| km2, RuA1, kkA1
| Lesser Supermajor Seventh, Diptolemaic Diminished Octave
| C#↑, Db↑↑
| -8
| 9
|-
| 151
| 1139.6226415
| SM7, kUM7, Ud8
| Greater Supermajor Seventh, Narrow Infraoctave, Ultra-Diminished Octave
| Dd<, C#↑/
| -8
| 10
|-
| 152
| 1147.1698113
| u8, RkUM7
| Infraoctave, Wide Supermajor Seventh
| Dd>, Ct#>↓
| -9
| 10
|-
| 153
| 1154.7169811
| KKM7, rUM7, Ru8
| Narrow Ultramajor Seventh, Wide Infraoctave
| C#↑↑, Dd>/
| -9
| 10
|-
| 154
| 1162.2641509
| UM7, rKu8
| Ultramajor Seventh, Wide Superprime
| Ct#<, Dd<↑
| -9
| 10
|-
| 155
| 1169.8113208
| s8, Ku8
| Lesser Suboctave, Wide Ultramajor Seventh
| Ct#>, Dd>↑
| -10
| 3
|-
| 156
| 1177.3584906
| k8
| Greater Suboctave
| D↓
| -10
| -3
|-
| 157
| 1184.9056604
| Rk8
| Wide Suboctave
| D↓/
| -10
| -10
|-
| 158
| 1192.4528302
| r8
| Narrow Octave
| D\
| 0
| 0
|-
| 159
| 1200
| P8
| Perfect Octave
| D
| 10
| 10
|}

== Trines and Tetrachords ==
159edo has multiple types of trine and tetrachord. While trines are important in the aspects of 159edo music theory derived from Medieval and Neo-Medieval music theory, the concept of tetrachords is significantly older, as it can be traced back to Ancient Greece.

First, the trines shall be covered, for they serve as one of the fundamental units framing scales, melodies and harmonies.
{| class="wikitable"
|+Table of 159edo Trines
|-
! Name
! Notation (from D)
! Steps
! Approximate JI
! Notes
|-
| Otonal Perfect
| D, A, D
| 0, 93, 0
| 2:3:4
| This is the first of two trines that can be considered fully-resolved in Medieval and Neo-Medieval harmony
|-
| Utonal Perfect
| D, G, D
| 0, 66, 0
| 1/(2:3:4)
| This is the second of two trines that can be considered fully-resolved in Medieval and Neo-Medieval harmony
|-
| Otonal Archagall
| D, G\, D
| 0, 65, 0
| 64:85:128
| This trine is the first of two that are often used in the extended harmony of t<IV chords and is considered a dissonance
|-
| Utonal Archagall
| D, A/, D
| 0, 94, 0
| 1/(64:85:128)
| This trine is the second of two that are often used in the extended harmony of t<IV chords and is considered a dissonance
|-
| Bass-Up Marvelous
| D, A\, D
| 0, 92, 0
| 75:112:150
| This dissonant trine is the first of two that are formed from stacking identical approximations of the LCJI neutral third
|-
| Treble-Down Marvelous
| D, G/, D
| 0, 67, 0
| 1/(75:112:150)
| This dissonant trine is the second of two that are formed from stacking identical approximations of the LCJI neutral third
|-
| Narrow Supernaiadic
| D, G↓\, D
| 0, 62, 0
| 16:21:32
| This dissonant trine is common in essentially tempered chords
|-
| Wide Subcocytic
| D, A↑/, D
| 0, 97, 0
| 1/(16:21:32)
| This dissonant trine is common in essentially tempered chords
|-
| Subcocytic
| D, A↑, D
| 0, 96, 0
| 160:243:320
| This dissonant trine is common in essentially tempered chords
|-
| Supernaiadic
| D, G↓, D
| 0, 63, 0
| 1/(160:243:320)
| This dissonant trine is common in essentially tempered chords
|-
| Wide Supernaiadic
| D, G↓/, D
| 0, 64, 0
| 25:33:50
| This dissonant trine is on the outer edge of the diatonic range and is common in essentially tempered chords
|-
| Narrow Subcocytic
| D, A↑\, D
| 0, 95, 0
| 1/(25:33:50)
| This dissonant trine is on the outer edge of the diatonic range and is common in essentially tempered chords
|-
| Wide Naiadic
| D, Gd<↑, D
| 0, 61, 0
| 135:176:270
| This dissonant trine is among the more consistently complex
|-
| Narrow Cocytic
| D, At>↓, D
| 0, 98, 0
| 1/(135:176:270)
| This dissonant trine is among the more consistently complex
|-
| Naiadic
| D, Gd>/, D
| 0, 60, 0
| 10:13:20
| This dissonant trine is relatively simple and thus expected to be rather common
|-
| Cocytic
| D, At<\, D
| 0, 99, 0
| 1/(10:13:20)
| This dissonant trine is relatively simple and thus expected to be rather common
|-
| Wide Cocytic
| D, At<, D
| 0, 100, 0
| 11:17:22
| This essentially tempered trine is very likely to be used as a basis for cocytic triads
|-
| Narrow Niadic
| D, Gd>, D
| 0, 59, 0
| 1/(11:17:22)
| This essentially tempered trine is very likely to be used as a partial basis for suspended chords
|-
| Narrow Superdusthumic
| D, Ad<↑, D
| 0, 89, 0
| 128:189:256
| This dissonant trine is common in essentially tempered chords
|-
| Wide Subagallic
| D, Gt>↓, D
| 0, 70, 0
| 1/(128:189:256)
| This dissonant trine is common in essentially tempered chords
|-
| Subagallic
| D, G↑, D
| 0, 69, 0
| 20:27:40
| This dissonant trine is very likely to show up in non-meantone diatonic contexts
|-
| Superdusthumic
| D, A↓, D
| 0, 90, 0
| 1/(20:27:40)
| This dissonant trine is very likely to show up in non-meantone diatonic contexts
|-
| Narrow Subagallic
| D, G↑\, D
| 0, 68, 0
| 90:121:180
| This dissonant trine is on the outer edge of the diatonic range
|-
| Wide Superdusthumic
| D, A↓/, D
| 0, 91, 0
| 1/(90:121:180)
| This dissonant trine is on the outer edge of the diatonic range
|-
| Wide Agallic
| D, Gt<, D
| 0, 73, 0
| 8:11:16
| This ambisonant trine is very likely to be used as a partial basis for suspended chords
|-
| Narrow Dusthumic
| D, Ad>, D
| 0, 86, 0
| 1/(8:11:16)
| This ambisonant trine is very likely to be used as a basis for dusthumic triads
|-
| Dusthumic
| D, Ad<\, D
| 0, 87, 0
| 128:187:256
| This dissonant trine is common in essentially tempered chords
|-
| Agallic
| D, Gt<\, D
| 0, 72, 0
| 1/(128:187:256)
| This dissonant trine is common in essentially tempered chords
|-
| Narrow Agallic
| D, Gt>↓, D
| 0, 71, 0
| 11:15:22
| This trine is very likely to be used as a partial basis for suspended chords
|-
| Wide Dusthumic
| D, Ad<↑, D
| 0, 88, 0
| 1/(11:15:22)
| This trine is very likely to be used as a basis for dusthumic triads
|-
| Wide Subdusthumic
| D, Ad<, D
| 0, 85, 0
| 56:81:112
| This essentially tempered trine is likely to be used as a basis for subdusthumic triads
|-
| Narrow Superagallic
| D, Gt>, D
| 0, 74, 0
| 1/(56:81:112)
| This essentially tempered trine is likely to be used as a partial basis for suspended chords
|-
| Subdusthumic
| D, Ab↑↑, D
| 0, 84, 0
| 9:13:18
| This essentially tempered trine is very likely to be used as a basis for subdusthumic triads
|-
| Superagallic
| D, G#↓↓, D
| 0, 75, 0
| 1/(9:13:18)
| This essentially tempered trine is very likely to be used as a partial basis for suspended chords
|-
| Wide Superagallic
| D, Gt<↑, D
| 0, 76, 0
| 256:357:512
| This essentially tempered trine is very likely to be used as a partial basis for suspended chords
|-
| Narrow Subdusthumic
| D, Ad>↓, D
| 0, 83, 0
| 1/(256:357:512)
| This essentially tempered trine is very likely to be used as a basis for subdusthumic triads
|-
| Narrow Hyperquartal
| D, Gt>↑, D
| 0, 77, 0
| 5:7:10
| This ambisonant trine is very common as a basis for diminished chords, and is very likely to be used as a partial basis for suspended chords
|-
| Wide Hypoquintal
| D, Ad<↓, D
| 0, 82, 0
| 1/(5:7:10)
| This ambisonant trine is very common as a basis for diminished chords, and is very likely to be used as a partial basis for suspended chords
|-
| Hyperquartal
| D, G#↓, D
| 0, 78, 0
| 32:45:64
| This trine is very likely to be used as a partial basis for suspended chords
|-
| Hypoquintal
| D, Ab↑, D
| 0, 81, 0
| 1/(32:45:64)
| This trine is very common as a basis for diminished chords
|-
| Narrow Hypoquintal
| D, Ab↑\, D
| 0, 80, 0
| 12:17:24
| This trine is very common as a basis for diminished chords
|-
| Wide Hyperquartal
| D, G#↓/, D
| 0, 79, 0
| 1/(12:17:24)
| This trine is very likely to be used as a partial basis for suspended chords
|}

Next, the tetrachords need to be covered, for tetrachords are extremely useful in framing scales and melodies on a smaller scale than trines. Note that for tetrachords, the fourth that bounds it will be referred to by the same names used for varieties of trine in which the fourth is the first interval encountered, since tetrachords are always built by dividing a fourth.

User:Aura/On 159edo Music Theory (Part 1)

2026-03-30T23:45:45Z

Aura:

Of all the multiples of [[53edo]], [[159edo]] is the lowest multiple that is noteworthy for being accurate in the 2.3.5.11.17 subgroup while having structural compromises in the 7.13.19.23.29 subgroup. Despite the number of pitches in this tuning system making it perhaps best fit for digital instruments of various kinds in actual performance, it is nevertheless also useful as an interval classification scheme.

== Intervals and Notation ==
159edo contains all the intervals of 53edo and can be thought of as having three fields of 53edo each separated by a third of 53edo's step. However, as some of the interpretations differ due 159edo having different mappings for certain primes, those differences show up in how harmonies are constructed.

159edo has its own variation on the [[dinner party rules]]— represented here by the Harmonic Compatibility Rating and Melodic Compatibility Rating columns in the following chart, where 10 is a full-blown friend relative to the root and −10 if a full-blown enemy relative to the root. Note that the Harmonic Compatibility and Melodic Compatibility ratings are based on octave-equivalence, and that some of the ratings are still speculative.

{| class="wikitable"
|+159edo Interval Names and Compatibility Ratings
! rowspan="2" | Step
! rowspan="2" | Cents
! colspan="3" | Interval and Note names
! colspan="2" | Compatibility rating
|-
! SKULO-based interval names
! Pythagorean-commatic-based interval names
! SRS notation
! Harmonic
! Melodic
|-
| 0
| 0
| P1
| Perfect Unison
| D
| 10
| 10
|-
| 1
| 7.5471698
| R1
| Wide Unison
| D/
| 0
| 0
|-
| 2
| 15.0943396
| rK1
| Narrow Superunison
| D↑\
| -10
| -10
|-
| 3
| 22.6415094
| K1
| Lesser Superunison
| D↑
| -10
| -3
|-
| 4
| 30.1886792
| S1, kU1
| Greater Superunison, Narrow Inframinor Second
| Edb<, Dt<↓
| -10
| 3
|-
| 5
| 37.7358491
| um2, RkU1
| Inframinor Second, Wide Superunison
| Edb>, Dt>↓
| -9
| 10
|-
| 6
| 45.2830189
| kkm2, Rum2, rU1
| Wide Inframinor Second, Narrow Ultraunison
| Eb↓↓, Dt<\
| -9
| 10
|-
| 7
| 52.8301887
| U1, rKum2
| Ultraunison, Narrow Subminor Second
| Dt<, Edb<↑
| -9
| 10
|-
| 8
| 60.3773585
| sm2, Kum2, uA1
| Lesser Subminor Second, Wide Ultraunison, Infra-Augmented Unison
| Dt>, Eb↓\
| -8
| 10
|-
| 9
| 67.9245283
| km2, RuA1, kkA1
| Greater Subminor Second, Diptolemaic Augmented Unison
| Eb↓, D#↓↓
| -8
| 9
|-
| 10
| 75.4716981
| Rkm2, rKuA1
| Wide Subminor Second, Lesser Sub-Augmented Unison
| Eb↓/, Dt<↑
| -7
| 9
|-
| 11
| 83.0188679
| rm2, KuA1
| Narrow Minor Second, Greater Sub-Augmented Unison
| Eb\, Dt>↑
| -7
| 9
|-
| 12
| 90.5660377
| m2, kA1
| Pythagorean Minor Second, Ptolemaic Augmented Unison
| Eb, D#↓
| -6
| 10
|-
| 13
| 98.1132075
| Rm2, RkA1
| Artomean Minor Second, Artomean Augmented Unison
| Eb/, D#↓/
| -6
| 10
|-
| 14
| 105.6603774
| rKm2, rA1
| Tendomean Minor Second, Tendomean Augmented Unison
| D#\, Eb↑\
| -5
| 10
|-
| 15
| 113.2075472
| Km2, A1
| Ptolemaic Minor Second, Pythagorean Augmented Unison
| D#, Eb↑
| -5
| 10
|-
| 16
| 120.7547170
| RKm2, kn2, RA1
| Wide Minor Second, Artoretromean Augmented Unison
| Ed<↓, Eb↑/, D#/
| -5
| 9
|-
| 17
| 128.3018868
| kN2, rKA1
| Lesser Supraminor Second, Tendoretromean Augmented Unison
| Ed>↓, D#↑\
| -6
| 8
|-
| 18
| 135.8490566
| KKm2, rn2, KA1
| Greater Supraminor Second, Diptolemaic Limma, Retroptolemaic Augmented Unison
| Ed<\, Eb↑↑, D#↑
| -7
| 6
|-
| 19
| 143.3962264
| n2, SA1
| Artoneutral Second, Lesser Super-Augmented Unison
| Ed<, Dt#<↓
| -8
| 5
|-
| 20
| 150.9433962
| N2, RkUA1
| Tendoneutral Second, Greater Super-Augmented Unison
| Ed>, Dt#>↓
| -7
| 6
|-
| 21
| 158.4905660
| kkM2, RN2, rUA1
| Lesser Submajor Second, Retrodiptolemaic Augmented Unison
| Ed>/, E↓↓, Dt#>↓/, D#↑↑
| -6
| 8
|-
| 22
| 166.0377358
| Kn2, UA1
| Greater Submajor Second, Ultra-Augmented Unison
| Ed<↑, Dt#<, Fb↓/
| -5
| 9
|-
| 23
| 173.5849057
| rkM2, KN2
| Narrow Major Second
| Ed>↑, E↓\, Dt#>, Fb\
| -4
| 10
|-
| 24
| 181.1320755
| kM2
| Ptolemaic Major Second
| E↓, Fb
| -3
| 10
|-
| 25
| 188.6792458
| RkM2
| Artomean Major Second
| E↓/, Fb/
| -3
| 10
|-
| 26
| 196.2264151
| rM2
| Tendomean Major Second
| E\, Fb↑\
| -2
| 10
|-
| 27
| 203.7735849
| M2
| Pythagorean Major Second
| E, Fb↑
| -2
| 10
|-
| 28
| 211.3207547
| RM2
| Wide Major Second
| E/, Fd<↓
| -1
| 10
|-
| 29
| 218.8679245
| rKM2
| Narrow Supermajor Second
| E↑\, Fd>↓
| -1
| 10
|-
| 30
| 226.4150943
| KM2
| Lesser Supermajor Second
| E↑, Fd<\, Fb↑↑, Dx
| -1
| 9
|-
| 31
| 233.9622642
| SM2, kUM2
| Greater Supermajor Second, Narrow Inframinor Third
| Fd<, Et<↓, E↑/
| 0
| 9
|-
| 32
| 241.5094340
| um3, RkUM2
| Inframinor Third, Wide Supermajor Second
| Fd>, Et>↓
| -1
| 8
|-
| 33
| 249.0566038
| kkm3, KKM2, Rum3, rUM2
| Wide Inframinor Third, Narrow Ultramajor Second, Semifourth
| Fd>/, Et<\, F↓↓, E↑↑
| 0
| 8
|-
| 34
| 256.6037736
| UM2, rKum3
| Ultramajor Second, Narrow Subminor Third
| Et<, Fd<↑
| -1
| 7
|-
| 35
| 264.1509434
| sm3, Kum3
| Lesser Subminor Third, Wide Ultramajor Second
| Et>, Fd>↑, F↓\
| 0
| 7
|-
| 36
| 271.6981132
| km3
| Greater Subminor Third
| F↓, Et>/, E#↓↓, Gbb
| -1
| 7
|-
| 37
| 279.2452830
| Rkm3
| Wide Subminor Third
| F↓/, Et<↑
| -1
| 8
|-
| 38
| 286.7924528
| rm3
| Narrow Minor Third
| F\, Et>↑
| 0
| 8
|-
| 39
| 294.3396226
| m3
| Pythagorean Minor Third
| F
| -1
| 9
|-
| 40
| 301.8867925
| Rm3
| Artomean Minor Third
| F/
| 1
| 9
|-
| 41
| 309.4339622
| rKm3
| Tendomean Minor Third
| F↑\
| 4
| 10
|-
| 42
| 316.9811321
| Km3
| Ptolemaic Minor Third
| F↑, E#
| 7
| 10
|-
| 43
| 324.5283019
| RKm3, kn3
| Wide Minor Third
| Ft<↓, F↑/, Gdb<
| 4
| 9
|-
| 44
| 332.0754717
| kN3, ud4
| Lesser Supraminor Third, Infra-Diminished Fourth
| Ft>↓, Gdb>
| 1
| 9
|-
| 45
| 339.6226415
| KKm3, rn3, Rud4
| Greater Supraminor Third, Retrodiptolemaic Diminished Fourth
| Ft<\, F↑↑, Gdb<↑\, Gb↓↓
| -1
| 8
|-
| 46
| 347.1698113
| n3, rKud4
| Artoneutral Third, Lesser Sub-Diminished Fourth
| Ft<, Gdb<↑
| 0
| 7
|-
| 47
| 354.7169811
| N3, sd4, Kud4
| Tendoneutral Third, Greater Sub-Diminished Fourth
| Ft>, Gdb>↑
| -1
| 7
|-
| 48
| 362.2641509
| kkM3, RN3, kd4
| Lesser Submajor Third, Retroptolemaic Diminished Fourth
| Ft>/, F#↓↓, Gb↓
| 0
| 8
|-
| 49
| 369.8113208
| Kn3, Rkd4
| Greater Submajor Third, Artoretromean Diminished Fourth
| Ft<↑, Gb↓/
| -1
| 9
|-
| 50
| 377.3584906
| rkM3, KN3, rd4
| Narrow Major Third, Tendoretromean Diminished Fourth
| Ft>↑, F#↓\, Gb\
| 3
| 9
|-
| 51
| 384.9056604
| kM3, d4
| Ptolemaic Major Third, Pythagorean Diminished Fourth
| Gb, F#↓
| 8
| 10
|-
| 52
| 392.4528302
| RkM3, Rd4
| Artomean Major Third, Artomean Diminished Fourth
| Gb/, F#↓/
| 4
| 10
|-
| 53
| 400
| rM3, rKd4
| Tendomean Major Third, Tendomean Diminished Fourth
| F#\, Gb↑\
| 1
| 9
|-
| 54
| 407.5471698
| M3, Kd4
| Pythagorean Major Third, Ptolemaic Diminished Fourth
| F#, Gb↑
| -1
| 9
|-
| 55
| 415.0943396
| RM3, kUd4
| Wide Major Third, Lesser Super-Diminished Fourth
| F#/, Gd<↓, Gb↑/
| 0
| 8
|-
| 56
| 422.6415094
| rKM3, RkUd4
| Narrow Supermajor Third, Greater Super-Diminished Fourth
| F#↑\, Gd>↓
| -1
| 7
|-
| 57
| 430.1886792
| KM3, rUd4, KKd4
| Lesser Supermajor Third, Diptolemaic Diminished Fourth
| F#↑, Gd<\, Gb↑↑
| -1
| 6
|-
| 58
| 437.7358491
| SM3, kUM3, rm4, Ud4
| Greater Supermajor Third, Ultra-Diminished Fourth
| Gd<, F#↑/
| 0
| 5
|-
| 59
| 445.2830189
| m4, RkUM3
| Paraminor Fourth, Wide Supermajor Third
| Gd>, Ft#>↓
| -1
| 3
|-
| 60
| 452.8301887
| Rm4, KKM3, rUM3
| Wide Paraminor Fourth, Narrow Ultramajor Third
| Gd>/, F#↑↑, G↓↓
| -2
| 1
|-
| 61
| 460.3773585
| UM3, rKm4
| Ultramajor Third, Narrow Grave Fourth
| Gd<↑, Ft#<
| -4
| -2
|-
| 62
| 467.9245283
| s4, Km4
| Lesser Grave Fourth, Wide Ultramajor Third
| Gd>↑, G↓\
| -7
| -4
|-
| 63
| 475.4716981
| k4
| Greater Grave Fourth
| G↓, Abb
| -6
| -5
|-
| 64
| 483.0188679
| Rk4
| Wide Grave Fourth
| G↓/
| -4
| 0
|-
| 65
| 490.5660377
| r4
| Narrow Fourth
| G\
| 1
| 5
|-
| 66
| 498.1132075
| P4
| Perfect Fourth
| G
| 9
| 10
|-
| 67
| 505.6603774
| R4
| Wide Fourth
| G/
| 1
| 8
|-
| 68
| 513.2075472
| rK4
| Narrow Acute Fourth
| G↑\
| -3
| 6
|-
| 69
| 520.7547170
| K4
| Lesser Acute Fourth
| G↑
| -5
| 5
|-
| 70
| 528.3018868
| S4, kM4
| Greater Acute Fourth
| Gt<↓, G↑/, Adb<
| -3
| 5
|-
| 71
| 535.8490566
| RkM4, ud5
| Wide Acute Fourth, Infra-Diminished Fifth
| Gt>↓, Adb>
| -2
| 5
|-
| 72
| 543.3962264
| rM4, Rud5
| Narrow Paramajor Fourth, Retrodiptolemaic Diminished Fifth
| Gt<\, G↑↑, Ab↓↓
| -1
| 6
|-
| 73
| 550.9433962
| M4, rKud5
| Paramajor Fourth, Lesser Sub-Diminished Fifth
| Gt<, Adb<↑
| 0
| 7
|-
| 74
| 558.4905660
| RM4, uA4, Kud5
| Infra-Augmented Fourth, Greater Sub-Diminished Fifth
| Gt>, Adb>↑
| -2
| 5
|-
| 75
| 566.0377358
| kkA4, RuA4, kd5
| Diptolemaic Augmented Fourth, Retroptolemaic Diminished Fifth
| Gt>/, G#↓↓, Ab↓
| -3
| 4
|-
| 76
| 573.5849057
| rKuA4, Rkd5
| Lesser Sub-Augmented Fourth, Artoretromean Diminished Fifth
| Gt<↑, Ab↓/
| -2
| 4
|-
| 77
| 581.1320755
| KuA4, rd5
| Greater Sub-Augmented Fourth, Tendoretromean Diminished Fifth
| Gt>↑, Ab\
| 0
| 5
|-
| 78
| 588.6792458
| kA4, d5
| Ptolemaic Augmented Fourth, Pythagorean Diminished Fifth
| Ab, G#↓
| -5
| 6
|-
| 79
| 596.2264151
| RkA4, Rd5
| Artomean Augmented Fourth, Artomean Diminished Fifth
| G#↓/, Ab/
| -9
| 7
|-
| 80
| 603.7735849
| rKd5, rA4
| Tendomean Diminished Fifth, Tendomean Augmented Fourth
| Ab↑\, G#\
| -9
| 7
|-
| 81
| 611.3207547
| Kd5, A4
| Ptolemaic Diminished Fifth, Pythagorean Augmented Fourth
| Ab↑, G#
| -5
| 6
|-
| 82
| 618.8679245
| kUd5, RA4
| Lesser Super-Diminished Fifth, Artoretromean Augmented Fourth
| Ad<↓, G#/
| 0
| 5
|-
| 83
| 626.4150943
| RkUd5, rKA4
| Greater Super-Diminished Fifth, Tendoretromean Augmented Fourth
| Ad>↓, G#↑\
| -2
| 4
|-
| 84
| 633.9622642
| KKd5, rUDd5, KA4
| Diptolemaic Diminished Fifth, Retroptolemaic Augmented Fourth
| Ad<\, Ab↑↑, G#↑
| -3
| 4
|-
| 85
| 641.5094340
| rm5, Ud5, kUA4
| Ultra-Diminished Fifth, Lesser Super-Augmented Fourth
| Ad<, Gt#<↓
| -2
| 5
|-
| 86
| 649.0566038
| m5, RkUA4
| Paraminor Fifth, Greater Super-Augmented Fourth
| Ad>, Gt#>↓
| 0
| 7
|-
| 87
| 656.6037736
| Rm5, rUA4
| Wide Paraminor Fifth, Retrodiptolemaic Augmented Fourth
| Ad>/, G#↑, Ab↑↑
| -1
| 6
|-
| 88
| 664.1509434
| rKm5, UA4
| Narrow Grave Fifth, Ultra-Augmented Fourth
| Ad<↑, Gt#<
| -2
| 5
|-
| 89
| 671.6981132
| s5, Km5
| Lesser Grave Fifth
| Ad>↑, A↓\, Gt#>
| -3
| 5
|-
| 90
| 679.2452830
| k5
| Greater Grave Fifth
| A↓
| -5
| 5
|-
| 91
| 686.7924528
| Rk5
| Wide Grave Fifth
| A↓/
| -3
| 6
|-
| 92
| 694.3396226
| r5
| Narrow Fifth
| A\
| 1
| 8
|-
| 93
| 701.8867925
| P5
| Perfect Fifth
| A
| 9
| 10
|-
| 94
| 709.4339622
| R5
| Wide Fifth
| A/
| 1
| 5
|-
| 95
| 716.9811321
| rK5
| Narrow Acute Fifth
| A↑\
| -4
| 0
|-
| 96
| 724.5283019
| K5
| Lesser Acute Fifth
| A↑, Gx
| -6
| -5
|-
| 97
| 732.0754717
| S5, kM5
| Greater Acute Fifth, Narrow Inframinor Sixth
| At<↓, A↑/
| -7
| -4
|-
| 98
| 739.6226415
| um6, RkM5
| Inframinor Sixth, Wide Acute Fifth
| At>↓, Bdb>
| -4
| -2
|-
| 99
| 747.1698113
| Rm4, KKM3, rUM3
| Narrow Paramajor Fifth, Wide Inframinor Sixth
| At<\, Bb↓↓, A↑↑
| -2
| 1
|-
| 100
| 754.7169811
| M5, rKum6
| Paramajor Fifth, Narrow Subminor Sixth
| At<, Bdb<↑
| -1
| 3
|-
| 101
| 762.2641509
| sm6, Kum6, RM5, uA5
| Lesser Subminor Sixth, Infra-Augmented Fifth
| At>, Bb↓\
| 0
| 5
|-
| 102
| 769.8113208
| km6, RuA5, kkA5
| Greater Subminor Sixth, Diptolemaic Augmented Fifth
| Bb↓, At>/, A#↓↓
| -1
| 6
|-
| 103
| 777.3584906
| Rkm6, rKuA5
| Wide Subminor Sixth, Lesser Sub-Augmented Fifth
| Bb↓/, At<↑
| -1
| 7
|-
| 104
| 784.9056604
| rm6, KuA5
| Narrow Minor Sixth, Greater Sub-Augmented Fifth
| Bb\, At>↑, A#↓\
| 0
| 8
|-
| 105
| 792.4528302
| m6, kA5
| Pythagorean Minor Sixth, Ptolemaic Augmented Fifth
| Bb, A#↓
| -1
| 9
|-
| 106
| 800
| Rm6, RkA5
| Artomean Minor Sixth, Artomean Augmented Fifth
| Bb/, A#↓/
| 1
| 9
|-
| 107
| 807.5471698
| rKm6, rA5
| Tendomean Minor Sixth, Tendomean Augmented Fifth
| A#\, Bb↑\
| 4
| 10
|-
| 108
| 815.0943396
| Km6, A5
| Ptolemaic Minor Sixth, Pythagorean Augmented Fifth
| A#, Bb↑
| 8
| 10
|-
| 109
| 822.6415094
| RKm6, kn6, RA5
|Wide Minor Sixth, Artoretromean Augmented Fifth
| Bd<↓, Bb↑/, A#/
| 3
| 9
|-
| 110
| 830.1886792
| kN6, rKA5
| Lesser Supraminor Sixth, Tendoretromean Augmented Fifth
| Bd>↓, A#↑\
| -1
| 9
|-
| 111
| 837.7358491
| KKm6, rn6, KA5
| Greater Supraminor Sixth, Retroptolemaic Augmented Fifth
| Bd<\, Bb↑↑, A#↑
| 0
| 8
|-
| 112
| 845.2830189
| n6, SA5, kUA5
| Artoneutral Sixth, Lesser Super-Augmented Fifth
| Bd<, At#<↓
| -1
| 7
|-
| 113
| 852.8301887
| N6, RkUA5
| Tendoneutral Sixth, Greater Super-Augmented Fifth
| Bd>, At#>↓
| 0
| 7
|-
| 114
| 860.3773585
| kkM6, RN6, rUA5
| Lesser Submajor Sixth, Retrodiptolemaic Augmented Fifth
| Bd>/, B↓↓, At#>↓/, A#↑↑
| -1
| 8
|-
| 115
| 867.9245283
| Kn6, UA5
| Greater Submajor Sixth, Ultra-Augmented Fifth
| Bd<↑, At#<
| 1
| 9
|-
| 116
| 875.4716981
| rkM6, KN6
| Narrow Major Sixth
| Bd>↑, B↓\, At#>
| 4
| 9
|-
| 117
| 883.0188679
| kM6
| Ptolemaic Major Sixth
| B↓, Cb
| 7
| 10
|-
| 118
| 890.5660377
| RkM6
| Artomean Major Sixth
| B↓/
| 4
| 10
|-
| 119
| 898.1132075
| rM6
| Tendomean Major Sixth
| B\
| 1
| 9
|-
| 120
| 905.6603774
| M6
| Pythagorean Major Sixth
| B
| -1
| 9
|-
| 121
| 913.2075472
| RM6
| Wide Major Sixth
| B/, Cd<↓
| 0
| 8
|-
| 122
| 920.7547170
| rKM6
| Narrow Supermajor Sixth
| B↑\, Cd>↓
| -1
| 8
|-
| 123
| 928.3018868
| KM6
| Lesser Supermajor Sixth
| B↑, Cd<\, Cb↑↑, Ax
| -1
| 7
|-
| 124
| 935.8490566
| SM6, kUM6
| Greater Supermajor Second, Narrow Inframinor Seventh
| Cd<, Bt<↓, B↑/
| 0
| 7
|-
| 125
| 943.3962264
| um7, RkUM6
| Inframinor Seventh, Wide Supermajor Sixth
| Cd>, Bt>↓
| -1
| 7
|-
| 126
| 950.9433962
| KKM6, kkm7, rUM6, Rum7
| Narrow Ultramajor Sixth, Wide Inframinor Seventh, Semitwelfth
| Bt<\, Cd>/, B↑↑, C↓↓
| 0
| 8
|-
| 127
| 958.4905660
| UM6, rKum7
| Ultramajor Sixth, Narrow Subminor Seventh
| Bt<, Cd<↑
| -1
| 8
|-
| 128
| 966.0377358
| sm7, Kum7
| Lesser Subminor Seventh, Wide Ultramajor Sixth
| Bt>, Cd>↑, C↓\
| 0
| 9
|-
| 129
| 973.5849057
| km7
| Greater Subminor Seventh
| C↓, Bt>/, B#↓↓, Dbb
| -1
| 9
|-
| 130
| 981.1320755
| Rkm7
| Wide Subminor Seventh
| C↓/, Bt<↑
| -1
| 10
|-
| 131
| 988.6792458
| rm7
| Narrow Minor Seventh
| C\, Bt>↑
| -1
| 10
|-
| 132
| 996.2264151
| m7
| Pythagorean Minor Seventh
| C, B#↓
| -2
| 10
|-
| 133
| 1003.7735849
| Rm7
| Artomean Minor Seventh
| C/, B#↓/
| -2
| 10
|-
| 134
| 1011.3207547
| rKm7
| Tendomean Minor Seventh
| C↑\, B#\
| -3
| 10
|-
| 135
| 1018.8679245
| kM2
| Ptolemaic Minor Seventh
| C↑, B#
| -3
| 10
|-
| 136
| 1026.4150943
| RKm7, kn7
| Wide Minor Seventh
| Ct<↓, C↑/, Ddb<, B#/
| -4
| 10
|-
| 137
| 1033.9622642
| kN7, ud8
| Lesser Supraminor Seventh, Infra-Diminished Octave
| Ct>↓, Ddb>, B#↑\
| -5
| 9
|-
| 138
| 1041.5094340
| KKm7, rn7, Rud8
| Greater Supraminor Seventh, Retrodiptolemaic Diminished Octave
| Ct<\, C↑↑, Ddb<↑\, Db↓↓
| -6
| 8
|-
| 139
| 1049.0566038
| n7, rKud8
| Artoneutral Seventh, Lesser Sub-Diminished Octave
| Ct<, Ddb<↑
| -7
| 6
|-
| 140
| 1056.6037736
| N7, sd8
| Tendoneutral Seventh, Greater Sub-Diminished Octave
| Ct>, Ddb>↑
| -8
| 5
|-
| 141
| 1064.1509434
| kkM7, RN7, kd8
| Lesser Submajor Seventh, Diptolemaic Major Seventh, Retroptolemaic Diminished Octave
| Ct>/, C#↓↓, Db↓
| -7
| 6
|-
| 142
| 1071.6981132
| Kn7, Rkd8
| Greater Submajor Seventh, Artoretromean Diminished Octave
| Ct<↑, Db↓/
| -6
| 8
|-
| 143
| 1079.2452830
| rkM7, KN7, rd8
| Narrow Major Seventh, Tendoretromean Diminished Octave
| Ct>↑, C#↓\, Db\
| -5
| 9
|-
| 144
| 1086.7924528
| kM7, d8
| Ptolemaic Major Seventh, Pythagorean Diminished Octave
| Db, C#↓
| -5
| 10
|-
| 145
| 1094.3396226
| RkM7, Rd8
| Artomean Major Seventh, Artomean Diminished Octave
| Db/, C#↓/
| -5
| 10
|-
| 146
| 1101.8867925
| rM7, rKd8
| Tendomean Major Seventh, Tendomean Diminished Octave
| C#\, Db↑\
| -6
| 10
|-
| 147
| 1109.4339622
| M7, Kd8
| Pythagorean Major Seventh, Ptolemaic Diminished Octave
| C#, Db↑
| -6
| 10
|-
| 148
| 1116.9811321
| RM7, kUd8
| Wide Major Seventh, Lesser Super-Diminished Octave
| C#/, Dd<↓
| -7
| 9
|-
| 149
| 1124.5283019
| rKM7, RkUd8
| Narrow Supermajor Seventh, Greater Super-Diminished Octave
| C#↑\, Dd>↓
| -7
| 9
|-
| 150
| 1132.0754717
| km2, RuA1, kkA1
| Lesser Supermajor Seventh, Diptolemaic Diminished Octave
| C#↑, Db↑↑
| -8
| 9
|-
| 151
| 1139.6226415
| SM7, kUM7, Ud8
| Greater Supermajor Seventh, Narrow Infraoctave, Ultra-Diminished Octave
| Dd<, C#↑/
| -8
| 10
|-
| 152
| 1147.1698113
| u8, RkUM7
| Infraoctave, Wide Supermajor Seventh
| Dd>, Ct#>↓
| -9
| 10
|-
| 153
| 1154.7169811
| KKM7, rUM7, Ru8
| Narrow Ultramajor Seventh, Wide Infraoctave
| C#↑↑, Dd>/
| -9
| 10
|-
| 154
| 1162.2641509
| UM7, rKu8
| Ultramajor Seventh, Wide Superprime
| Ct#<, Dd<↑
| -9
| 10
|-
| 155
| 1169.8113208
| s8, Ku8
| Lesser Suboctave, Wide Ultramajor Seventh
| Ct#>, Dd>↑
| -10
| 3
|-
| 156
| 1177.3584906
| k8
| Greater Suboctave
| D↓
| -10
| -3
|-
| 157
| 1184.9056604
| Rk8
| Wide Suboctave
| D↓/
| -10
| -10
|-
| 158
| 1192.4528302
| r8
| Narrow Octave
| D\
| 0
| 0
|-
| 159
| 1200
| P8
| Perfect Octave
| D
| 10
| 10
|}

== Trines and Tetrachords ==
159edo has multiple types of trine and tetrachord. While trines are important in the aspects of 159edo music theory derived from Medieval and Neo-Medieval music theory, the concept of tetrachords is significantly older, as it can be traced back to Ancient Greece.

First, the trines shall be covered, for they serve as one of the fundamental units framing scales, melodies and harmonies.
{| class="wikitable"
|+Table of 159edo Trines
|-
! Name
! Notation (from D)
! Steps
! Approximate JI
! Notes
|-
| Otonal Perfect
| D, A, D
| 0, 93, 0
| 2:3:4
| This is the first of two trines that can be considered fully-resolved in Medieval and Neo-Medieval harmony
|-
| Utonal Perfect
| D, G, D
| 0, 66, 0
| 1/(2:3:4)
| This is the second of two trines that can be considered fully-resolved in Medieval and Neo-Medieval harmony
|-
| Otonal Archagall
| D, G\, D
| 0, 65, 0
| 64:85:128
| This trine is the first of two that are often used in the extended harmony of t<IV chords and is considered a dissonance
|-
| Utonal Archagall
| D, A/, D
| 0, 94, 0
| 1/(64:85:128)
| This trine is the second of two that are often used in the extended harmony of t<IV chords and is considered a dissonance
|-
| Bass-Up Marvelous
| D, A\, D
| 0, 92, 0
| 75:112:150
| This dissonant trine is the first of two that are formed from stacking identical approximations of the LCJI neutral third
|-
| Treble-Down Marvelous
| D, G/, D
| 0, 67, 0
| 1/(75:112:150)
| This dissonant trine is the second of two that are formed from stacking identical approximations of the LCJI neutral third
|-
| Narrow Supernaiadic
| D, G↓\, D
| 0, 62, 0
| 16:21:32
| This dissonant trine is common in essentially tempered chords
|-
| Wide Subcocytic
| D, A↑/, D
| 0, 97, 0
| 1/(16:21:32)
| This dissonant trine is common in essentially tempered chords
|-
| Subcocytic
| D, A↑, D
| 0, 96, 0
| 160:243:320
| This dissonant trine is common in essentially tempered chords
|-
| Supernaiadic
| D, G↓, D
| 0, 63, 0
| 1/(160:243:320)
| This dissonant trine is common in essentially tempered chords
|-
| Wide Supernaiadic
| D, G↓/, D
| 0, 64, 0
| 25:33:50
| This dissonant trine is on the outer edge of the diatonic range and is common in essentially tempered chords
|-
| Narrow Subcocytic
| D, A↑\, D
| 0, 95, 0
| 1/(25:33:50)
| This dissonant trine is on the outer edge of the diatonic range and is common in essentially tempered chords
|-
| Wide Naiadic
| D, Gd<↑, D
| 0, 61, 0
| 135:176:270
| This dissonant trine is among the more consistently complex
|-
| Narrow Cocytic
| D, At>↓, D
| 0, 98, 0
| 1/(135:176:270)
| This dissonant trine is among the more consistently complex
|-
| Naiadic
| D, Gd>/, D
| 0, 60, 0
| 10:13:20
| This dissonant trine is relatively simple and thus expected to be rather common
|-
| Cocytic
| D, At<\, D
| 0, 99, 0
| 1/(10:13:20)
| This dissonant trine is relatively simple and thus expected to be rather common
|-
| Wide Cocytic
| D, At<, D
| 0, 100, 0
| 11:17:22
| This essentially tempered trine is very likely to be used as a basis for cocytic triads
|-
| Narrow Niadic
| D, Gd>, D
| 0, 59, 0
| 1/(11:17:22)
| This essentially tempered trine is very likely to be used as a partial basis for suspended chords
|-
| Narrow Superdusthumic
| D, Ad<↑, D
| 0, 89, 0
| 128:189:256
| This dissonant trine is common in essentially tempered chords
|-
| Wide Subagallic
| D, Gt>↓, D
| 0, 70, 0
| 1/(128:189:256)
| This dissonant trine is common in essentially tempered chords
|-
| Subagallic
| D, G↑, D
| 0, 69, 0
| 20:27:40
| This dissonant trine is very likely to show up in non-meantone diatonic contexts
|-
| Superdusthumic
| D, A↓, D
| 0, 90, 0
| 1/(20:27:40)
| This dissonant trine is very likely to show up in non-meantone diatonic contexts
|-
| Narrow Subagallic
| D, G↑\, D
| 0, 68, 0
| 90:121:180
| This dissonant trine is on the outer edge of the diatonic range
|-
| Wide Superdusthumic
| D, A↓/, D
| 0, 91, 0
| 1/(90:121:180)
| This dissonant trine is on the outer edge of the diatonic range
|-
| Wide Agallic
| D, Gt<, D
| 0, 73, 0
| 8:11:16
| This ambisonant trine is very likely to be used as a partial basis for suspended chords
|-
| Narrow Dusthumic
| D, Ad>, D
| 0, 86, 0
| 1/(8:11:16)
| This ambisonant trine is very likely to be used as a basis for dusthumic triads
|-
| Dusthumic
| D, Ad<\, D
| 0, 87, 0
| 128:187:256
| This dissonant trine is common in essentially tempered chords
|-
| Agallic
| D, Gt<\, D
| 0, 72, 0
| 1/(128:187:256)
| This dissonant trine is common in essentially tempered chords
|-
| Narrow Agallic
| D, Gt>↓, D
| 0, 71, 0
| 11:15:22
| This trine is very likely to be used as a partial basis for suspended chords
|-
| Wide Dusthumic
| D, Ad<↑, D
| 0, 88, 0
| 1/(11:15:22)
| This trine is very likely to be used as a basis for dusthumic triads
|-
| Wide Subdusthumic
| D, Ad<, D
| 0, 85, 0
| 56:81:112
| This essentially tempered trine is likely to be used as a basis for subdusthumic triads
|-
| Narrow Superagallic
| D, Gt>, D
| 0, 74, 0
| 1/(56:81:112)
| This essentially tempered trine is likely to be used as a partial basis for suspended chords
|-
| Subdusthumic
| D, Ab↑↑, D
| 0, 84, 0
| 9:13:18
| This essentially tempered trine is very likely to be used as a basis for subdusthumic triads
|-
| Superagallic
| D, G#↓↓, D
| 0, 75, 0
| 1/(9:13:18)
| This essentially tempered trine is very likely to be used as a partial basis for suspended chords
|-
| Wide Superagallic
| D, Gt<↑, D
| 0, 76, 0
| 256:357:512
| This essentially tempered trine is very likely to be used as a partial basis for suspended chords
|-
| Narrow Subdusthumic
| D, Ad>↓, D
| 0, 83, 0
| 1/(256:357:512)
| This essentially tempered trine is very likely to be used as a basis for subdusthumic triads
|-
| Narrow Hyperquartal
| D, Gt>↑, D
| 0, 77, 0
| 5:7:10
| This ambisonant trine is very common as a basis for diminished chords, and is very likely to be used as a partial basis for suspended chords
|-
| Wide Hypoquintal
| D, Ad<↓, D
| 0, 82, 0
| 1/(5:7:10)
| This ambisonant trine is very common as a basis for diminished chords, and is very likely to be used as a partial basis for suspended chords
|-
| Hyperquartal
| D, G#↓, D
| 0, 78, 0
| 32:45:64
| This trine is very likely to be used as a partial basis for suspended chords
|-
| Hypoquintal
| D, Ab↑, D
| 0, 81, 0
| 1/(32:45:64)
| This trine is very common as a basis for diminished chords
|-
| Narrow Hypoquintal
| D, Ab↑\, D
| 0, 80, 0
| 12:17:24
| This trine is very common as a basis for diminished chords
|-
| Wide Hyperquartal
| D, G#↓/, D
| 0, 79, 0
| 1/(12:17:24)
| This trine is very likely to be used as a partial basis for suspended chords
|}

Next, the tetrachords need to be covered, for tetrachords are extremely useful in framing scales and melodies on a smaller scale than trines.

{| class="wikitable"
|+Table of 159edo Tetrachords
|-
! rowspan="2" |Genus
! colspan="3" |Step Sizes
! rowspan="2" |Notes
|-
! Large Step
! Medium Step
! Small Step
|-
|Pythagorean Diatonic
|Pythagorean Whole Tone (2)
|
|Pythagorean Limma
|
|-
|Ptolemaic Diatonic
|Pythagorean Whole Tone
|Ptolemaic Whole Tone
|Ptolemaic Limma
|
|-
|
|Ptolemaic Whole Tone (2)
|
|Diptolemaic Limma
|
|}
Note that the number of instances of a given step is given in parentheses if that quantity is greater than one.

User:Aura/On 159edo Music Theory (Part 1)

2026-03-30T23:41:12Z

Aura: Started the tetrachords chart. There's a lot of work to do here...

Of all the multiples of [[53edo]], [[159edo]] is the lowest multiple that is noteworthy for being accurate in the 2.3.5.11.17 subgroup while having structural compromises in the 7.13.19.23.29 subgroup. Despite the number of pitches in this tuning system making it perhaps best fit for digital instruments of various kinds in actual performance, it is nevertheless also useful as an interval classification scheme.

== Intervals and Notation ==
159edo contains all the intervals of 53edo and can be thought of as having three fields of 53edo each separated by a third of 53edo's step. However, as some of the interpretations differ due 159edo having different mappings for certain primes, those differences show up in how harmonies are constructed.

159edo has its own variation on the [[dinner party rules]]— represented here by the Harmonic Compatibility Rating and Melodic Compatibility Rating columns in the following chart, where 10 is a full-blown friend relative to the root and −10 if a full-blown enemy relative to the root. Note that the Harmonic Compatibility and Melodic Compatibility ratings are based on octave-equivalence, and that some of the ratings are still speculative.

{| class="wikitable"
|+159edo Interval Names and Compatibility Ratings
! rowspan="2" | Step
! rowspan="2" | Cents
! colspan="3" | Interval and Note names
! colspan="2" | Compatibility rating
|-
! SKULO-based interval names
! Pythagorean-commatic-based interval names
! SRS notation
! Harmonic
! Melodic
|-
| 0
| 0
| P1
| Perfect Unison
| D
| 10
| 10
|-
| 1
| 7.5471698
| R1
| Wide Unison
| D/
| 0
| 0
|-
| 2
| 15.0943396
| rK1
| Narrow Superunison
| D↑\
| -10
| -10
|-
| 3
| 22.6415094
| K1
| Lesser Superunison
| D↑
| -10
| -3
|-
| 4
| 30.1886792
| S1, kU1
| Greater Superunison, Narrow Inframinor Second
| Edb<, Dt<↓
| -10
| 3
|-
| 5
| 37.7358491
| um2, RkU1
| Inframinor Second, Wide Superunison
| Edb>, Dt>↓
| -9
| 10
|-
| 6
| 45.2830189
| kkm2, Rum2, rU1
| Wide Inframinor Second, Narrow Ultraunison
| Eb↓↓, Dt<\
| -9
| 10
|-
| 7
| 52.8301887
| U1, rKum2
| Ultraunison, Narrow Subminor Second
| Dt<, Edb<↑
| -9
| 10
|-
| 8
| 60.3773585
| sm2, Kum2, uA1
| Lesser Subminor Second, Wide Ultraunison, Infra-Augmented Unison
| Dt>, Eb↓\
| -8
| 10
|-
| 9
| 67.9245283
| km2, RuA1, kkA1
| Greater Subminor Second, Diptolemaic Augmented Unison
| Eb↓, D#↓↓
| -8
| 9
|-
| 10
| 75.4716981
| Rkm2, rKuA1
| Wide Subminor Second, Lesser Sub-Augmented Unison
| Eb↓/, Dt<↑
| -7
| 9
|-
| 11
| 83.0188679
| rm2, KuA1
| Narrow Minor Second, Greater Sub-Augmented Unison
| Eb\, Dt>↑
| -7
| 9
|-
| 12
| 90.5660377
| m2, kA1
| Pythagorean Minor Second, Ptolemaic Augmented Unison
| Eb, D#↓
| -6
| 10
|-
| 13
| 98.1132075
| Rm2, RkA1
| Artomean Minor Second, Artomean Augmented Unison
| Eb/, D#↓/
| -6
| 10
|-
| 14
| 105.6603774
| rKm2, rA1
| Tendomean Minor Second, Tendomean Augmented Unison
| D#\, Eb↑\
| -5
| 10
|-
| 15
| 113.2075472
| Km2, A1
| Ptolemaic Minor Second, Pythagorean Augmented Unison
| D#, Eb↑
| -5
| 10
|-
| 16
| 120.7547170
| RKm2, kn2, RA1
| Wide Minor Second, Artoretromean Augmented Unison
| Ed<↓, Eb↑/, D#/
| -5
| 9
|-
| 17
| 128.3018868
| kN2, rKA1
| Lesser Supraminor Second, Tendoretromean Augmented Unison
| Ed>↓, D#↑\
| -6
| 8
|-
| 18
| 135.8490566
| KKm2, rn2, KA1
| Greater Supraminor Second, Diptolemaic Limma, Retroptolemaic Augmented Unison
| Ed<\, Eb↑↑, D#↑
| -7
| 6
|-
| 19
| 143.3962264
| n2, SA1
| Artoneutral Second, Lesser Super-Augmented Unison
| Ed<, Dt#<↓
| -8
| 5
|-
| 20
| 150.9433962
| N2, RkUA1
| Tendoneutral Second, Greater Super-Augmented Unison
| Ed>, Dt#>↓
| -7
| 6
|-
| 21
| 158.4905660
| kkM2, RN2, rUA1
| Lesser Submajor Second, Retrodiptolemaic Augmented Unison
| Ed>/, E↓↓, Dt#>↓/, D#↑↑
| -6
| 8
|-
| 22
| 166.0377358
| Kn2, UA1
| Greater Submajor Second, Ultra-Augmented Unison
| Ed<↑, Dt#<, Fb↓/
| -5
| 9
|-
| 23
| 173.5849057
| rkM2, KN2
| Narrow Major Second
| Ed>↑, E↓\, Dt#>, Fb\
| -4
| 10
|-
| 24
| 181.1320755
| kM2
| Ptolemaic Major Second
| E↓, Fb
| -3
| 10
|-
| 25
| 188.6792458
| RkM2
| Artomean Major Second
| E↓/, Fb/
| -3
| 10
|-
| 26
| 196.2264151
| rM2
| Tendomean Major Second
| E\, Fb↑\
| -2
| 10
|-
| 27
| 203.7735849
| M2
| Pythagorean Major Second
| E, Fb↑
| -2
| 10
|-
| 28
| 211.3207547
| RM2
| Wide Major Second
| E/, Fd<↓
| -1
| 10
|-
| 29
| 218.8679245
| rKM2
| Narrow Supermajor Second
| E↑\, Fd>↓
| -1
| 10
|-
| 30
| 226.4150943
| KM2
| Lesser Supermajor Second
| E↑, Fd<\, Fb↑↑, Dx
| -1
| 9
|-
| 31
| 233.9622642
| SM2, kUM2
| Greater Supermajor Second, Narrow Inframinor Third
| Fd<, Et<↓, E↑/
| 0
| 9
|-
| 32
| 241.5094340
| um3, RkUM2
| Inframinor Third, Wide Supermajor Second
| Fd>, Et>↓
| -1
| 8
|-
| 33
| 249.0566038
| kkm3, KKM2, Rum3, rUM2
| Wide Inframinor Third, Narrow Ultramajor Second, Semifourth
| Fd>/, Et<\, F↓↓, E↑↑
| 0
| 8
|-
| 34
| 256.6037736
| UM2, rKum3
| Ultramajor Second, Narrow Subminor Third
| Et<, Fd<↑
| -1
| 7
|-
| 35
| 264.1509434
| sm3, Kum3
| Lesser Subminor Third, Wide Ultramajor Second
| Et>, Fd>↑, F↓\
| 0
| 7
|-
| 36
| 271.6981132
| km3
| Greater Subminor Third
| F↓, Et>/, E#↓↓, Gbb
| -1
| 7
|-
| 37
| 279.2452830
| Rkm3
| Wide Subminor Third
| F↓/, Et<↑
| -1
| 8
|-
| 38
| 286.7924528
| rm3
| Narrow Minor Third
| F\, Et>↑
| 0
| 8
|-
| 39
| 294.3396226
| m3
| Pythagorean Minor Third
| F
| -1
| 9
|-
| 40
| 301.8867925
| Rm3
| Artomean Minor Third
| F/
| 1
| 9
|-
| 41
| 309.4339622
| rKm3
| Tendomean Minor Third
| F↑\
| 4
| 10
|-
| 42
| 316.9811321
| Km3
| Ptolemaic Minor Third
| F↑, E#
| 7
| 10
|-
| 43
| 324.5283019
| RKm3, kn3
| Wide Minor Third
| Ft<↓, F↑/, Gdb<
| 4
| 9
|-
| 44
| 332.0754717
| kN3, ud4
| Lesser Supraminor Third, Infra-Diminished Fourth
| Ft>↓, Gdb>
| 1
| 9
|-
| 45
| 339.6226415
| KKm3, rn3, Rud4
| Greater Supraminor Third, Retrodiptolemaic Diminished Fourth
| Ft<\, F↑↑, Gdb<↑\, Gb↓↓
| -1
| 8
|-
| 46
| 347.1698113
| n3, rKud4
| Artoneutral Third, Lesser Sub-Diminished Fourth
| Ft<, Gdb<↑
| 0
| 7
|-
| 47
| 354.7169811
| N3, sd4, Kud4
| Tendoneutral Third, Greater Sub-Diminished Fourth
| Ft>, Gdb>↑
| -1
| 7
|-
| 48
| 362.2641509
| kkM3, RN3, kd4
| Lesser Submajor Third, Retroptolemaic Diminished Fourth
| Ft>/, F#↓↓, Gb↓
| 0
| 8
|-
| 49
| 369.8113208
| Kn3, Rkd4
| Greater Submajor Third, Artoretromean Diminished Fourth
| Ft<↑, Gb↓/
| -1
| 9
|-
| 50
| 377.3584906
| rkM3, KN3, rd4
| Narrow Major Third, Tendoretromean Diminished Fourth
| Ft>↑, F#↓\, Gb\
| 3
| 9
|-
| 51
| 384.9056604
| kM3, d4
| Ptolemaic Major Third, Pythagorean Diminished Fourth
| Gb, F#↓
| 8
| 10
|-
| 52
| 392.4528302
| RkM3, Rd4
| Artomean Major Third, Artomean Diminished Fourth
| Gb/, F#↓/
| 4
| 10
|-
| 53
| 400
| rM3, rKd4
| Tendomean Major Third, Tendomean Diminished Fourth
| F#\, Gb↑\
| 1
| 9
|-
| 54
| 407.5471698
| M3, Kd4
| Pythagorean Major Third, Ptolemaic Diminished Fourth
| F#, Gb↑
| -1
| 9
|-
| 55
| 415.0943396
| RM3, kUd4
| Wide Major Third, Lesser Super-Diminished Fourth
| F#/, Gd<↓, Gb↑/
| 0
| 8
|-
| 56
| 422.6415094
| rKM3, RkUd4
| Narrow Supermajor Third, Greater Super-Diminished Fourth
| F#↑\, Gd>↓
| -1
| 7
|-
| 57
| 430.1886792
| KM3, rUd4, KKd4
| Lesser Supermajor Third, Diptolemaic Diminished Fourth
| F#↑, Gd<\, Gb↑↑
| -1
| 6
|-
| 58
| 437.7358491
| SM3, kUM3, rm4, Ud4
| Greater Supermajor Third, Ultra-Diminished Fourth
| Gd<, F#↑/
| 0
| 5
|-
| 59
| 445.2830189
| m4, RkUM3
| Paraminor Fourth, Wide Supermajor Third
| Gd>, Ft#>↓
| -1
| 3
|-
| 60
| 452.8301887
| Rm4, KKM3, rUM3
| Wide Paraminor Fourth, Narrow Ultramajor Third
| Gd>/, F#↑↑, G↓↓
| -2
| 1
|-
| 61
| 460.3773585
| UM3, rKm4
| Ultramajor Third, Narrow Grave Fourth
| Gd<↑, Ft#<
| -4
| -2
|-
| 62
| 467.9245283
| s4, Km4
| Lesser Grave Fourth, Wide Ultramajor Third
| Gd>↑, G↓\
| -7
| -4
|-
| 63
| 475.4716981
| k4
| Greater Grave Fourth
| G↓, Abb
| -6
| -5
|-
| 64
| 483.0188679
| Rk4
| Wide Grave Fourth
| G↓/
| -4
| 0
|-
| 65
| 490.5660377
| r4
| Narrow Fourth
| G\
| 1
| 5
|-
| 66
| 498.1132075
| P4
| Perfect Fourth
| G
| 9
| 10
|-
| 67
| 505.6603774
| R4
| Wide Fourth
| G/
| 1
| 8
|-
| 68
| 513.2075472
| rK4
| Narrow Acute Fourth
| G↑\
| -3
| 6
|-
| 69
| 520.7547170
| K4
| Lesser Acute Fourth
| G↑
| -5
| 5
|-
| 70
| 528.3018868
| S4, kM4
| Greater Acute Fourth
| Gt<↓, G↑/, Adb<
| -3
| 5
|-
| 71
| 535.8490566
| RkM4, ud5
| Wide Acute Fourth, Infra-Diminished Fifth
| Gt>↓, Adb>
| -2
| 5
|-
| 72
| 543.3962264
| rM4, Rud5
| Narrow Paramajor Fourth, Retrodiptolemaic Diminished Fifth
| Gt<\, G↑↑, Ab↓↓
| -1
| 6
|-
| 73
| 550.9433962
| M4, rKud5
| Paramajor Fourth, Lesser Sub-Diminished Fifth
| Gt<, Adb<↑
| 0
| 7
|-
| 74
| 558.4905660
| RM4, uA4, Kud5
| Infra-Augmented Fourth, Greater Sub-Diminished Fifth
| Gt>, Adb>↑
| -2
| 5
|-
| 75
| 566.0377358
| kkA4, RuA4, kd5
| Diptolemaic Augmented Fourth, Retroptolemaic Diminished Fifth
| Gt>/, G#↓↓, Ab↓
| -3
| 4
|-
| 76
| 573.5849057
| rKuA4, Rkd5
| Lesser Sub-Augmented Fourth, Artoretromean Diminished Fifth
| Gt<↑, Ab↓/
| -2
| 4
|-
| 77
| 581.1320755
| KuA4, rd5
| Greater Sub-Augmented Fourth, Tendoretromean Diminished Fifth
| Gt>↑, Ab\
| 0
| 5
|-
| 78
| 588.6792458
| kA4, d5
| Ptolemaic Augmented Fourth, Pythagorean Diminished Fifth
| Ab, G#↓
| -5
| 6
|-
| 79
| 596.2264151
| RkA4, Rd5
| Artomean Augmented Fourth, Artomean Diminished Fifth
| G#↓/, Ab/
| -9
| 7
|-
| 80
| 603.7735849
| rKd5, rA4
| Tendomean Diminished Fifth, Tendomean Augmented Fourth
| Ab↑\, G#\
| -9
| 7
|-
| 81
| 611.3207547
| Kd5, A4
| Ptolemaic Diminished Fifth, Pythagorean Augmented Fourth
| Ab↑, G#
| -5
| 6
|-
| 82
| 618.8679245
| kUd5, RA4
| Lesser Super-Diminished Fifth, Artoretromean Augmented Fourth
| Ad<↓, G#/
| 0
| 5
|-
| 83
| 626.4150943
| RkUd5, rKA4
| Greater Super-Diminished Fifth, Tendoretromean Augmented Fourth
| Ad>↓, G#↑\
| -2
| 4
|-
| 84
| 633.9622642
| KKd5, rUDd5, KA4
| Diptolemaic Diminished Fifth, Retroptolemaic Augmented Fourth
| Ad<\, Ab↑↑, G#↑
| -3
| 4
|-
| 85
| 641.5094340
| rm5, Ud5, kUA4
| Ultra-Diminished Fifth, Lesser Super-Augmented Fourth
| Ad<, Gt#<↓
| -2
| 5
|-
| 86
| 649.0566038
| m5, RkUA4
| Paraminor Fifth, Greater Super-Augmented Fourth
| Ad>, Gt#>↓
| 0
| 7
|-
| 87
| 656.6037736
| Rm5, rUA4
| Wide Paraminor Fifth, Retrodiptolemaic Augmented Fourth
| Ad>/, G#↑, Ab↑↑
| -1
| 6
|-
| 88
| 664.1509434
| rKm5, UA4
| Narrow Grave Fifth, Ultra-Augmented Fourth
| Ad<↑, Gt#<
| -2
| 5
|-
| 89
| 671.6981132
| s5, Km5
| Lesser Grave Fifth
| Ad>↑, A↓\, Gt#>
| -3
| 5
|-
| 90
| 679.2452830
| k5
| Greater Grave Fifth
| A↓
| -5
| 5
|-
| 91
| 686.7924528
| Rk5
| Wide Grave Fifth
| A↓/
| -3
| 6
|-
| 92
| 694.3396226
| r5
| Narrow Fifth
| A\
| 1
| 8
|-
| 93
| 701.8867925
| P5
| Perfect Fifth
| A
| 9
| 10
|-
| 94
| 709.4339622
| R5
| Wide Fifth
| A/
| 1
| 5
|-
| 95
| 716.9811321
| rK5
| Narrow Acute Fifth
| A↑\
| -4
| 0
|-
| 96
| 724.5283019
| K5
| Lesser Acute Fifth
| A↑, Gx
| -6
| -5
|-
| 97
| 732.0754717
| S5, kM5
| Greater Acute Fifth, Narrow Inframinor Sixth
| At<↓, A↑/
| -7
| -4
|-
| 98
| 739.6226415
| um6, RkM5
| Inframinor Sixth, Wide Acute Fifth
| At>↓, Bdb>
| -4
| -2
|-
| 99
| 747.1698113
| Rm4, KKM3, rUM3
| Narrow Paramajor Fifth, Wide Inframinor Sixth
| At<\, Bb↓↓, A↑↑
| -2
| 1
|-
| 100
| 754.7169811
| M5, rKum6
| Paramajor Fifth, Narrow Subminor Sixth
| At<, Bdb<↑
| -1
| 3
|-
| 101
| 762.2641509
| sm6, Kum6, RM5, uA5
| Lesser Subminor Sixth, Infra-Augmented Fifth
| At>, Bb↓\
| 0
| 5
|-
| 102
| 769.8113208
| km6, RuA5, kkA5
| Greater Subminor Sixth, Diptolemaic Augmented Fifth
| Bb↓, At>/, A#↓↓
| -1
| 6
|-
| 103
| 777.3584906
| Rkm6, rKuA5
| Wide Subminor Sixth, Lesser Sub-Augmented Fifth
| Bb↓/, At<↑
| -1
| 7
|-
| 104
| 784.9056604
| rm6, KuA5
| Narrow Minor Sixth, Greater Sub-Augmented Fifth
| Bb\, At>↑, A#↓\
| 0
| 8
|-
| 105
| 792.4528302
| m6, kA5
| Pythagorean Minor Sixth, Ptolemaic Augmented Fifth
| Bb, A#↓
| -1
| 9
|-
| 106
| 800
| Rm6, RkA5
| Artomean Minor Sixth, Artomean Augmented Fifth
| Bb/, A#↓/
| 1
| 9
|-
| 107
| 807.5471698
| rKm6, rA5
| Tendomean Minor Sixth, Tendomean Augmented Fifth
| A#\, Bb↑\
| 4
| 10
|-
| 108
| 815.0943396
| Km6, A5
| Ptolemaic Minor Sixth, Pythagorean Augmented Fifth
| A#, Bb↑
| 8
| 10
|-
| 109
| 822.6415094
| RKm6, kn6, RA5
|Wide Minor Sixth, Artoretromean Augmented Fifth
| Bd<↓, Bb↑/, A#/
| 3
| 9
|-
| 110
| 830.1886792
| kN6, rKA5
| Lesser Supraminor Sixth, Tendoretromean Augmented Fifth
| Bd>↓, A#↑\
| -1
| 9
|-
| 111
| 837.7358491
| KKm6, rn6, KA5
| Greater Supraminor Sixth, Retroptolemaic Augmented Fifth
| Bd<\, Bb↑↑, A#↑
| 0
| 8
|-
| 112
| 845.2830189
| n6, SA5, kUA5
| Artoneutral Sixth, Lesser Super-Augmented Fifth
| Bd<, At#<↓
| -1
| 7
|-
| 113
| 852.8301887
| N6, RkUA5
| Tendoneutral Sixth, Greater Super-Augmented Fifth
| Bd>, At#>↓
| 0
| 7
|-
| 114
| 860.3773585
| kkM6, RN6, rUA5
| Lesser Submajor Sixth, Retrodiptolemaic Augmented Fifth
| Bd>/, B↓↓, At#>↓/, A#↑↑
| -1
| 8
|-
| 115
| 867.9245283
| Kn6, UA5
| Greater Submajor Sixth, Ultra-Augmented Fifth
| Bd<↑, At#<
| 1
| 9
|-
| 116
| 875.4716981
| rkM6, KN6
| Narrow Major Sixth
| Bd>↑, B↓\, At#>
| 4
| 9
|-
| 117
| 883.0188679
| kM6
| Ptolemaic Major Sixth
| B↓, Cb
| 7
| 10
|-
| 118
| 890.5660377
| RkM6
| Artomean Major Sixth
| B↓/
| 4
| 10
|-
| 119
| 898.1132075
| rM6
| Tendomean Major Sixth
| B\
| 1
| 9
|-
| 120
| 905.6603774
| M6
| Pythagorean Major Sixth
| B
| -1
| 9
|-
| 121
| 913.2075472
| RM6
| Wide Major Sixth
| B/, Cd<↓
| 0
| 8
|-
| 122
| 920.7547170
| rKM6
| Narrow Supermajor Sixth
| B↑\, Cd>↓
| -1
| 8
|-
| 123
| 928.3018868
| KM6
| Lesser Supermajor Sixth
| B↑, Cd<\, Cb↑↑, Ax
| -1
| 7
|-
| 124
| 935.8490566
| SM6, kUM6
| Greater Supermajor Second, Narrow Inframinor Seventh
| Cd<, Bt<↓, B↑/
| 0
| 7
|-
| 125
| 943.3962264
| um7, RkUM6
| Inframinor Seventh, Wide Supermajor Sixth
| Cd>, Bt>↓
| -1
| 7
|-
| 126
| 950.9433962
| KKM6, kkm7, rUM6, Rum7
| Narrow Ultramajor Sixth, Wide Inframinor Seventh, Semitwelfth
| Bt<\, Cd>/, B↑↑, C↓↓
| 0
| 8
|-
| 127
| 958.4905660
| UM6, rKum7
| Ultramajor Sixth, Narrow Subminor Seventh
| Bt<, Cd<↑
| -1
| 8
|-
| 128
| 966.0377358
| sm7, Kum7
| Lesser Subminor Seventh, Wide Ultramajor Sixth
| Bt>, Cd>↑, C↓\
| 0
| 9
|-
| 129
| 973.5849057
| km7
| Greater Subminor Seventh
| C↓, Bt>/, B#↓↓, Dbb
| -1
| 9
|-
| 130
| 981.1320755
| Rkm7
| Wide Subminor Seventh
| C↓/, Bt<↑
| -1
| 10
|-
| 131
| 988.6792458
| rm7
| Narrow Minor Seventh
| C\, Bt>↑
| -1
| 10
|-
| 132
| 996.2264151
| m7
| Pythagorean Minor Seventh
| C, B#↓
| -2
| 10
|-
| 133
| 1003.7735849
| Rm7
| Artomean Minor Seventh
| C/, B#↓/
| -2
| 10
|-
| 134
| 1011.3207547
| rKm7
| Tendomean Minor Seventh
| C↑\, B#\
| -3
| 10
|-
| 135
| 1018.8679245
| kM2
| Ptolemaic Minor Seventh
| C↑, B#
| -3
| 10
|-
| 136
| 1026.4150943
| RKm7, kn7
| Wide Minor Seventh
| Ct<↓, C↑/, Ddb<, B#/
| -4
| 10
|-
| 137
| 1033.9622642
| kN7, ud8
| Lesser Supraminor Seventh, Infra-Diminished Octave
| Ct>↓, Ddb>, B#↑\
| -5
| 9
|-
| 138
| 1041.5094340
| KKm7, rn7, Rud8
| Greater Supraminor Seventh, Retrodiptolemaic Diminished Octave
| Ct<\, C↑↑, Ddb<↑\, Db↓↓
| -6
| 8
|-
| 139
| 1049.0566038
| n7, rKud8
| Artoneutral Seventh, Lesser Sub-Diminished Octave
| Ct<, Ddb<↑
| -7
| 6
|-
| 140
| 1056.6037736
| N7, sd8
| Tendoneutral Seventh, Greater Sub-Diminished Octave
| Ct>, Ddb>↑
| -8
| 5
|-
| 141
| 1064.1509434
| kkM7, RN7, kd8
| Lesser Submajor Seventh, Diptolemaic Major Seventh, Retroptolemaic Diminished Octave
| Ct>/, C#↓↓, Db↓
| -7
| 6
|-
| 142
| 1071.6981132
| Kn7, Rkd8
| Greater Submajor Seventh, Artoretromean Diminished Octave
| Ct<↑, Db↓/
| -6
| 8
|-
| 143
| 1079.2452830
| rkM7, KN7, rd8
| Narrow Major Seventh, Tendoretromean Diminished Octave
| Ct>↑, C#↓\, Db\
| -5
| 9
|-
| 144
| 1086.7924528
| kM7, d8
| Ptolemaic Major Seventh, Pythagorean Diminished Octave
| Db, C#↓
| -5
| 10
|-
| 145
| 1094.3396226
| RkM7, Rd8
| Artomean Major Seventh, Artomean Diminished Octave
| Db/, C#↓/
| -5
| 10
|-
| 146
| 1101.8867925
| rM7, rKd8
| Tendomean Major Seventh, Tendomean Diminished Octave
| C#\, Db↑\
| -6
| 10
|-
| 147
| 1109.4339622
| M7, Kd8
| Pythagorean Major Seventh, Ptolemaic Diminished Octave
| C#, Db↑
| -6
| 10
|-
| 148
| 1116.9811321
| RM7, kUd8
| Wide Major Seventh, Lesser Super-Diminished Octave
| C#/, Dd<↓
| -7
| 9
|-
| 149
| 1124.5283019
| rKM7, RkUd8
| Narrow Supermajor Seventh, Greater Super-Diminished Octave
| C#↑\, Dd>↓
| -7
| 9
|-
| 150
| 1132.0754717
| km2, RuA1, kkA1
| Lesser Supermajor Seventh, Diptolemaic Diminished Octave
| C#↑, Db↑↑
| -8
| 9
|-
| 151
| 1139.6226415
| SM7, kUM7, Ud8
| Greater Supermajor Seventh, Narrow Infraoctave, Ultra-Diminished Octave
| Dd<, C#↑/
| -8
| 10
|-
| 152
| 1147.1698113
| u8, RkUM7
| Infraoctave, Wide Supermajor Seventh
| Dd>, Ct#>↓
| -9
| 10
|-
| 153
| 1154.7169811
| KKM7, rUM7, Ru8
| Narrow Ultramajor Seventh, Wide Infraoctave
| C#↑↑, Dd>/
| -9
| 10
|-
| 154
| 1162.2641509
| UM7, rKu8
| Ultramajor Seventh, Wide Superprime
| Ct#<, Dd<↑
| -9
| 10
|-
| 155
| 1169.8113208
| s8, Ku8
| Lesser Suboctave, Wide Ultramajor Seventh
| Ct#>, Dd>↑
| -10
| 3
|-
| 156
| 1177.3584906
| k8
| Greater Suboctave
| D↓
| -10
| -3
|-
| 157
| 1184.9056604
| Rk8
| Wide Suboctave
| D↓/
| -10
| -10
|-
| 158
| 1192.4528302
| r8
| Narrow Octave
| D\
| 0
| 0
|-
| 159
| 1200
| P8
| Perfect Octave
| D
| 10
| 10
|}

== Trines and Tetrachords ==
159edo has multiple types of trine and tetrachord. While trines are important in the aspects of 159edo music theory derived from Medieval and Neo-Medieval music theory, the concept of tetrachords is significantly older, as it can be traced back to Ancient Greece.

First, the trines shall be covered, for they serve as one of the fundamental units framing scales, melodies and harmonies.
{| class="wikitable"
|+Table of 159edo Trines
|-
! Name
! Notation (from D)
! Steps
! Approximate JI
! Notes
|-
| Otonal Perfect
| D, A, D
| 0, 93, 0
| 2:3:4
| This is the first of two trines that can be considered fully-resolved in Medieval and Neo-Medieval harmony
|-
| Utonal Perfect
| D, G, D
| 0, 66, 0
| 1/(2:3:4)
| This is the second of two trines that can be considered fully-resolved in Medieval and Neo-Medieval harmony
|-
| Otonal Archagall
| D, G\, D
| 0, 65, 0
| 64:85:128
| This trine is the first of two that are often used in the extended harmony of t<IV chords and is considered a dissonance
|-
| Utonal Archagall
| D, A/, D
| 0, 94, 0
| 1/(64:85:128)
| This trine is the second of two that are often used in the extended harmony of t<IV chords and is considered a dissonance
|-
| Bass-Up Marvelous
| D, A\, D
| 0, 92, 0
| 75:112:150
| This dissonant trine is the first of two that are formed from stacking identical approximations of the LCJI neutral third
|-
| Treble-Down Marvelous
| D, G/, D
| 0, 67, 0
| 1/(75:112:150)
| This dissonant trine is the second of two that are formed from stacking identical approximations of the LCJI neutral third
|-
| Narrow Supernaiadic
| D, G↓\, D
| 0, 62, 0
| 16:21:32
| This dissonant trine is common in essentially tempered chords
|-
| Wide Subcocytic
| D, A↑/, D
| 0, 97, 0
| 1/(16:21:32)
| This dissonant trine is common in essentially tempered chords
|-
| Subcocytic
| D, A↑, D
| 0, 96, 0
| 160:243:320
| This dissonant trine is common in essentially tempered chords
|-
| Supernaiadic
| D, G↓, D
| 0, 63, 0
| 1/(160:243:320)
| This dissonant trine is common in essentially tempered chords
|-
| Wide Supernaiadic
| D, G↓/, D
| 0, 64, 0
| 25:33:50
| This dissonant trine is on the outer edge of the diatonic range and is common in essentially tempered chords
|-
| Narrow Subcocytic
| D, A↑\, D
| 0, 95, 0
| 1/(25:33:50)
| This dissonant trine is on the outer edge of the diatonic range and is common in essentially tempered chords
|-
| Wide Naiadic
| D, Gd<↑, D
| 0, 61, 0
| 135:176:270
| This dissonant trine is among the more consistently complex
|-
| Narrow Cocytic
| D, At>↓, D
| 0, 98, 0
| 1/(135:176:270)
| This dissonant trine is among the more consistently complex
|-
| Naiadic
| D, Gd>/, D
| 0, 60, 0
| 10:13:20
| This dissonant trine is relatively simple and thus expected to be rather common
|-
| Cocytic
| D, At<\, D
| 0, 99, 0
| 1/(10:13:20)
| This dissonant trine is relatively simple and thus expected to be rather common
|-
| Wide Cocytic
| D, At<, D
| 0, 100, 0
| 11:17:22
| This essentially tempered trine is very likely to be used as a basis for cocytic triads
|-
| Narrow Niadic
| D, Gd>, D
| 0, 59, 0
| 1/(11:17:22)
| This essentially tempered trine is very likely to be used as a partial basis for suspended chords
|-
| Narrow Superdusthumic
| D, Ad<↑, D
| 0, 89, 0
| 128:189:256
| This dissonant trine is common in essentially tempered chords
|-
| Wide Subagallic
| D, Gt>↓, D
| 0, 70, 0
| 1/(128:189:256)
| This dissonant trine is common in essentially tempered chords
|-
| Subagallic
| D, G↑, D
| 0, 69, 0
| 20:27:40
| This dissonant trine is very likely to show up in non-meantone diatonic contexts
|-
| Superdusthumic
| D, A↓, D
| 0, 90, 0
| 1/(20:27:40)
| This dissonant trine is very likely to show up in non-meantone diatonic contexts
|-
| Narrow Subagallic
| D, G↑\, D
| 0, 68, 0
| 90:121:180
| This dissonant trine is on the outer edge of the diatonic range
|-
| Wide Superdusthumic
| D, A↓/, D
| 0, 91, 0
| 1/(90:121:180)
| This dissonant trine is on the outer edge of the diatonic range
|-
| Wide Agallic
| D, Gt<, D
| 0, 73, 0
| 8:11:16
| This ambisonant trine is very likely to be used as a partial basis for suspended chords
|-
| Narrow Dusthumic
| D, Ad>, D
| 0, 86, 0
| 1/(8:11:16)
| This ambisonant trine is very likely to be used as a basis for dusthumic triads
|-
| Dusthumic
| D, Ad<\, D
| 0, 87, 0
| 128:187:256
| This dissonant trine is common in essentially tempered chords
|-
| Agallic
| D, Gt<\, D
| 0, 72, 0
| 1/(128:187:256)
| This dissonant trine is common in essentially tempered chords
|-
| Narrow Agallic
| D, Gt>↓, D
| 0, 71, 0
| 11:15:22
| This trine is very likely to be used as a partial basis for suspended chords
|-
| Wide Dusthumic
| D, Ad<↑, D
| 0, 88, 0
| 1/(11:15:22)
| This trine is very likely to be used as a basis for dusthumic triads
|-
| Wide Subdusthumic
| D, Ad<, D
| 0, 85, 0
| 56:81:112
| This essentially tempered trine is likely to be used as a basis for subdusthumic triads
|-
| Narrow Superagallic
| D, Gt>, D
| 0, 74, 0
| 1/(56:81:112)
| This essentially tempered trine is likely to be used as a partial basis for suspended chords
|-
| Subdusthumic
| D, Ab↑↑, D
| 0, 84, 0
| 9:13:18
| This essentially tempered trine is very likely to be used as a basis for subdusthumic triads
|-
| Superagallic
| D, G#↓↓, D
| 0, 75, 0
| 1/(9:13:18)
| This essentially tempered trine is very likely to be used as a partial basis for suspended chords
|-
| Wide Superagallic
| D, Gt<↑, D
| 0, 76, 0
| 256:357:512
| This essentially tempered trine is very likely to be used as a partial basis for suspended chords
|-
| Narrow Subdusthumic
| D, Ad>↓, D
| 0, 83, 0
| 1/(256:357:512)
| This essentially tempered trine is very likely to be used as a basis for subdusthumic triads
|-
| Narrow Hyperquartal
| D, Gt>↑, D
| 0, 77, 0
| 5:7:10
| This ambisonant trine is very common as a basis for diminished chords, and is very likely to be used as a partial basis for suspended chords
|-
| Wide Hypoquintal
| D, Ad<↓, D
| 0, 82, 0
| 1/(5:7:10)
| This ambisonant trine is very common as a basis for diminished chords, and is very likely to be used as a partial basis for suspended chords
|-
| Hyperquartal
| D, G#↓, D
| 0, 78, 0
| 32:45:64
| This trine is very likely to be used as a partial basis for suspended chords
|-
| Hypoquintal
| D, Ab↑, D
| 0, 81, 0
| 1/(32:45:64)
| This trine is very common as a basis for diminished chords
|-
| Narrow Hypoquintal
| D, Ab↑\, D
| 0, 80, 0
| 12:17:24
| This trine is very common as a basis for diminished chords
|-
| Wide Hyperquartal
| D, G#↓/, D
| 0, 79, 0
| 1/(12:17:24)
| This trine is very likely to be used as a partial basis for suspended chords
|}

Next, the tetrachords need to be covered, for tetrachords are extremely useful in framing scales and melodies on a smaller scale than trines.

{| class="wikitable"
|+Table of 159edo Tetrachords
|-
! rowspan="2" |Genus
! colspan="3" |Step Sizes
! rowspan="2" |Notes
|-
! Large Step
! Medium Step
! Small Step
|-
|Pythagorean Diatonic
|Pythagorean Whole Tone (2)
|
|Pythagorean Diatonic Semitone
|
|-
|Ptolemaic Diatonic
|Pythagorean Whole Tone
|Ptolemaic Whole Tone
|Ptolemaic Diatonic Semitone
|
|}

User:Aura/On 159edo Music Theory (Part 1)

2026-03-30T22:53:49Z

Aura: Added the table of trines- this is also more or less copied from the Xenharmonic Wiki, but this time from a 159edo subpage

Of all the multiples of [[53edo]], [[159edo]] is the lowest multiple that is noteworthy for being accurate in the 2.3.5.11.17 subgroup while having structural compromises in the 7.13.19.23.29 subgroup. Despite the number of pitches in this tuning system making it perhaps best fit for digital instruments of various kinds in actual performance, it is nevertheless also useful as an interval classification scheme.

== Intervals and Notation ==
159edo contains all the intervals of 53edo and can be thought of as having three fields of 53edo each separated by a third of 53edo's step. However, as some of the interpretations differ due 159edo having different mappings for certain primes, those differences show up in how harmonies are constructed.

159edo has its own variation on the [[dinner party rules]]— represented here by the Harmonic Compatibility Rating and Melodic Compatibility Rating columns in the following chart, where 10 is a full-blown friend relative to the root and −10 if a full-blown enemy relative to the root. Note that the Harmonic Compatibility and Melodic Compatibility ratings are based on octave-equivalence, and that some of the ratings are still speculative.

{| class="wikitable"
|+159edo Interval Names and Compatibility Ratings
! rowspan="2" | Step
! rowspan="2" | Cents
! colspan="3" | Interval and Note names
! colspan="2" | Compatibility rating
|-
! SKULO-based interval names
! Pythagorean-commatic-based interval names
! SRS notation
! Harmonic
! Melodic
|-
| 0
| 0
| P1
| Perfect Unison
| D
| 10
| 10
|-
| 1
| 7.5471698
| R1
| Wide Prime
| D/
| 0
| 0
|-
| 2
| 15.0943396
| rK1
| Narrow Superprime
| D↑\
| -10
| -10
|-
| 3
| 22.6415094
| K1
| Lesser Superprime
| D↑
| -10
| -3
|-
| 4
| 30.1886792
| S1, kU1
| Greater Superprime, Narrow Inframinor Second
| Edb<, Dt<↓
| -10
| 3
|-
| 5
| 37.7358491
| um2, RkU1
| Inframinor Second, Wide Superprime
| Edb>, Dt>↓
| -9
| 10
|-
| 6
| 45.2830189
| kkm2, Rum2, rU1
| Wide Inframinor Second, Narrow Ultraprime
| Eb↓↓, Dt<\
| -9
| 10
|-
| 7
| 52.8301887
| U1, rKum2
| Ultraprime, Narrow Subminor Second
| Dt<, Edb<↑
| -9
| 10
|-
| 8
| 60.3773585
| sm2, Kum2, uA1
| Lesser Subminor Second, Wide Ultraprime, Infra-Augmented Prime
| Dt>, Eb↓\
| -8
| 10
|-
| 9
| 67.9245283
| km2, RuA1, kkA1
| Greater Subminor Second, Diptolemaic Augmented Prime
| Eb↓, D#↓↓
| -8
| 9
|-
| 10
| 75.4716981
| Rkm2, rKuA1
| Wide Subminor Second, Lesser Sub-Augmented Prime
| Eb↓/, Dt<↑
| -7
| 9
|-
| 11
| 83.0188679
| rm2, KuA1
| Narrow Minor Second, Greater Sub-Augmented Prime
| Eb\, Dt>↑
| -7
| 9
|-
| 12
| 90.5660377
| m2, kA1
| Pythagorean Minor Second, Ptolemaic Augmented Prime
| Eb, D#↓
| -6
| 10
|-
| 13
| 98.1132075
| Rm2, RkA1
| Artomean Minor Second, Artomean Augmented Prime
| Eb/, D#↓/
| -6
| 10
|-
| 14
| 105.6603774
| rKm2, rA1
| Tendomean Minor Second, Tendomean Augmented Prime
| D#\, Eb↑\
| -5
| 10
|-
| 15
| 113.2075472
| Km2, A1
| Ptolemaic Minor Second, Pythagorean Augmented Prime
| D#, Eb↑
| -5
| 10
|-
| 16
| 120.7547170
| RKm2, kn2, RA1
| Wide Minor Second, Artoretromean Augmented Prime
| Ed<↓, Eb↑/, D#/
| -5
| 9
|-
| 17
| 128.3018868
| kN2, rKA1
| Lesser Supraminor Second, Tendoretromean Augmented Prime
| Ed>↓, D#↑\
| -6
| 8
|-
| 18
| 135.8490566
| KKm2, rn2, KA1
| Greater Supraminor Second, Diptolemaic Limma, Retroptolemaic Augmented Prime
| Ed<\, Eb↑↑, D#↑
| -7
| 6
|-
| 19
| 143.3962264
| n2, SA1
| Artoneutral Second, Lesser Super-Augmented Prime
| Ed<, Dt#<↓
| -8
| 5
|-
| 20
| 150.9433962
| N2, RkUA1
| Tendoneutral Second, Greater Super-Augmented Prime
| Ed>, Dt#>↓
| -7
| 6
|-
| 21
| 158.4905660
| kkM2, RN2, rUA1
| Lesser Submajor Second, Retrodiptolemaic Augmented Prime
| Ed>/, E↓↓, Dt#>↓/, D#↑↑
| -6
| 8
|-
| 22
| 166.0377358
| Kn2, UA1
| Greater Submajor Second, Ultra-Augmented Prime
| Ed<↑, Dt#<, Fb↓/
| -5
| 9
|-
| 23
| 173.5849057
| rkM2, KN2
| Narrow Major Second
| Ed>↑, E↓\, Dt#>, Fb\
| -4
| 10
|-
| 24
| 181.1320755
| kM2
| Ptolemaic Major Second
| E↓, Fb
| -3
| 10
|-
| 25
| 188.6792458
| RkM2
| Artomean Major Second
| E↓/, Fb/
| -3
| 10
|-
| 26
| 196.2264151
| rM2
| Tendomean Major Second
| E\, Fb↑\
| -2
| 10
|-
| 27
| 203.7735849
| M2
| Pythagorean Major Second
| E, Fb↑
| -2
| 10
|-
| 28
| 211.3207547
| RM2
| Wide Major Second
| E/, Fd<↓
| -1
| 10
|-
| 29
| 218.8679245
| rKM2
| Narrow Supermajor Second
| E↑\, Fd>↓
| -1
| 10
|-
| 30
| 226.4150943
| KM2
| Lesser Supermajor Second
| E↑, Fd<\, Fb↑↑, Dx
| -1
| 9
|-
| 31
| 233.9622642
| SM2, kUM2
| Greater Supermajor Second, Narrow Inframinor Third
| Fd<, Et<↓, E↑/
| 0
| 9
|-
| 32
| 241.5094340
| um3, RkUM2
| Inframinor Third, Wide Supermajor Second
| Fd>, Et>↓
| -1
| 8
|-
| 33
| 249.0566038
| kkm3, KKM2, Rum3, rUM2
| Wide Inframinor Third, Narrow Ultramajor Second, Semifourth
| Fd>/, Et<\, F↓↓, E↑↑
| 0
| 8
|-
| 34
| 256.6037736
| UM2, rKum3
| Ultramajor Second, Narrow Subminor Third
| Et<, Fd<↑
| -1
| 7
|-
| 35
| 264.1509434
| sm3, Kum3
| Lesser Subminor Third, Wide Ultramajor Second
| Et>, Fd>↑, F↓\
| 0
| 7
|-
| 36
| 271.6981132
| km3
| Greater Subminor Third
| F↓, Et>/, E#↓↓, Gbb
| -1
| 7
|-
| 37
| 279.2452830
| Rkm3
| Wide Subminor Third
| F↓/, Et<↑
| -1
| 8
|-
| 38
| 286.7924528
| rm3
| Narrow Minor Third
| F\, Et>↑
| 0
| 8
|-
| 39
| 294.3396226
| m3
| Pythagorean Minor Third
| F
| -1
| 9
|-
| 40
| 301.8867925
| Rm3
| Artomean Minor Third
| F/
| 1
| 9
|-
| 41
| 309.4339622
| rKm3
| Tendomean Minor Third
| F↑\
| 4
| 10
|-
| 42
| 316.9811321
| Km3
| Ptolemaic Minor Third
| F↑, E#
| 7
| 10
|-
| 43
| 324.5283019
| RKm3, kn3
| Wide Minor Third
| Ft<↓, F↑/, Gdb<
| 4
| 9
|-
| 44
| 332.0754717
| kN3, ud4
| Lesser Supraminor Third, Infra-Diminished Fourth
| Ft>↓, Gdb>
| 1
| 9
|-
| 45
| 339.6226415
| KKm3, rn3, Rud4
| Greater Supraminor Third, Retrodiptolemaic Diminished Fourth
| Ft<\, F↑↑, Gdb<↑\, Gb↓↓
| -1
| 8
|-
| 46
| 347.1698113
| n3, rKud4
| Artoneutral Third, Lesser Sub-Diminished Fourth
| Ft<, Gdb<↑
| 0
| 7
|-
| 47
| 354.7169811
| N3, sd4, Kud4
| Tendoneutral Third, Greater Sub-Diminished Fourth
| Ft>, Gdb>↑
| -1
| 7
|-
| 48
| 362.2641509
| kkM3, RN3, kd4
| Lesser Submajor Third, Retroptolemaic Diminished Fourth
| Ft>/, F#↓↓, Gb↓
| 0
| 8
|-
| 49
| 369.8113208
| Kn3, Rkd4
| Greater Submajor Third, Artoretromean Diminished Fourth
| Ft<↑, Gb↓/
| -1
| 9
|-
| 50
| 377.3584906
| rkM3, KN3, rd4
| Narrow Major Third, Tendoretromean Diminished Fourth
| Ft>↑, F#↓\, Gb\
| 3
| 9
|-
| 51
| 384.9056604
| kM3, d4
| Ptolemaic Major Third, Pythagorean Diminished Fourth
| Gb, F#↓
| 8
| 10
|-
| 52
| 392.4528302
| RkM3, Rd4
| Artomean Major Third, Artomean Diminished Fourth
| Gb/, F#↓/
| 4
| 10
|-
| 53
| 400
| rM3, rKd4
| Tendomean Major Third, Tendomean Diminished Fourth
| F#\, Gb↑\
| 1
| 9
|-
| 54
| 407.5471698
| M3, Kd4
| Pythagorean Major Third, Ptolemaic Diminished Fourth
| F#, Gb↑
| -1
| 9
|-
| 55
| 415.0943396
| RM3, kUd4
| Wide Major Third, Lesser Super-Diminished Fourth
| F#/, Gd<↓, Gb↑/
| 0
| 8
|-
| 56
| 422.6415094
| rKM3, RkUd4
| Narrow Supermajor Third, Greater Super-Diminished Fourth
| F#↑\, Gd>↓
| -1
| 7
|-
| 57
| 430.1886792
| KM3, rUd4, KKd4
| Lesser Supermajor Third, Diptolemaic Diminished Fourth
| F#↑, Gd<\, Gb↑↑
| -1
| 6
|-
| 58
| 437.7358491
| SM3, kUM3, rm4, Ud4
| Greater Supermajor Third, Ultra-Diminished Fourth
| Gd<, F#↑/
| 0
| 5
|-
| 59
| 445.2830189
| m4, RkUM3
| Paraminor Fourth, Wide Supermajor Third
| Gd>, Ft#>↓
| -1
| 3
|-
| 60
| 452.8301887
| Rm4, KKM3, rUM3
| Wide Paraminor Fourth, Narrow Ultramajor Third
| Gd>/, F#↑↑, G↓↓
| -2
| 1
|-
| 61
| 460.3773585
| UM3, rKm4
| Ultramajor Third, Narrow Grave Fourth
| Gd<↑, Ft#<
| -4
| -2
|-
| 62
| 467.9245283
| s4, Km4
| Lesser Grave Fourth, Wide Ultramajor Third
| Gd>↑, G↓\
| -7
| -4
|-
| 63
| 475.4716981
| k4
| Greater Grave Fourth
| G↓, Abb
| -6
| -5
|-
| 64
| 483.0188679
| Rk4
| Wide Grave Fourth
| G↓/
| -4
| 0
|-
| 65
| 490.5660377
| r4
| Narrow Fourth
| G\
| 1
| 5
|-
| 66
| 498.1132075
| P4
| Perfect Fourth
| G
| 9
| 10
|-
| 67
| 505.6603774
| R4
| Wide Fourth
| G/
| 1
| 8
|-
| 68
| 513.2075472
| rK4
| Narrow Acute Fourth
| G↑\
| -3
| 6
|-
| 69
| 520.7547170
| K4
| Lesser Acute Fourth
| G↑
| -5
| 5
|-
| 70
| 528.3018868
| S4, kM4
| Greater Acute Fourth
| Gt<↓, G↑/, Adb<
| -3
| 5
|-
| 71
| 535.8490566
| RkM4, ud5
| Wide Acute Fourth, Infra-Diminished Fifth
| Gt>↓, Adb>
| -2
| 5
|-
| 72
| 543.3962264
| rM4, Rud5
| Narrow Paramajor Fourth, Retrodiptolemaic Diminished Fifth
| Gt<\, G↑↑, Ab↓↓
| -1
| 6
|-
| 73
| 550.9433962
| M4, rKud5
| Paramajor Fourth, Lesser Sub-Diminished Fifth
| Gt<, Adb<↑
| 0
| 7
|-
| 74
| 558.4905660
| RM4, uA4, Kud5
| Infra-Augmented Fourth, Greater Sub-Diminished Fifth
| Gt>, Adb>↑
| -2
| 5
|-
| 75
| 566.0377358
| kkA4, RuA4, kd5
| Diptolemaic Augmented Fourth, Retroptolemaic Diminished Fifth
| Gt>/, G#↓↓, Ab↓
| -3
| 4
|-
| 76
| 573.5849057
| rKuA4, Rkd5
| Lesser Sub-Augmented Fourth, Artoretromean Diminished Fifth
| Gt<↑, Ab↓/
| -2
| 4
|-
| 77
| 581.1320755
| KuA4, rd5
| Greater Sub-Augmented Fourth, Tendoretromean Diminished Fifth
| Gt>↑, Ab\
| 0
| 5
|-
| 78
| 588.6792458
| kA4, d5
| Ptolemaic Augmented Fourth, Pythagorean Diminished Fifth
| Ab, G#↓
| -5
| 6
|-
| 79
| 596.2264151
| RkA4, Rd5
| Artomean Augmented Fourth, Artomean Diminished Fifth
| G#↓/, Ab/
| -9
| 7
|-
| 80
| 603.7735849
| rKd5, rA4
| Tendomean Diminished Fifth, Tendomean Augmented Fourth
| Ab↑\, G#\
| -9
| 7
|-
| 81
| 611.3207547
| Kd5, A4
| Ptolemaic Diminished Fifth, Pythagorean Augmented Fourth
| Ab↑, G#
| -5
| 6
|-
| 82
| 618.8679245
| kUd5, RA4
| Lesser Super-Diminished Fifth, Artoretromean Augmented Fourth
| Ad<↓, G#/
| 0
| 5
|-
| 83
| 626.4150943
| RkUd5, rKA4
| Greater Super-Diminished Fifth, Tendoretromean Augmented Fourth
| Ad>↓, G#↑\
| -2
| 4
|-
| 84
| 633.9622642
| KKd5, rUDd5, KA4
| Diptolemaic Diminished Fifth, Retroptolemaic Augmented Fourth
| Ad<\, Ab↑↑, G#↑
| -3
| 4
|-
| 85
| 641.5094340
| rm5, Ud5, kUA4
| Ultra-Diminished Fifth, Lesser Super-Augmented Fourth
| Ad<, Gt#<↓
| -2
| 5
|-
| 86
| 649.0566038
| m5, RkUA4
| Paraminor Fifth, Greater Super-Augmented Fourth
| Ad>, Gt#>↓
| 0
| 7
|-
| 87
| 656.6037736
| Rm5, rUA4
| Wide Paraminor Fifth, Retrodiptolemaic Augmented Fourth
| Ad>/, G#↑, Ab↑↑
| -1
| 6
|-
| 88
| 664.1509434
| rKm5, UA4
| Narrow Grave Fifth, Ultra-Augmented Fourth
| Ad<↑, Gt#<
| -2
| 5
|-
| 89
| 671.6981132
| s5, Km5
| Lesser Grave Fifth
| Ad>↑, A↓\, Gt#>
| -3
| 5
|-
| 90
| 679.2452830
| k5
| Greater Grave Fifth
| A↓
| -5
| 5
|-
| 91
| 686.7924528
| Rk5
| Wide Grave Fifth
| A↓/
| -3
| 6
|-
| 92
| 694.3396226
| r5
| Narrow Fifth
| A\
| 1
| 8
|-
| 93
| 701.8867925
| P5
| Perfect Fifth
| A
| 9
| 10
|-
| 94
| 709.4339622
| R5
| Wide Fifth
| A/
| 1
| 5
|-
| 95
| 716.9811321
| rK5
| Narrow Acute Fifth
| A↑\
| -4
| 0
|-
| 96
| 724.5283019
| K5
| Lesser Acute Fifth
| A↑, Gx
| -6
| -5
|-
| 97
| 732.0754717
| S5, kM5
| Greater Acute Fifth, Narrow Inframinor Sixth
| At<↓, A↑/
| -7
| -4
|-
| 98
| 739.6226415
| um6, RkM5
| Inframinor Sixth, Wide Acute Fifth
| At>↓, Bdb>
| -4
| -2
|-
| 99
| 747.1698113
| Rm4, KKM3, rUM3
| Narrow Paramajor Fifth, Wide Inframinor Sixth
| At<\, Bb↓↓, A↑↑
| -2
| 1
|-
| 100
| 754.7169811
| M5, rKum6
| Paramajor Fifth, Narrow Subminor Sixth
| At<, Bdb<↑
| -1
| 3
|-
| 101
| 762.2641509
| sm6, Kum6, RM5, uA5
| Lesser Subminor Sixth, Infra-Augmented Fifth
| At>, Bb↓\
| 0
| 5
|-
| 102
| 769.8113208
| km6, RuA5, kkA5
| Greater Subminor Sixth, Diptolemaic Augmented Fifth
| Bb↓, At>/, A#↓↓
| -1
| 6
|-
| 103
| 777.3584906
| Rkm6, rKuA5
| Wide Subminor Sixth, Lesser Sub-Augmented Fifth
| Bb↓/, At<↑
| -1
| 7
|-
| 104
| 784.9056604
| rm6, KuA5
| Narrow Minor Sixth, Greater Sub-Augmented Fifth
| Bb\, At>↑, A#↓\
| 0
| 8
|-
| 105
| 792.4528302
| m6, kA5
| Pythagorean Minor Sixth, Ptolemaic Augmented Fifth
| Bb, A#↓
| -1
| 9
|-
| 106
| 800
| Rm6, RkA5
| Artomean Minor Sixth, Artomean Augmented Fifth
| Bb/, A#↓/
| 1
| 9
|-
| 107
| 807.5471698
| rKm6, rA5
| Tendomean Minor Sixth, Tendomean Augmented Fifth
| A#\, Bb↑\
| 4
| 10
|-
| 108
| 815.0943396
| Km6, A5
| Ptolemaic Minor Sixth, Pythagorean Augmented Fifth
| A#, Bb↑
| 8
| 10
|-
| 109
| 822.6415094
| RKm6, kn6, RA5
|Wide Minor Sixth, Artoretromean Augmented Fifth
| Bd<↓, Bb↑/, A#/
| 3
| 9
|-
| 110
| 830.1886792
| kN6, rKA5
| Lesser Supraminor Sixth, Tendoretromean Augmented Fifth
| Bd>↓, A#↑\
| -1
| 9
|-
| 111
| 837.7358491
| KKm6, rn6, KA5
| Greater Supraminor Sixth, Retroptolemaic Augmented Fifth
| Bd<\, Bb↑↑, A#↑
| 0
| 8
|-
| 112
| 845.2830189
| n6, SA5, kUA5
| Artoneutral Sixth, Lesser Super-Augmented Fifth
| Bd<, At#<↓
| -1
| 7
|-
| 113
| 852.8301887
| N6, RkUA5
| Tendoneutral Sixth, Greater Super-Augmented Fifth
| Bd>, At#>↓
| 0
| 7
|-
| 114
| 860.3773585
| kkM6, RN6, rUA5
| Lesser Submajor Sixth, Retrodiptolemaic Augmented Fifth
| Bd>/, B↓↓, At#>↓/, A#↑↑
| -1
| 8
|-
| 115
| 867.9245283
| Kn6, UA5
| Greater Submajor Sixth, Ultra-Augmented Fifth
| Bd<↑, At#<
| 1
| 9
|-
| 116
| 875.4716981
| rkM6, KN6
| Narrow Major Sixth
| Bd>↑, B↓\, At#>
| 4
| 9
|-
| 117
| 883.0188679
| kM6
| Ptolemaic Major Sixth
| B↓, Cb
| 7
| 10
|-
| 118
| 890.5660377
| RkM6
| Artomean Major Sixth
| B↓/
| 4
| 10
|-
| 119
| 898.1132075
| rM6
| Tendomean Major Sixth
| B\
| 1
| 9
|-
| 120
| 905.6603774
| M6
| Pythagorean Major Sixth
| B
| -1
| 9
|-
| 121
| 913.2075472
| RM6
| Wide Major Sixth
| B/, Cd<↓
| 0
| 8
|-
| 122
| 920.7547170
| rKM6
| Narrow Supermajor Sixth
| B↑\, Cd>↓
| -1
| 8
|-
| 123
| 928.3018868
| KM6
| Lesser Supermajor Sixth
| B↑, Cd<\, Cb↑↑, Ax
| -1
| 7
|-
| 124
| 935.8490566
| SM6, kUM6
| Greater Supermajor Second, Narrow Inframinor Seventh
| Cd<, Bt<↓, B↑/
| 0
| 7
|-
| 125
| 943.3962264
| um7, RkUM6
| Inframinor Seventh, Wide Supermajor Sixth
| Cd>, Bt>↓
| -1
| 7
|-
| 126
| 950.9433962
| KKM6, kkm7, rUM6, Rum7
| Narrow Ultramajor Sixth, Wide Inframinor Seventh, Semitwelfth
| Bt<\, Cd>/, B↑↑, C↓↓
| 0
| 8
|-
| 127
| 958.4905660
| UM6, rKum7
| Ultramajor Sixth, Narrow Subminor Seventh
| Bt<, Cd<↑
| -1
| 8
|-
| 128
| 966.0377358
| sm7, Kum7
| Lesser Subminor Seventh, Wide Ultramajor Sixth
| Bt>, Cd>↑, C↓\
| 0
| 9
|-
| 129
| 973.5849057
| km7
| Greater Subminor Seventh
| C↓, Bt>/, B#↓↓, Dbb
| -1
| 9
|-
| 130
| 981.1320755
| Rkm7
| Wide Subminor Seventh
| C↓/, Bt<↑
| -1
| 10
|-
| 131
| 988.6792458
| rm7
| Narrow Minor Seventh
| C\, Bt>↑
| -1
| 10
|-
| 132
| 996.2264151
| m7
| Pythagorean Minor Seventh
| C, B#↓
| -2
| 10
|-
| 133
| 1003.7735849
| Rm7
| Artomean Minor Seventh
| C/, B#↓/
| -2
| 10
|-
| 134
| 1011.3207547
| rKm7
| Tendomean Minor Seventh
| C↑\, B#\
| -3
| 10
|-
| 135
| 1018.8679245
| kM2
| Ptolemaic Minor Seventh
| C↑, B#
| -3
| 10
|-
| 136
| 1026.4150943
| RKm7, kn7
| Wide Minor Seventh
| Ct<↓, C↑/, Ddb<, B#/
| -4
| 10
|-
| 137
| 1033.9622642
| kN7, ud8
| Lesser Supraminor Seventh, Infra-Diminished Octave
| Ct>↓, Ddb>, B#↑\
| -5
| 9
|-
| 138
| 1041.5094340
| KKm7, rn7, Rud8
| Greater Supraminor Seventh, Retrodiptolemaic Diminished Octave
| Ct<\, C↑↑, Ddb<↑\, Db↓↓
| -6
| 8
|-
| 139
| 1049.0566038
| n7, rKud8
| Artoneutral Seventh, Lesser Sub-Diminished Octave
| Ct<, Ddb<↑
| -7
| 6
|-
| 140
| 1056.6037736
| N7, sd8
| Tendoneutral Seventh, Greater Sub-Diminished Octave
| Ct>, Ddb>↑
| -8
| 5
|-
| 141
| 1064.1509434
| kkM7, RN7, kd8
| Lesser Submajor Seventh, Diptolemaic Major Seventh, Retroptolemaic Diminished Octave
| Ct>/, C#↓↓, Db↓
| -7
| 6
|-
| 142
| 1071.6981132
| Kn7, Rkd8
| Greater Submajor Seventh, Artoretromean Diminished Octave
| Ct<↑, Db↓/
| -6
| 8
|-
| 143
| 1079.2452830
| rkM7, KN7, rd8
| Narrow Major Seventh, Tendoretromean Diminished Octave
| Ct>↑, C#↓\, Db\
| -5
| 9
|-
| 144
| 1086.7924528
| kM7, d8
| Ptolemaic Major Seventh, Pythagorean Diminished Octave
| Db, C#↓
| -5
| 10
|-
| 145
| 1094.3396226
| RkM7, Rd8
| Artomean Major Seventh, Artomean Diminished Octave
| Db/, C#↓/
| -5
| 10
|-
| 146
| 1101.8867925
| rM7, rKd8
| Tendomean Major Seventh, Tendomean Diminished Octave
| C#\, Db↑\
| -6
| 10
|-
| 147
| 1109.4339622
| M7, Kd8
| Pythagorean Major Seventh, Ptolemaic Diminished Octave
| C#, Db↑
| -6
| 10
|-
| 148
| 1116.9811321
| RM7, kUd8
| Wide Major Seventh, Lesser Super-Diminished Octave
| C#/, Dd<↓
| -7
| 9
|-
| 149
| 1124.5283019
| rKM7, RkUd8
| Narrow Supermajor Seventh, Greater Super-Diminished Octave
| C#↑\, Dd>↓
| -7
| 9
|-
| 150
| 1132.0754717
| km2, RuA1, kkA1
| Lesser Supermajor Seventh, Diptolemaic Diminished Octave
| C#↑, Db↑↑
| -8
| 9
|-
| 151
| 1139.6226415
| SM7, kUM7, Ud8
| Greater Supermajor Seventh, Narrow Infraoctave, Ultra-Diminished Octave
| Dd<, C#↑/
| -8
| 10
|-
| 152
| 1147.1698113
| u8, RkUM7
| Infraoctave, Wide Supermajor Seventh
| Dd>, Ct#>↓
| -9
| 10
|-
| 153
| 1154.7169811
| KKM7, rUM7, Ru8
| Narrow Ultramajor Seventh, Wide Infraoctave
| C#↑↑, Dd>/
| -9
| 10
|-
| 154
| 1162.2641509
| UM7, rKu8
| Ultramajor Seventh, Wide Superprime
| Ct#<, Dd<↑
| -9
| 10
|-
| 155
| 1169.8113208
| s8, Ku8
| Lesser Suboctave, Wide Ultramajor Seventh
| Ct#>, Dd>↑
| -10
| 3
|-
| 156
| 1177.3584906
| k8
| Greater Suboctave
| D↓
| -10
| -3
|-
| 157
| 1184.9056604
| Rk8
| Wide Suboctave
| D↓/
| -10
| -10
|-
| 158
| 1192.4528302
| r8
| Narrow Octave
| D\
| 0
| 0
|-
| 159
| 1200
| P8
| Perfect Octave
| D
| 10
| 10
|}

== Trines and Tetrachords ==
159edo has multiple types of trine and tetrachord. While trines are important in the aspects of 159edo music theory derived from Medieval and Neo-Medieval music theory, the concept of tetrachords is significantly older.

{| class="wikitable"
|+Table of 159edo Trines
|-
! Name
! Notation (from D)
! Steps
! Approximate JI
! Notes
|-
| Otonal Perfect
| D, A, D
| 0, 93, 0
| 2:3:4
| This is the first of two trines that can be considered fully-resolved in Medieval and Neo-Medieval harmony
|-
| Utonal Perfect
| D, G, D
| 0, 66, 0
| 1/(2:3:4)
| This is the second of two trines that can be considered fully-resolved in Medieval and Neo-Medieval harmony
|-
| Otonal Archagall
| D, G\, D
| 0, 65, 0
| 64:85:128
| This trine is the first of two that are often used in the extended harmony of t<IV chords and is considered a dissonance
|-
| Utonal Archagall
| D, A/, D
| 0, 94, 0
| 1/(64:85:128)
| This trine is the second of two that are often used in the extended harmony of t<IV chords and is considered a dissonance
|-
| Bass-Up Marvelous
| D, A\, D
| 0, 92, 0
| 75:112:150
| This dissonant trine is the first of two that are formed from stacking identical approximations of the LCJI neutral third
|-
| Treble-Down Marvelous
| D, G/, D
| 0, 67, 0
| 1/(75:112:150)
| This dissonant trine is the second of two that are formed from stacking identical approximations of the LCJI neutral third
|-
| Narrow Supranaiadic
| D, G↓\, D
| 0, 62, 0
| 16:21:32
| This dissonant trine is common in essentially tempered chords
|-
| Wide Subcocytic
| D, A↑/, D
| 0, 97, 0
| 1/(16:21:32)
| This dissonant trine is common in essentially tempered chords
|-
| Subcocytic
| D, A↑, D
| 0, 96, 0
| 160:243:320
| This dissonant trine is common in essentially tempered chords
|-
| Supranaiadic
| D, G↓, D
| 0, 63, 0
| 1/(160:243:320)
| This dissonant trine is common in essentially tempered chords
|-
| Wide Supranaiadic
| D, G↓/, D
| 0, 64, 0
| 25:33:50
| This dissonant trine is on the outer edge of the diatonic range and is common in essentially tempered chords
|-
| Narrow Subcocytic
| D, A↑\, D
| 0, 95, 0
| 1/(25:33:50)
| This dissonant trine is on the outer edge of the diatonic range and is common in essentially tempered chords
|-
| Wide Naiadic
| D, Gd<↑, D
| 0, 61, 0
| 135:176:270
| This dissonant trine is among the more consistently complex
|-
| Narrow Cocytic
| D, At>↓, D
| 0, 98, 0
| 1/(135:176:270)
| This dissonant trine is among the more consistently complex
|-
| Naiadic
| D, Gd>/, D
| 0, 60, 0
| 10:13:20
| This dissonant trine is relatively simple and thus expected to be rather common
|-
| Cocytic
| D, At<\, D
| 0, 99, 0
| 1/(10:13:20)
| This dissonant trine is relatively simple and thus expected to be rather common
|-
| Wide Cocytic
| D, At<, D
| 0, 100, 0
| 11:17:22
| This essentially tempered trine is very likely to be used as a basis for cocytic triads
|-
| Narrow Niadic
| D, Gd>, D
| 0, 59, 0
| 1/(11:17:22)
| This essentially tempered trine is very likely to be used as a partial basis for suspended chords
|-
| Narrow Supradusthumic
| D, Ad<↑, D
| 0, 89, 0
| 128:189:256
| This dissonant trine is common in essentially tempered chords
|-
| Wide Subagallic
| D, Gt>↓, D
| 0, 70, 0
| 1/(128:189:256)
| This dissonant trine is common in essentially tempered chords
|-
| Subagallic
| D, G↑, D
| 0, 69, 0
| 20:27:40
| This dissonant trine is very likely to show up in non-meantone diatonic contexts
|-
| Supradusthumic
| D, A↓, D
| 0, 90, 0
| 1/(20:27:40)
| This dissonant trine is very likely to show up in non-meantone diatonic contexts
|-
| Narrow Subagallic
| D, G↑\, D
| 0, 68, 0
| 90:121:180
| This dissonant trine is on the outer edge of the diatonic range
|-
| Wide Supradusthumic
| D, A↓/, D
| 0, 91, 0
| 1/(90:121:180)
| This dissonant trine is on the outer edge of the diatonic range
|-
| Wide Agallic
| D, Gt<, D
| 0, 73, 0
| 8:11:16
| This ambisonant trine is very likely to be used as a partial basis for suspended chords
|-
| Narrow Dusthumic
| D, Ad>, D
| 0, 86, 0
| 1/(8:11:16)
| This ambisonant trine is very likely to be used as a basis for dusthumic triads
|-
| Dusthumic
| D, Ad<\, D
| 0, 87, 0
| 128:187:256
| This dissonant trine is common in essentially tempered chords
|-
| Agallic
| D, Gt<\, D
| 0, 72, 0
| 1/(128:187:256)
| This dissonant trine is common in essentially tempered chords
|-
| Narrow Agallic
| D, Gt>↓, D
| 0, 71, 0
| 11:15:22
| This trine is very likely to be used as a partial basis for suspended chords
|-
| Wide Dusthumic
| D, Ad<↑, D
| 0, 88, 0
| 1/(11:15:22)
| This trine is very likely to be used as a basis for dusthumic triads
|-
| Wide Subdusthumic
| D, Ad<, D
| 0, 85, 0
| 56:81:112
| This essentially tempered trine is likely to be used as a basis for subdusthumic triads
|-
| Narrow Supraagallic
| D, Gt>, D
| 0, 74, 0
| 1/(56:81:112)
| This essentially tempered trine is likely to be used as a partial basis for suspended chords
|-
| Subdusthumic
| D, Ab↑↑, D
| 0, 84, 0
| 9:13:18
| This essentially tempered trine is very likely to be used as a basis for subdusthumic triads
|-
| Supraagallic
| D, G#↓↓, D
| 0, 75, 0
| 1/(9:13:18)
| This essentially tempered trine is very likely to be used as a partial basis for suspended chords
|-
| Wide Supraagallic
| D, Gt<↑, D
| 0, 76, 0
| 256:357:512
| This essentially tempered trine is very likely to be used as a partial basis for suspended chords
|-
| Narrow Subdusthumic
| D, Ad>↓, D
| 0, 83, 0
| 1/(256:357:512)
| This essentially tempered trine is very likely to be used as a basis for subdusthumic triads
|-
| Narrow Hyperquartal
| D, Gt>↑, D
| 0, 77, 0
| 5:7:10
| This ambisonant trine is very common as a basis for diminished chords, and is very likely to be used as a partial basis for suspended chords
|-
| Wide Hypoquintal
| D, Ad<↓, D
| 0, 82, 0
| 1/(5:7:10)
| This ambisonant trine is very common as a basis for diminished chords, and is very likely to be used as a partial basis for suspended chords
|-
| Hyperquartal
| D, G#↓, D
| 0, 78, 0
| 32:45:64
| This trine is very likely to be used as a partial basis for suspended chords
|-
| Hypoquintal
| D, Ab↑, D
| 0, 81, 0
| 1/(32:45:64)
| This trine is very common as a basis for diminished chords
|-
| Narrow Hypoquintal
| D, Ab↑\, D
| 0, 80, 0
| 12:17:24
| This trine is very common as a basis for diminished chords
|-
| Wide Hyperquartal
| D, G#↓/, D
| 0, 79, 0
| 1/(12:17:24)
| This trine is very likely to be used as a partial basis for suspended chords
|}

User:Aura/On 159edo Music Theory (Part 1)

2026-03-30T22:41:39Z

Aura: Starting this page- might as well copy much of the first table's information from my userspace on the Xenharmonic Wiki

User:Aura

2026-03-30T22:17:27Z

Aura:

'''Aura''', short for '''DaffodilAura''', is a user who has most of his music on either YouTube or the Xenharmonic Wiki. He's one of a handful of people who specialize in [[159edo]]. He also has idea for how to extend functional harmony into the realm of microtonality, and is one of the main developers of [[Syntonic-Rastmic Subchroma notation]] or SRS notation for short.

'''On 159edo Music Theory'''

[[User:Aura/On 159edo Music Theory (Part 1)|Part 1]]

[[User:Aura/On 159edo Music Theory (Part 2)|Part 2]]

User:Aura

2026-03-30T22:14:19Z

Aura: Time to get stuff started

'''Aura''', short for '''DaffodilAura''', is a user who has most of his music on either YouTube or the Xenharmonic Wiki. He's one of a handful of people who specialize in [[159edo]]. He also has idea for how to extend functional harmony into the realm of microtonality.

'''On 159edo Music Theory'''

[[User:Aura/On 159edo Music Theory (Part 1)|Part 1]]

[[User:Aura/On 159edo Music Theory (Part 2)|Part 2]]

53edo

2026-03-29T03:05:46Z

Aura: /* 159edo */

'''53edo''', or 53 equal divisions of the octave, is the equal tuning featuring steps of (1200/53) ~= 22.64 cents, 53 of which stack to the perfect octave [[2/1]]. 53edo is an excellent approximation to Pythagorean tuning (stacking pure 3/2 fifths). Theoretical interest in this tuning system goes back to antiquity.

== Theory ==
Unless one has a set of accidentals for the syntonic comma (see the Notation section) one is left in the unenviable position of having to label a Ptolemaic major third the same way as the Pythagorean diminished fourth, for example. Apart from that issue, 53edo is very useful for 5-limit music.

==== Edostep interpretations ====
53edo's edostep has the following interpretations in the 2.3.5.7.13 subgroup:

* 65/64, the difference between the 13-limit tendoneutral third 16/13 and the classical major third 5/4
* 81/80 (the syntonic comma), the difference between 5/4 and the diatonic major third
* The Pythagorean comma, the difference between the Pythagorean diatonic and chromatic semitones
* 91/90, the difference between the 13-limit ultramajor third 13/10 and the septimal supermajor third 9/7
* 64/63, the difference between the diatonic major third and 9/7
* 512/507, the difference between the 13-limit neutral thirds

53edo tempers out the following commas:

* The schisma (the difference between 5/4 and the Pythagorean diminished fourth)
* The vulture comma (the difference between four 320/243 intervals and the tritave)
* The amiton (the difference between a stack of five 10/9 intervals and 27/16)
* The kleisma (the difference between a stack of three 25/24 intervals and 9/8)
* The semicomma (the difference between a stack of three 75/64 intervals and 8/5)
* 225/224 (the difference between 15/14 and 16/15)
* 385/384 (the difference between 77/64 and 6/5)
* 121/120 (the difference between 12/11 and 11/10)
* 625/624 (the difference between 25/24 and 26/25)
* 676/675 (the difference between a stack of two 15/13 intervals and the perfect fourth)

==== JI approximation ====
53edo is most usefully seen as a 2.3.5.7.13 tuning, but the 2.3.5.13 restriction is more accurate and shared with a number of its multiples, such as [[159edo]]. Because it is not a Meantone system, there are actually multiple potential diatonic scales to use for 5-limit harmony, one of which is the Zarlino diatonic scale (LMsLMLs), tuned in 53edo as 9-8-5-9-8-9-5, though this particular scale is arguably best used for Lydian or Locrian modes. There's also the Didymic diatonic scale, tuned in 53edo as 9-8-5-9-9-8-5, which is better suited for Ionian mode and Major tonality in general. However, 53edo also features a MOS diatonic of 9-9-4-9-9-9-4, which is basically the Pythagorean diatonic scale.

{{Harmonics in ED|53|31|0}}
{| class="wikitable"
|+Thirds in 53edo
!Quality
|Inframinor
|Subminor
|'''Farminor'''
|Nearminor
|Supraminor
|Submajor
|Nearmajor
|'''Farmajor'''
|Supermajor
|Ultramajor
|-
!Cents
|249
|272
|'''294'''
|317
|340
|362
|385
|'''408'''
|430
|453
|-
!Just interpretation
|15/13
|7/6, 75/64
|'''32/27'''
|6/5
|39/32
|16/13
|5/4
|'''81/64'''
|9/7, 32/25
|13/10
|-
!Steps
|11
|12
|'''13'''
|14
|15
|16
|17
|'''18'''
|19
|20
|}
Diatonic thirds are bolded.

=== Chords ===
53edo has four different flavors of minor and major intervals as well as supraminor and submajor intervals. Its inframinor and ultramajor thirds approximate 15/13 and 13/10 respectively. At the same time, 53edo's subminor and supermajor intervals approximate 7/6 and 9/7. Then there's the novaminor and novamajor thirds, which are extremely close approximations of Pythagorean minor and major thirds and can be referred to as such. There are also the pentaminor and pentamajor thirds, which are very close approximations of the Ptolemaic minor and major thirds and can also be referred to as such. Finally, the supraminor and submajor thirds approximate 39/32 and 16/13. For fourth-bounded triads, there's only really five options. The first two, which involve the approximations of 9/8 and 32/27, have a marked propensity to cause crowding, and thus are dissonant. Then there's the next two, the latal triads, which involve the approximations of 8/7 and 7/6, and which, due to their tuning are markedly less dissonant, but still dissonant. Finally, the last option, which splits the perfect fourth cleanly in half, is an ambisonance- that is, an interval that is halfway between the extremes of consonance and dissonance.

== Notation ==
This section provides some of the options for notating 53edo.

=== Pythagorean notation ===
In 53edo, the space between each of the notes that is separated by 2 steps in 12edo is instead 9 steps; notes separated by a single step in 12edo have to be distinguished from each other as the Pythagorean diatonic semitone is 4 steps while the Pythagorean chromatic semitone is 5 steps. Furthermore, the Pythagorean comma is a single step in 53edo, unlike in 12edo where it's tempered out. It is important to understand the usage of enharmonic equivalence here; unlike in systems such as 31edo where each note has an easily derivable "canonical" notation, it is important to understand the multiple faces of each of 53edo's pitches (which some might consider as a downside of using the Pythagorean system).

{| class="wikitable"
|+
!
!
|-
| colspan="2" |D
|-
|^^Ebb
|^D
|-
|vvEb
|^^D
|-
|vEb
|vvD#
|-
|Eb
|vD#
|-
|^Eb
|D#
|-
|^^Eb
|^D#
|-
|vvE
|^^D#
|-
|vE
|vvDx
|-
| colspan="2" |E
|}

==== Ups and Downs ====
Ups and downs naturally reflect 53edo's structure, as 5/4 is downmajor, 81/64 is major, and 9/7 is upmajor.

==== Syntonic-Rastmic Subchroma notation ====
Syntonic-Rastmic Subchroma notation, or SRS notation for short, uses '''synsharp''' and '''synflat''' as accidentals to cover the syntonic comma. However, while SRS notation is a 2.3.5.11 notation, only the 2.3.5 portion of the notation for 53edo is shared with multiples like 159edo.

==== Accidentals ====
53edo's accidentals, as mentioned and demonstrated previously, consist of sharps and flats, as well as either up and down accidentals, or, alternatively, synsharps and synflats and their derivatives.
{{Navbox EDO}}
{{Cat|Edos}}

== Multiples ==

===106edo===
106edo has inconsistent 11th and 17th harmonics, and also loses some ability to be consistent that 53edo has due to inconsistencies in the 7-odd-limit.

{{Harmonics in ED|106|31|0}}

===159edo===
''Main article: [[159edo]]''

159edo corrects 53edo's approximate 11th and 17th harmonics to near-just qualities, and also improves the 7th harmonic to a lesser extent, resulting in it being consistent to the 17-odd-limit. If you go further than that, you're forced to choose between the 17the harmonic on one hand and both the 19th and 29th harmonics on the other, but you do get a good 23rd harmonic regardless. In addition, you also gain access to a set of intervals that approximates those of simpler systems such as [[10edo]], [[12edo]], [[13edo]], [[14edo]], [[17edo]], [[19edo]], [[22edo]], [[24edo]] and [[31edo]] among others, opening up additional compositional techniques.

{{Harmonics in ED|159|31|0}}

159edo

2026-03-29T03:03:28Z

Aura: Perhaps using the term "intervals" is better than using the term "pitches" here due to the latter being a bit of an oversimplification...

'''159edo''', or 159 equal divisions of the octave, is the equal tuning featuring steps of (1200/159) ~= 7.55 cents, 159 of which stack to the perfect octave [[2/1]]. Like [[53edo]], 159edo is an excellent approximation to Pythagorean tuning (stacking pure 3/2 fifths), however it includes near-just approximations of the 11th and 17th harmonics, and a slightly more accurate 7th harmonic, resulting in consistency up to the 17-odd-limit. The step-size, being slightly above the melodic just noticeable difference (JND) of 5 cents as well as more than twice the harmonic JND of the average trained musician at 3.5 cents, enables one to perform fluid modulations by means of [[comma pump]]s as well as by step substitutions. Furthermore, 159edo, like a number of higher edos, is characterized by being capable of imitating the intervals of smaller tuning systems - in this case, you get detemperings of [[10edo]], [[12edo]], [[13edo]], [[14edo]], [[17edo]], [[19edo]], [[22edo]], [[24edo]] and [[31edo]] among others with errors smaller than the melodic JND.

The interval qualities supported by 159edo are many, however, while every step of 159edo can be interpreted harmonically or subharmonically as being a 17-limit interval or simpler, some of the intervals you get have rather complex interpretations in terms of odd-limit. While the [[perfect fifth]] is really only divisible by three due to 159edo being the three-fold multiple of 53edo- resulting in [[slendric]] temperament and hence 159edo's distinction from 53 in the 7-limit- the [[perfect fourth]] has a little more to offer in terms of divisions. For starters, the perfect fourth can be divided into two instances of 15/13, giving us [[island]] temperament. Dividing the perfect fourth into three instances of 11/10 gives us [[pine]] temperament. Dividing the perfect fourth into six instances of an interval which can be interpreted as 21/20 and 22/21 tempered together gives us [[sextilifourths]] temperament. The perfect fourth can also be cut into eleven intervals which, individually, are half of a Pythagorean limma, giving us a number of temperaments based on the exact interpretation of the semilimma.

On top of all that, there are a number of microtemperament-based structures also supported by 159edo, each of which provides some decent, unexpected melodic possibilities. For instance, one can split the Pythagorean minor third into three instances of 128/121 which gives us [[nexus]] temperament- a temperament which also happens to split the Pythagorean diatonic semitone into two, and the octave into three. One can also split the Ptolemaic minor third into three instances of 17/16, giving us [[archagall]] temperament, which is named for certain tunings found in other temperaments producing fractal-like acoustics. There's also the ability to split the Ptolemaic major sixth into six instances of 11/9, leading to [[parimic]] temperament. In addition, there's also the ability to split the septimal supermajor third into two instances of 17/15, leading to [[fidesmic]] temperament, which acts like a more accurate rendition of [[archy]] temperament in a different subgroup, and this can be exploited for modulation purposes. As if that weren't enough, there's the possibility of splitting the septimal subminor third into five instances of 33/32, leading to [[quartismic]] temperament. Furthermore, there's the possibility of splitting the greater tridecimal neutral tenth into three instances of 27/20, producing [[phaotismic]] temperament, and the list goes on.

== Theory ==
159edo was first used for maqams by Ozan Yarman. It was later put to use by Aura for its ability to handle near-just quartertones derived from the 2.3.11 subgroup on top of the 5-limit foundation provided by 53edo.

==== Edostep interpretations ====
159edo's edostep has the following interpretations in the 2.3.5.11.17 subgroup:

* 243/242, the difference between the 11-limit artoneutral third 11/9, and the 11-limit tendoneutral third 27/22
* 256/255, the difference between 16/15 and 17/16
* 289/288, the difference between 17/16 and 18/17

159edo tempers out the following commas in the 17-limit:

* The schisma (the difference between 5/4 and the Pythagorean diminished fourth)
* The vulture comma (the difference between four 320/243 intervals and the tritave)
* The amiton (the difference between a stack of five 10/9 intervals and 27/16)
* The kleisma (the difference between a stack of three 25/24 intervals and 9/8)
* The semicomma (the difference between a stack of three 75/64 intervals and 8/5)
* The gamelisma (the difference between a stack of three 8/7 intervals and 3/2)
* 385/384 (the difference between 77/64 and 6/5)
* The pine comma (the difference between a stack of three 11/10 intervals and 4/3)
* 625/624 (the difference between 25/24 and 26/25)
* 676/675 (the difference between a stack of two 15/13 intervals and the perfect fourth)
* The twosquare comma (the difference between a stack of two 33/32 intervals and 17/16)

==== JI approximation ====
Although 159edo inherits its approximations of the 5-limit from 53edo, the 5th harmonic can nonetheless be stacked twice without accumulating too much error, rendering it sufficient for Western Classical usage. While the 7th harmonic is technically more accurate in terms of absolute error than in 53edo, the relative error doesn't allow one to stack more than one instance of 7/4 without excessive error accumulation, and the same is true with 13/8. Regardless, the inconsistency remains less than 10 cents even when either interval is stacked three times, and 13/7 or 14/13 is tuned almost perfectly. As a whole, 159edo is characterized by its combination of accuracy in the 2.3.5.11.17 subgroup, and a series of compromises in the 7.13.19.23.29 subgroup- among the compromises are the slendric, [[marveltwin]], [[nestoria]], [[minor semivicemic]] and [[brunisimic]] temperaments.
{{Harmonics in ED|159|31|0}}

Currently, the [[ADIN]] system used for interval naming elsewhere on this site fails for 159edo, so another set of interval names will be used here, though the ADIN names will be referenced in places. Furthermore, because there are so many thirds, what is usually a single chart will be split into five.
{| class="wikitable"
|+Inframinor and Subminor Thirds in 159edo
!Quality
|Narrow Inframinor
|Inframinor
|Wide Inframinor
|Narrow Subminor
|Lesser Subminor (Septiminor)
|Greater Subminor
|Wide Subminor
|-
!Cents
|234
|242
|249
|257
|264
|272
|279
|-
!Just interpretation
|340/297
|1024/891
|15/13
|51/44
|7/6
|117/100
|20/17
|-
!Steps
|31
|32
|33
|34
|35
|36
|37
|}
{| class="wikitable"
|+Minor Thirds in 159edo
!Quality
|Narrow Minor (Gothminor)
|'''Pythagorean Minor (Triminor)'''
|Artomean Minor
|Tendomean Minor
|Ptolemaic Minor (Pentaminor)
|Wide Minor
|-
!Cents
|287
|'''294'''
|302
|309
|317
|325
|-
!Just interpretation
|33/28, 13/11
|'''32/27'''
|25/21
|153/128
|6/5
|135/112
|-
!Steps
|38
|'''39'''
|40
|41
|42
|43
|}
{| class="wikitable"
|+Supraminor, Neutral and Submajor Thirds in 159edo
!Quality
|Lesser Supraminor
|Greater Supraminor
|Artoneutral
|Tendoneutral
|Lesser Submajor
|Greater Submajor
|-
!Cents
|332
|340
|347
|355
|362
|370
|-
!Just interpretation
|40/33, 63/52
|39/32, 17/14
|11/9
|27/22
|16/13, 21/17
|99/80, 26/21
|-
!Steps
|44
|45
|46
|47
|48
|49
|}
{| class="wikitable"
|+Major Thirds in 159edo
!Quality
|Narrow Major
|Ptolemaic Major (Pentamajor)
|Artomean Major
|Tendomean Major
|'''Pythagorean Major (Trimajor)'''
|Wide Major (Gothmajor)
|-
!Cents
|377
|385
|392
|400
|'''408'''
|415
|-
!Just interpretation
|56/45
|5/4
|64/51
|63/50
|'''81/64'''
|14/11, 33/26
|-
!Steps
|50
|51
|52
|53
|'''54'''
|55
|}
{| class="wikitable"
|+Supermajor and Ultramajor Thirds in 159edo
!Quality
|Narrow Supermajor
|Lesser Supermajor
|Greater Supermajor (Septimajor)
|Wide Supermajor
|Narrow Ultramajor
|Ultramajor
|Wide Ultramajor
|-
!Cents
|423
|430
|438
|445
|453
|460
|468
|-
!Just interpretation
|51/40
|50/39
|9/7
|22/17
|13/10
|2673/2048
|891/680
|-
!Steps
|56
|57
|58
|59
|60
|61
|62
|}
Regular diatonic thirds are bolded.

=== Chords ===
159edo has a vast array of triads at its disposal, both fifth-bounded and fourth-bounded. However, the JI interpretation will tend to inform the usage of the various triads offered. The main exceptions to this rule involve chords that serve as detemperings of other, smaller tuning systems.

Talk:159edo

2026-03-29T02:45:43Z

Aura: /* Pre-emptively: */

== Pre-emptively: ==

- "pitch-hue" needs to be explained somewhere
- do not use "7-prime", use "prime 7" for stylistic consistency
-- [[User:Vector|Vector]] ([[User talk:Vector|talk]]) 00:58, 29 March 2026 (UTC)

Pitch hue, for lack of a better term is the psychoacoustic phenomenon that makes a C a C and a D a D, at least to those with some form of octave-equivalent absolute pitch. This may need to be explained in its own article. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 02:44, 29 March 2026 (UTC)

Talk:159edo

2026-03-29T02:44:35Z

Aura: /* Pre-emptively: */

== Pre-emptively: ==

- "pitch-hue" needs to be explained somewhere
- do not use "7-prime", use "prime 7" for stylistic consistency
-- [[User:Vector|Vector]] ([[User talk:Vector|talk]]) 00:58, 29 March 2026 (UTC)

Pitch hue, for lack of a better term is the psychoacoustic phenomenon that makes a C a C and a D a D, at least to those with some form of absolute pitch. This may need to be explained in its own article. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 02:44, 29 March 2026 (UTC)

159edo

2026-03-29T02:39:12Z

Aura:

'''159edo''', or 159 equal divisions of the octave, is the equal tuning featuring steps of (1200/159) ~= 7.55 cents, 159 of which stack to the perfect octave [[2/1]]. Like [[53edo]], 159edo is an excellent approximation to Pythagorean tuning (stacking pure 3/2 fifths), however it includes near-just approximations of the 11th and 17th harmonics, and a slightly more accurate 7th harmonic, resulting in consistency up to the 17-odd-limit. The step-size, being slightly above the melodic just noticeable difference (JND) of 5 cents as well as more than twice the harmonic JND of the average trained musician at 3.5 cents, enables one to perform fluid modulations by means of [[comma pump]]s as well as by step substitutions. Furthermore, 159edo, like a number of higher edos, is characterized by having a pitch hue palette that's capable of imitating the pitch-hue palettes of smaller tuning systems- in this case, you get detemperings of [[10edo]], [[12edo]], [[13edo]], [[14edo]], [[17edo]], [[19edo]], [[22edo]], [[24edo]] and [[31edo]] among others with errors smaller than the melodic JND.

The interval qualities supported by 159edo are many, however, while every step of 159edo can be interpreted harmonically or subharmonically as being a 17-limit interval or simpler, some of the intervals you get have rather complex interpretations in terms of odd-limit. While the [[perfect fifth]] is really only divisible by three due to 159edo being the three-fold multiple of 53edo- resulting in [[slendric]] temperament and hence 159edo's distinction from 53 in the 7-limit- the [[perfect fourth]] has a little more to offer in terms of divisions. For starters, the perfect fourth can be divided into two instances of 15/13, giving us [[island]] temperament. Dividing the perfect fourth into three instances of 11/10 gives us [[pine]] temperament. Dividing the perfect fourth into six instances of an interval which can be interpreted as 21/20 and 22/21 tempered together gives us [[sextilifourths]] temperament. The perfect fourth can also be cut into eleven intervals which, individually, are half of a Pythagorean limma, giving us a number of temperaments based on the exact interpretation of the semilimma.

On top of all that, there are a number of microtemperament-based structures also supported by 159edo, each of which provides some decent, unexpected melodic possibilities. For instance, one can split the Pythagorean minor third into three instances of 128/121 which gives us [[nexus]] temperament- a temperament which also happens to split the Pythagorean diatonic semitone into two, and the octave into three. One can also split the Ptolemaic minor third into three instances of 17/16, giving us [[archagall]] temperament, which is named for certain tunings found in other temperaments producing fractal-like acoustics. There's also the ability to split the Ptolemaic major sixth into six instances of 11/9, leading to [[parimic]] temperament. In addition, there's also the ability to split the septimal supermajor third into two instances of 17/15, leading to [[fidesmic]] temperament, which acts like a more accurate rendition of [[archy]] temperament in a different subgroup, and this can be exploited for modulation purposes. As if that weren't enough, there's the possibility of splitting the septimal subminor third into five instances of 33/32, leading to [[quartismic]] temperament. Furthermore, there's the possibility of splitting the greater tridecimal neutral tenth into three instances of 27/20, producing [[phaotismic]] temperament, and the list goes on.

== Theory ==
159edo was first used for maqams by Ozan Yarman. It was later put to use by Aura for its ability to handle near-just quartertones derived from the 2.3.11 subgroup on top of the 5-limit foundation provided by 53edo.

==== Edostep interpretations ====
159edo's edostep has the following interpretations in the 2.3.5.11.17 subgroup:

* 243/242, the difference between the 11-limit artoneutral third 11/9, and the 11-limit tendoneutral third 27/22
* 256/255, the difference between 16/15 and 17/16
* 289/288, the difference between 17/16 and 18/17

159edo tempers out the following commas in the 17-limit:

* The schisma (the difference between 5/4 and the Pythagorean diminished fourth)
* The vulture comma (the difference between four 320/243 intervals and the tritave)
* The amiton (the difference between a stack of five 10/9 intervals and 27/16)
* The kleisma (the difference between a stack of three 25/24 intervals and 9/8)
* The semicomma (the difference between a stack of three 75/64 intervals and 8/5)
* The gamelisma (the difference between a stack of three 8/7 intervals and 3/2)
* 385/384 (the difference between 77/64 and 6/5)
* The pine comma (the difference between a stack of three 11/10 intervals and 4/3)
* 625/624 (the difference between 25/24 and 26/25)
* 676/675 (the difference between a stack of two 15/13 intervals and the perfect fourth)
* The twosquare comma (the difference between a stack of two 33/32 intervals and 17/16)

==== JI approximation ====
Although 159edo inherits its approximations of the 5-limit from 53edo, the 5th harmonic can nonetheless be stacked twice without accumulating too much error, rendering it sufficient for Western Classical usage. While the 7th harmonic is technically more accurate in terms of absolute error than in 53edo, the relative error doesn't allow one to stack more than one instance of 7/4 without excessive error accumulation, and the same is true with 13/8. Regardless, the inconsistency remains less than 10 cents even when either interval is stacked three times, and 13/7 or 14/13 is tuned almost perfectly. As a whole, 159edo is characterized by its combination of accuracy in the 2.3.5.11.17 subgroup, and a series of compromises in the 7.13.19.23.29 subgroup- among the compromises are the slendric, [[marveltwin]], [[nestoria]], [[minor semivicemic]] and [[brunisimic]] temperaments.
{{Harmonics in ED|159|31|0}}

Currently, the [[ADIN]] system used for interval naming elsewhere on this site fails for 159edo, so another set of interval names will be used here, though the ADIN names will be referenced in places. Furthermore, because there are so many thirds, what is usually a single chart will be split into five.
{| class="wikitable"
|+Inframinor and Subminor Thirds in 159edo
!Quality
|Narrow Inframinor
|Inframinor
|Wide Inframinor
|Narrow Subminor
|Lesser Subminor (Septiminor)
|Greater Subminor
|Wide Subminor
|-
!Cents
|234
|242
|249
|257
|264
|272
|279
|-
!Just interpretation
|340/297
|1024/891
|15/13
|51/44
|7/6
|117/100
|20/17
|-
!Steps
|31
|32
|33
|34
|35
|36
|37
|}
{| class="wikitable"
|+Minor Thirds in 159edo
!Quality
|Narrow Minor (Gothminor)
|'''Pythagorean Minor (Triminor)'''
|Artomean Minor
|Tendomean Minor
|Ptolemaic Minor (Pentaminor)
|Wide Minor
|-
!Cents
|287
|'''294'''
|302
|309
|317
|325
|-
!Just interpretation
|33/28, 13/11
|'''32/27'''
|25/21
|153/128
|6/5
|135/112
|-
!Steps
|38
|'''39'''
|40
|41
|42
|43
|}
{| class="wikitable"
|+Supraminor, Neutral and Submajor Thirds in 159edo
!Quality
|Lesser Supraminor
|Greater Supraminor
|Artoneutral
|Tendoneutral
|Lesser Submajor
|Greater Submajor
|-
!Cents
|332
|340
|347
|355
|362
|370
|-
!Just interpretation
|40/33, 63/52
|39/32, 17/14
|11/9
|27/22
|16/13, 21/17
|99/80, 26/21
|-
!Steps
|44
|45
|46
|47
|48
|49
|}
{| class="wikitable"
|+Major Thirds in 159edo
!Quality
|Narrow Major
|Ptolemaic Major (Pentamajor)
|Artomean Major
|Tendomean Major
|'''Pythagorean Major (Trimajor)'''
|Wide Major (Gothmajor)
|-
!Cents
|377
|385
|392
|400
|'''408'''
|415
|-
!Just interpretation
|56/45
|5/4
|64/51
|63/50
|'''81/64'''
|14/11, 33/26
|-
!Steps
|50
|51
|52
|53
|'''54'''
|55
|}
{| class="wikitable"
|+Supermajor and Ultramajor Thirds in 159edo
!Quality
|Narrow Supermajor
|Lesser Supermajor
|Greater Supermajor (Septimajor)
|Wide Supermajor
|Narrow Ultramajor
|Ultramajor
|Wide Ultramajor
|-
!Cents
|423
|430
|438
|445
|453
|460
|468
|-
!Just interpretation
|51/40
|50/39
|9/7
|22/17
|13/10
|2673/2048
|891/680
|-
!Steps
|56
|57
|58
|59
|60
|61
|62
|}
Regular diatonic thirds are bolded.

=== Chords ===
159edo has a vast array of triads at its disposal, both fifth-bounded and fourth-bounded. However, the JI interpretation will tend to inform the usage of the various triads offered. The main exceptions to this rule involve chords that serve as detemperings of other, smaller tuning systems.

159edo

2026-03-29T02:34:15Z

Aura: Fleshed out what the JND abbreviation means

'''159edo''', or 159 equal divisions of the octave, is the equal tuning featuring steps of (1200/159) ~= 7.55 cents, 159 of which stack to the perfect octave [[2/1]]. Like [[53edo]], 159edo is an excellent approximation to Pythagorean tuning (stacking pure 3/2 fifths), however it includes near-just approximations of the 11th and 17th harmonics, and a slightly more accurate 7th harmonic, resulting in consistency up to the 17-odd-limit. The step-size, being slightly above the melodic just noticeable difference (JND) of 5 cents as well as more than twice the harmonic JND of the average trained musician at 3.5 cents, enables one to perform fluid modulations by means of [[comma pump]]s as well as by step substitutions. Furthermore, 159edo, like a number of higher edos, is characterized by having a pitch hue palette that's capable of imitating the pitch-hue palettes of smaller tuning systems- in this case, you get detemperings of [[10edo]], [[12edo]], [[13edo]], [[14edo]], [[17edo]], [[19edo]], [[22edo]], [[24edo]] and [[31edo]] among others with errors smaller than the melodic JND.

The interval qualities supported by 159edo are many, however, while every step of 159edo can be interpreted harmonically or subharmonically as being a 17-limit interval or simpler, some of the intervals you get have rather complex interpretations in terms of odd-limit. While the [[perfect fifth]] is really only divisible by three due to 159edo being the three-fold multiple of 53edo- resulting in [[slendric]] temperament and hence 159edo's distinction from 53 in the 7-limit- the [[perfect fourth]] has a little more to offer in terms of divisions. For starters, the perfect fourth can be divided into two instances of 15/13, giving us [[island]] temperament. Dividing the perfect fourth into three instances of 11/10 gives us [[pine]] temperament. Dividing the perfect fourth into six instances of an interval which can be interpreted as 21/20 and 22/21 tempered together gives us [[sextilifourths]] temperament. The perfect fourth can also be cut into eleven intervals which, individually, are half of a Pythagorean limma, giving us a number of temperaments based on the exact interpretation of the semilimma.

On top of all that, there are a number of microtemperament-based structures also supported by 159edo, each of which provides some decent, unexpected melodic possibilities. For instance, one can split the Pythagorean minor third into three instances of 128/121 which gives us [[nexus]] temperament- a temperament which also happens to split the Pythagorean diatonic semitone into two, and the octave into three. One can also split the Ptolemaic minor third into three instances of 17/16, giving us [[archagall]] temperament, which is named for certain tunings found in other temperaments producing fractal-like acoustics. There's also the ability to split the Ptolemaic major sixth into six instances of 11/9, leading to [[parimic]] temperament. In addition, there's also the ability to split the septimal supermajor third into two instances of 17/15, leading to [[fidesmic]] temperament, which acts like a more accurate rendition of [[archy]] temperament in a different subgroup, and this can be exploited for modulation purposes. As if that weren't enough, there's the possibility of splitting the septimal subminor third into five instances of 33/32, leading to [[quartismic]] temperament. Furthermore, there's the possibility of splitting the greater tridecimal neutral tenth into three instances of 27/20, producing [[phaotismic]] temperament, and the list goes on.

== Theory ==
159edo was first used for maqams by Ozan Yarman. It was later put to use by Aura for its ability to handle near-just quartertones derived from the 2.3.11 subgroup on top of the 5-limit foundation provided by 53edo.

==== Edostep interpretations ====
159edo's edostep has the following interpretations in the 2.3.5.11.17 subgroup:

* 243/242, the difference between the 11-limit artoneutral third 11/9, and the 11-limit tendoneutral third 27/22
* 256/255, the difference between 16/15 and 17/16
* 289/288, the difference between 17/16 and 18/17

159edo tempers out the following commas in the 17-limit:

* The schisma (the difference between 5/4 and the Pythagorean diminished fourth)
* The vulture comma (the difference between four 320/243 intervals and the tritave)
* The amiton (the difference between a stack of five 10/9 intervals and 27/16)
* The kleisma (the difference between a stack of three 25/24 intervals and 9/8)
* The semicomma (the difference between a stack of three 75/64 intervals and 8/5)
* The gamelisma (the difference between a stack of three 8/7 intervals and 3/2)
* 385/384 (the difference between 77/64 and 6/5)
* The pine comma (the difference between a stack of three 11/10 intervals and 4/3)
* 625/624 (the difference between 25/24 and 26/25)
* 676/675 (the difference between a stack of two 15/13 intervals and the perfect fourth)
* The twosquare comma (the difference between a stack of two 33/32 intervals and 17/16)

==== JI approximation ====
Although 159edo inherits its approximations of the 5-limit from 53edo, the 5th harmonic can nonetheless be stacked twice without accumulating too much error, rendering it sufficient for Western Classical usage. While the 7th harmonic is technically more accurate in terms of absolute error than in 53edo, the relative error doesn't allow one to stack more than one instance of 7/4 without excessive error accumulation, and the same is true with 13/8; regardless, the inconsistency remains less than 10 cents even when either interval is stacked three times, and 13/7 or 14/13 is tuned almost perfectly. As a whole, 159edo is characterized by its combination of accuracy in the 2.3.5.11.17 subgroup, and a series of compromises in the 7.13.19.23.29 subgroup- among the compromises are the slendric, [[marveltwin]], [[nestoria]], [[minor semivicemic]] and [[brunisimic]] temperaments.
{{Harmonics in ED|159|31|0}}

Currently, the [[ADIN]] system used for interval naming elsewhere on this site fails for 159edo, so another set of interval names will be used here, though the ADIN names will be referenced in places. Furthermore, because there are so many thirds, what is usually a single chart will be split into five.
{| class="wikitable"
|+Inframinor and Subminor Thirds in 159edo
!Quality
|Narrow Inframinor
|Inframinor
|Wide Inframinor
|Narrow Subminor
|Lesser Subminor (Septiminor)
|Greater Subminor
|Wide Subminor
|-
!Cents
|234
|242
|249
|257
|264
|272
|279
|-
!Just interpretation
|340/297
|1024/891
|15/13
|51/44
|7/6
|117/100
|20/17
|-
!Steps
|31
|32
|33
|34
|35
|36
|37
|}
{| class="wikitable"
|+Minor Thirds in 159edo
!Quality
|Narrow Minor (Gothminor)
|'''Pythagorean Minor (Triminor)'''
|Artomean Minor
|Tendomean Minor
|Ptolemaic Minor (Pentaminor)
|Wide Minor
|-
!Cents
|287
|'''294'''
|302
|309
|317
|325
|-
!Just interpretation
|33/28, 13/11
|'''32/27'''
|25/21
|153/128
|6/5
|135/112
|-
!Steps
|38
|'''39'''
|40
|41
|42
|43
|}
{| class="wikitable"
|+Supraminor, Neutral and Submajor Thirds in 159edo
!Quality
|Lesser Supraminor
|Greater Supraminor
|Artoneutral
|Tendoneutral
|Lesser Submajor
|Greater Submajor
|-
!Cents
|332
|340
|347
|355
|362
|370
|-
!Just interpretation
|40/33, 63/52
|39/32, 17/14
|11/9
|27/22
|16/13, 21/17
|99/80, 26/21
|-
!Steps
|44
|45
|46
|47
|48
|49
|}
{| class="wikitable"
|+Major Thirds in 159edo
!Quality
|Narrow Major
|Ptolemaic Major (Pentamajor)
|Artomean Major
|Tendomean Major
|'''Pythagorean Major (Trimajor)'''
|Wide Major (Gothmajor)
|-
!Cents
|377
|385
|392
|400
|'''408'''
|415
|-
!Just interpretation
|56/45
|5/4
|64/51
|63/50
|'''81/64'''
|14/11, 33/26
|-
!Steps
|50
|51
|52
|53
|'''54'''
|55
|}
{| class="wikitable"
|+Supermajor and Ultramajor Thirds in 159edo
!Quality
|Narrow Supermajor
|Lesser Supermajor
|Greater Supermajor (Septimajor)
|Wide Supermajor
|Narrow Ultramajor
|Ultramajor
|Wide Ultramajor
|-
!Cents
|423
|430
|438
|445
|453
|460
|468
|-
!Just interpretation
|51/40
|50/39
|9/7
|22/17
|13/10
|2673/2048
|891/680
|-
!Steps
|56
|57
|58
|59
|60
|61
|62
|}
Regular diatonic thirds are bolded.

=== Chords ===
159edo has a vast array of triads at its disposal, both fifth-bounded and fourth-bounded. However, the JI interpretation will tend to inform the usage of the various triads offered. The main exceptions to this rule involve chords that serve as detemperings of other, smaller tuning systems.

159edo

2026-03-29T02:30:11Z

Aura:

'''159edo''', or 159 equal divisions of the octave, is the equal tuning featuring steps of (1200/159) ~= 7.55 cents, 159 of which stack to the perfect octave [[2/1]]. Like [[53edo]], 159edo is an excellent approximation to Pythagorean tuning (stacking pure 3/2 fifths), however it includes near-just approximations of the 11th and 17th harmonics, and a slightly more accurate 7th harmonic, resulting in consistency up to the 17-odd-limit. The step-size, being slightly above the melodic JND of 5 cents as well as more than twice the harmonic JND of the average trained musician at 3.5 cents, enables one to perform fluid modulations by means of [[comma pump]]s as well as by step substitutions. Furthermore, 159edo, like a number of higher edos, is characterized by having a pitch hue palette that's capable of imitating the pitch-hue palettes of smaller tuning systems- in this case, you get detemperings of [[10edo]], [[12edo]], [[13edo]], [[14edo]], [[17edo]], [[19edo]], [[22edo]], [[24edo]] and [[31edo]] among others with errors smaller than the melodic JND.

The interval qualities supported by 159edo are many, however, while every step of 159edo can be interpreted harmonically or subharmonically as being a 17-limit interval or simpler, some of the intervals you get have rather complex interpretations in terms of odd-limit. While the [[perfect fifth]] is really only divisible by three due to 159edo being the three-fold multiple of 53edo- resulting in [[slendric]] temperament and hence 159edo's distinction from 53 in the 7-limit- the [[perfect fourth]] has a little more to offer in terms of divisions. For starters, the perfect fourth can be divided into two instances of 15/13, giving us [[island]] temperament. Dividing the perfect fourth into three instances of 11/10 gives us [[pine]] temperament. Dividing the perfect fourth into six instances of an interval which can be interpreted as 21/20 and 22/21 tempered together gives us [[sextilifourths]] temperament. The perfect fourth can also be cut into eleven intervals which, individually, are half of a Pythagorean limma, giving us a number of temperaments based on the exact interpretation of the semilimma.

On top of all that, there are a number of microtemperament-based structures also supported by 159edo, each of which provides some decent, unexpected melodic possibilities. For instance, one can split the Pythagorean minor third into three instances of 128/121 which gives us [[nexus]] temperament- a temperament which also happens to split the Pythagorean diatonic semitone into two, and the octave into three. Once can also split the Ptolemaic minor third into three instances of 17/16, giving us [[archagall]] temperament, which is named for certain tunings found in other temperaments producing fractal-like acoustics. There's also the ability to split the Ptolemaic major sixth into six instances of 11/9, leading to [[parimic]] temperament. In addition, there's also the ability to split the septimal supermajor third into two instances of 17/15, leading to [[fidesmic]] temperament, which acts like a more accurate rendition of [[archy]] temperament in a different subgroup, and this can be exploited for modulation purposes. As if that weren't enough, there's the possibility of splitting the septimal subminor third into five instances of 33/32, leading to [[quartismic]] temperament. Furthermore, there's the possibility of splitting the greater tridecimal neutral tenth into three instances of 27/20, producing [[phaotismic]] temperament, and the list goes on.

== Theory ==
159edo was first used for maqams by Ozan Yarman. It was later put to use by Aura for its ability to handle near-just quartertones derived from the 2.3.11 subgroup on top of the 5-limit foundation provided by 53edo.

==== Edostep interpretations ====
159edo's edostep has the following interpretations in the 2.3.5.11.17 subgroup:

* 243/242, the difference between the 11-limit artoneutral third 11/9, and the 11-limit tendoneutral third 27/22
* 256/255, the difference between 16/15 and 17/16
* 289/288, the difference between 17/16 and 18/17

159edo tempers out the following commas in the 17-limit:

* The schisma (the difference between 5/4 and the Pythagorean diminished fourth)
* The vulture comma (the difference between four 320/243 intervals and the tritave)
* The amiton (the difference between a stack of five 10/9 intervals and 27/16)
* The kleisma (the difference between a stack of three 25/24 intervals and 9/8)
* The semicomma (the difference between a stack of three 75/64 intervals and 8/5)
* The gamelisma (the difference between a stack of three 8/7 intervals and 3/2)
* 385/384 (the difference between 77/64 and 6/5)
* The pine comma (the difference between a stack of three 11/10 intervals and 4/3)
* 625/624 (the difference between 25/24 and 26/25)
* 676/675 (the difference between a stack of two 15/13 intervals and the perfect fourth)
* The twosquare comma (the difference between a stack of two 33/32 intervals and 17/16)

==== JI approximation ====
Although 159edo inherits its approximations of the 5-limit from 53edo, the 5th harmonic can nonetheless be stacked twice without accumulating too much error, rendering it sufficient for Western Classical usage. While the 7th harmonic is technically more accurate in terms of absolute error than in 53edo, the relative error doesn't allow one to stack more than one instance of 7/4 without excessive error accumulation, and the same is true with 13/8; regardless, the inconsistency remains less than 10 cents even when either interval is stacked three times, and 13/7 or 14/13 is tuned almost perfectly. As a whole, 159edo is characterized by its combination of accuracy in the 2.3.5.11.17 subgroup, and a series of compromises in the 7.13.19.23.29 subgroup- among the compromises are the slendric, [[marveltwin]], [[nestoria]], [[minor semivicemic]] and [[brunisimic]] temperaments.
{{Harmonics in ED|159|31|0}}

Currently, the [[ADIN]] system used for interval naming elsewhere on this site fails for 159edo, so another set of interval names will be used here, though the ADIN names will be referenced in places. Furthermore, because there are so many thirds, what is usually a single chart will be split into five.
{| class="wikitable"
|+Inframinor and Subminor Thirds in 159edo
!Quality
|Narrow Inframinor
|Inframinor
|Wide Inframinor
|Narrow Subminor
|Lesser Subminor (Septiminor)
|Greater Subminor
|Wide Subminor
|-
!Cents
|234
|242
|249
|257
|264
|272
|279
|-
!Just interpretation
|340/297
|1024/891
|15/13
|51/44
|7/6
|117/100
|20/17
|-
!Steps
|31
|32
|33
|34
|35
|36
|37
|}
{| class="wikitable"
|+Minor Thirds in 159edo
!Quality
|Narrow Minor (Gothminor)
|'''Pythagorean Minor (Triminor)'''
|Artomean Minor
|Tendomean Minor
|Ptolemaic Minor (Pentaminor)
|Wide Minor
|-
!Cents
|287
|'''294'''
|302
|309
|317
|325
|-
!Just interpretation
|33/28, 13/11
|'''32/27'''
|25/21
|153/128
|6/5
|135/112
|-
!Steps
|38
|'''39'''
|40
|41
|42
|43
|}
{| class="wikitable"
|+Supraminor, Neutral and Submajor Thirds in 159edo
!Quality
|Lesser Supraminor
|Greater Supraminor
|Artoneutral
|Tendoneutral
|Lesser Submajor
|Greater Submajor
|-
!Cents
|332
|340
|347
|355
|362
|370
|-
!Just interpretation
|40/33, 63/52
|39/32, 17/14
|11/9
|27/22
|16/13, 21/17
|99/80, 26/21
|-
!Steps
|44
|45
|46
|47
|48
|49
|}
{| class="wikitable"
|+Major Thirds in 159edo
!Quality
|Narrow Major
|Ptolemaic Major (Pentamajor)
|Artomean Major
|Tendomean Major
|'''Pythagorean Major (Trimajor)'''
|Wide Major (Gothmajor)
|-
!Cents
|377
|385
|392
|400
|'''408'''
|415
|-
!Just interpretation
|56/45
|5/4
|64/51
|63/50
|'''81/64'''
|14/11, 33/26
|-
!Steps
|50
|51
|52
|53
|'''54'''
|55
|}
{| class="wikitable"
|+Supermajor and Ultramajor Thirds in 159edo
!Quality
|Narrow Supermajor
|Lesser Supermajor
|Greater Supermajor (Septimajor)
|Wide Supermajor
|Narrow Ultramajor
|Ultramajor
|Wide Ultramajor
|-
!Cents
|423
|430
|438
|445
|453
|460
|468
|-
!Just interpretation
|51/40
|50/39
|9/7
|22/17
|13/10
|2673/2048
|891/680
|-
!Steps
|56
|57
|58
|59
|60
|61
|62
|}
Regular diatonic thirds are bolded.

=== Chords ===
159edo has a vast array of triads at its disposal, both fifth-bounded and fourth-bounded. However, the JI interpretation will tend to inform the usage of the various triads offered. The main exceptions to this rule involve chords that serve as detemperings of other, smaller tuning systems.

Nestoria

2026-03-29T00:14:25Z

Aura: Added this redirect

#REDIRECT [[Schismic]]

Schismic

2026-03-29T00:13:25Z

Aura: Added more information on the 19-limit extension, Nestoria.

{{Infobox regtemp|Mapping=1; 1 -8 -3|Subgroup=2.3.5, 2.3.5.19|Edo join 1=12|Edo join 2=29|Generators=3/2|Comma basis=32805/32768 (2.3.5); 361/360, 513/512 (2.3.5.19)|Odd limit 1=5|Complexity 1=12|Generators tuning=701.7|Subgroups=2.3.5, 2.3.5.19|Mistuning 1=0.217|Optimization method=CWE|MOS scales=[[5L 2s]], [[5L 7s]], [[12L 5s]]}}
'''Schismic''' or '''Schismatic'''[a], [12 & 29], is the temperament that equates 5/4 to the Pythagorean diminished fourth. The difference between these intervals is 32805/32768, the ''schisma'', which is about 2 cents; this means that Schismic can be tuned to perfect [[Pythagorean tuning]] (and is considered by some to be the primary 5-limit interpretation of Pythagorean tuning), however it is technically optimal to flatten the fifth by a fraction of a cent. Schismic is one of the simplest microtemperaments.

The Pythagorean and syntonic commas are thus equated to a single comma-sized step. Due to 3/2 being very close to just, it is also natural to equate 19/16 with the diatonic minor third, tempering out 513/512, which is sometimes called Boethius' comma. As a consequence, 19/15 is equated with the diatonic major third, tempering out 1216/1215, which is sometimes called Eratosthenes' comma. Tempering out both 513/512 and 1216/1215 results in an extension called ''Nestoria''.

The primary MOS to use for Schismic is Schismic[17], which maps the 5-limit major and minor thirds to the same degree, ensuring like diatonic does in meantone that either one or the other is always accessible. 12edo and 29edo serve as the boundaries of the tuning range; in 12edo the Pythagorean comma and schisma (and thus the meantone comma) are tempered out, and in 29edo the Pythagorean/syntonic comma is inflated to the size of the 25/24 chromatic semitone, supporting [[porcupine]].

48edo and 58edo are the first two edos to observe the schisma - that is, to not support Schismic - while tuning the fifth within the aforementioned Schismic tuning range.

== Extensions ==

=== Prime 7 ===
The primary 7-limit extension of Schismic is Garibaldi (also 12 & 29), which equates 64/63 with the syntonic comma. This is not considered canonical due to a significant loss in accuracy (tuned best with a fifth slightly sharp of just) - that is, Garibaldi is not a microtemperament - but it is still more accurate than [[Meantone]] as well as distinguishing 5-limit, 7-limit, and Pythagorean intervals in any given interval category. Garibaldi is an intuitive way of organizing just intonation as it tempers together the defining ~20-30c commas of the 7-limit.

=== Primes 11 and 13 ===
A reasonable extension to the 11-limit assuming Garibaldi is Cassandra (41 & 53), which sets 33/32 to twice the Garibaldi comma; alternatively there is Andromeda, which instead sets 33/32 to the difference between that and the chroma and is best tuned sharp of 41edo. In either case, 11/9 is set to the opposite neutral third to 16/13 to extend to the 13-limit.

=== Prime 17 ===
Schismic generally does not have 17; two options are setting 17/16 equal to 16/15 and 15/14 (preferring flatter schismic tunings) and tempering together 17/16 and 18/17 (which results in a weak extension including 17/12 as the semioctave).

== Intervals ==
These interpretations assume Cassandra (in Pythagorean tuning).
{| class="wikitable"
|+
! colspan="3" |Up from the unison
! colspan="3" |Down from the octave
|-
!#
!Cents
!JI
!#
!Cents
!JI
|-
|'''0'''
|'''0.00'''
|'''1/1'''
|'''0'''
|'''1,200.00'''
|'''2/1'''
|-
|'''1'''
|'''701.96'''
|'''3/2'''
|1
|498.04
|4/3
|-
|2
|203.91
|9/8
|2
|996.09
|16/9
|-
|3
|905.87
|32/19, 27/16
|3
|294.13
|19/16
|-
|4
|407.82
|19/15
|4
|792.18
|19/12
|-
|5
|1,109.78
|19/10
|5
|90.22
|19/18, 20/19, 21/20
|-
|6
|611.73
|10/7
|6
|588.27
|7/5
|-
|7
|113.69
|16/15, 15/14
|7
|1,086.31
|15/8
|-
|8
|815.64
|8/5
|'''8'''
|'''384.36'''
|'''5/4'''
|-
|9
|317.60
|5/3
|9
|882.40
|6/5
|-
|10
|1,019.55
|9/5
|10
|180.45
|10/9
|-
|11
|521.51
|27/20
|11
|678.49
|40/27
|-
|12
|23.46
|50/49
|12
|1,176.54
|49/25
|-
|13
|725.42
|32/21
|13
|474.58
|21/16
|-
|14
|227.37
|8/7
|'''14'''
|'''972.63'''
|'''7/4'''
|-
|15
|929.33
|12/7
|15
|270.67
|7/6
|-
|16
|431.28
|9/7
|16
|768.72
|14/9
|-
|17
|1,133.24
|28/27
|17
|66.76
|27/14
|-
|18
|635.19
|13/9
|18
|564.81
|18/13
|-
|19
|137.15
|13/12
|19
|1,062.85
|24/13
|-
|'''20'''
|'''839.10'''
|'''13/8'''
|20
|360.90
|16/13
|-
|21
|341.06
|11/9
|21
|858.94
|18/11
|-
|22
|1,043.01
|11/6
|22
|156.99
|12/11
|-
|'''23'''
|'''544.97'''
|'''11/8'''
|23
|655.03
|16/11
|-
|24
|46.92
|33/32
|24
|1,153.08
|64/33
|-
|25
|748.88
|54/35
|25
|451.12
|35/27
|-
|26
|250.83
|81/70
|26
|949.17
|140/81
|-
|27
|952.79
|26/15
|27
|247.21
|15/13
|-
|28
|454.74
|13/10
|28
|745.26
|20/13
|-
|29
|1,156.70
|39/20
|29
|43.30
|40/39
|}

== Patent vals ==
The following patent vals up to 272edo support Schismic.
{| class="wikitable sortable mw-collapsible"
|+
!EDO
!Generator tuning
!Extension info
|-
|12
|700.0000
|Garibaldi
|-
|101
|700.9901
|
|-
|89
|701.1236
|
|-
|77
|701.2987
|
|-
|219
|701.3699
|
|-
|142
|701.4085
|
|-
|207
|701.4493
|
|-
|272
|701.4706
|
|-
|65
|701.5385
|
|-
|248
|701.6129
|
|-
|183
|701.6393
|
|-
|118
|701.6949
|
|-
|171
|701.7544
|
|-
|224
|701.7857
|(no Schismic 19)
|-
|53
|701.8868
|Garibaldi
|-
|253
|701.9763
|(no Schismic 19)
|-
|200
|702.0000
|(no Schismic 19)
|-
|147
|702.0408
|Garibaldi
|-
|94
|702.1277
|Garibaldi
|-
|135
|702.2222
|Garibaldi
|-
|41
|702.4390
|Garibaldi
|-
|29
|703.4483
|Garibaldi
|-
|17
|705.8824
|
|}

The full list of patent vals supporting schismic is below:
{| class="wikitable sortable mw-collapsible mw-collapsed"
|+
!EDO
!Generator tuning
!Extension info
|-
|12
|700.0000
|Garibaldi
|-
|101
|700.9901
|
|-
|89
|701.1236
|
|-
|77
|701.2987
|
|-
|219
|701.3699
|
|-
|142
|701.4085
|
|-
|207
|701.4493
|
|-
|272
|701.4706
|
|-
|65
|701.5385
|
|-
|508
|701.5748
|(no Schismic 19)
|-
|443
|701.5801
|(no Schismic 19)
|-
|378
|701.5873
|(no Schismic 19)
|-
|313
|701.5974
|(no Schismic 19)
|-
|561
|701.6043
|(no Schismic 19)
|-
|248
|701.6129
|
|-
|679
|701.6200
|(no Schismic 19)
|-
|431
|701.6241
|(no Schismic 19)
|-
|614
|701.6287
|(no Schismic 19)
|-
|797
|701.6311
|(no Schismic 19)
|-
|183
|701.6393
|
|-
|1033
|701.6457
|(no Schismic 19)
|-
|850
|701.6471
|(no Schismic 19)
|-
|667
|701.6492
|(no Schismic 19)
|-
|1151
|701.6507
|(no Schismic 19)
|-
|484
|701.6529
|(no Schismic 19)
|-
|1269
|701.6548
|(no Schismic 19)
|-
|785
|701.6561
|(no Schismic 19)
|-
|1086
|701.6575
|(no Schismic 19)
|-
|1387
|701.6583
|(no Schismic 19)
|-
|301
|701.6611
|(no Schismic 19)
|-
|1322
|701.6641
|(no Schismic 19)
|-
|1021
|701.6650
|(no Schismic 19)
|-
|720
|701.6667
|(no Schismic 19)
|-
|1139
|701.6681
|(no Schismic 19)
|-
|1558
|701.6688
|(no Schismic 19)
|-
|419
|701.6706
|(no Schismic 19)
|-
|1794
|701.6722
|(no Schismic 19)
|-
|1375
|701.6727
|(no Schismic 19)
|-
|956
|701.6736
|(no Schismic 19)
|-
|1493
|701.6745
|(no Schismic 19)
|-
|2030
|701.6749
|(no Schismic 19)
|-
|537
|701.6760
|(no Schismic 19)
|-
|1729
|701.6773
|(no Schismic 19)
|-
|1192
|701.6779
|(no Schismic 19)
|-
|1847
|701.6784
|(no Schismic 19)
|-
|655
|701.6794
|(no Schismic 19)
|-
|2083
|701.6803
|(no Schismic 19)
|-
|1428
|701.6807
|(no Schismic 19)
|-
|773
|701.6818
|(no Schismic 19)
|-
|1664
|701.6827
|(no Schismic 19)
|-
|891
|701.6835
|(no Schismic 19)
|-
|1900
|701.6842
|(no Schismic 19)
|-
|1009
|701.6848
|(no Schismic 19)
|-
|2136
|701.6854
|(no Schismic 19)
|-
|1127
|701.6859
|(no Schismic 19)
|-
|1245
|701.6867
|(no Schismic 19)
|-
|1363
|701.6875
|(no Schismic 19)
|-
|1481
|701.6880
|(no Schismic 19)
|-
|1599
|701.6886
|(no Schismic 19)
|-
|1717
|701.6890
|(no Schismic 19)
|-
|1835
|701.6894
|(no Schismic 19)
|-
|1953
|701.6897
|(no Schismic 19)
|-
|2071
|701.6900
|(no Schismic 19)
|-
|2189
|701.6903
|(no Schismic 19)
|-
|118
|701.6949
|
|-
|2295
|701.6993
|(no Schismic 19)
|-
|2177
|701.6996
|(no Schismic 19)
|-
|2059
|701.6999
|(no Schismic 19)
|-
|1941
|701.7002
|(no Schismic 19)
|-
|1823
|701.7005
|(no Schismic 19)
|-
|1705
|701.7009
|(no Schismic 19)
|-
|1587
|701.7013
|(no Schismic 19)
|-
|1469
|701.7018
|(no Schismic 19)
|-
|1351
|701.7024
|(no Schismic 19)
|-
|1233
|701.7032
|(no Schismic 19)
|-
|2348
|701.7036
|(no Schismic 19)
|-
|1115
|701.7040
|(no Schismic 19)
|-
|2112
|701.7045
|(no Schismic 19)
|-
|997
|701.7051
|(no Schismic 19)
|-
|1876
|701.7058
|(no Schismic 19)
|-
|879
|701.7065
|(no Schismic 19)
|-
|1640
|701.7073
|(no Schismic 19)
|-
|2401
|701.7076
|(no Schismic 19)
|-
|761
|701.7083
|(no Schismic 19)
|-
|2165
|701.7090
|(no Schismic 19)
|-
|1404
|701.7094
|(no Schismic 19)
|-
|2047
|701.7098
|(no Schismic 19)
|-
|643
|701.7107
|(no Schismic 19)
|-
|2454
|701.7115
|(no Schismic 19)
|-
|1811
|701.7118
|(no Schismic 19)
|-
|1168
|701.7123
|(no Schismic 19)
|-
|1693
|701.7129
|(no Schismic 19)
|-
|2218
|701.7133
|(no Schismic 19)
|-
|525
|701.7143
|(no Schismic 19)
|-
|1982
|701.7154
|(no Schismic 19)
|-
|1457
|701.7159
|(no Schismic 19)
|-
|2389
|701.7162
|(no Schismic 19)
|-
|932
|701.7167
|(no Schismic 19)
|-
|2271
|701.7173
|(no Schismic 19)
|-
|1339
|701.7177
|(no Schismic 19)
|-
|1746
|701.7182
|(no Schismic 19)
|-
|2153
|701.7185
|(no Schismic 19)
|-
|407
|701.7199
|(no Schismic 19)
|-
|2324
|701.7212
|(no Schismic 19)
|-
|1917
|701.7214
|(no Schismic 19)
|-
|1510
|701.7219
|(no Schismic 19)
|-
|1103
|701.7226
|(no Schismic 19)
|-
|1799
|701.7232
|(no Schismic 19)
|-
|2495
|701.7234
|(no Schismic 19)
|-
|696
|701.7241
|(no Schismic 19)
|-
|2377
|701.7249
|(no Schismic 19)
|-
|1681
|701.7252
|(no Schismic 19)
|-
|985
|701.7259
|(no Schismic 19)
|-
|2259
|701.7264
|(no Schismic 19)
|-
|1274
|701.7268
|(no Schismic 19)
|-
|1563
|701.7274
|(no Schismic 19)
|-
|1852
|701.7279
|(no Schismic 19)
|-
|2141
|701.7282
|(no Schismic 19)
|-
|2430
|701.7284
|(no Schismic 19)
|-
|289
|701.7301
|(no Schismic 19)
|-
|2483
|701.7318
|(no Schismic 19)
|-
|2194
|701.7320
|(no Schismic 19)
|-
|1905
|701.7323
|(no Schismic 19)
|-
|1616
|701.7327
|(no Schismic 19)
|-
|1327
|701.7332
|(no Schismic 19)
|-
|2365
|701.7336
|(no Schismic 19)
|-
|1038
|701.7341
|(no Schismic 19)
|-
|1787
|701.7348
|(no Schismic 19)
|-
|2536
|701.7350
|(no Schismic 19)
|-
|749
|701.7356
|(no Schismic 19)
|-
|2707
|701.7362
|(no Schismic 19)
|-
|1958
|701.7365
|(no Schismic 19)
|-
|1209
|701.7370
|(no Schismic 19)
|-
|1669
|701.7376
|(no Schismic 19)
|-
|2129
|701.7379
|(no Schismic 19)
|-
|2589
|701.7381
|(no Schismic 19)
|-
|460
|701.7391
|(no Schismic 19)
|-
|2471
|701.7402
|(no Schismic 19)
|-
|2011
|701.7404
|(no Schismic 19)
|-
|1551
|701.7408
|(no Schismic 19)
|-
|1091
|701.7415
|(no Schismic 19)
|-
|1722
|701.7422
|(no Schismic 19)
|-
|2353
|701.7425
|(no Schismic 19)
|-
|631
|701.7433
|(no Schismic 19)
|-
|2064
|701.7442
|(no Schismic 19)
|-
|1433
|701.7446
|(no Schismic 19)
|-
|802
|701.7456
|(no Schismic 19)
|-
|1775
|701.7465
|(no Schismic 19)
|-
|973
|701.7472
|(no Schismic 19)
|-
|1144
|701.7483
|(no Schismic 19)
|-
|1315
|701.7490
|(no Schismic 19)
|-
|1486
|701.7497
|(no Schismic 19)
|-
|1657
|701.7502
|(no Schismic 19)
|-
|1828
|701.7505
|(no Schismic 19)
|-
|171
|701.7544
|
|-
|1421
|701.7593
|(no Schismic 19)
|-
|1250
|701.7600
|(no Schismic 19)
|-
|1079
|701.7609
|(no Schismic 19)
|-
|908
|701.7621
|(no Schismic 19)
|-
|737
|701.7639
|(no Schismic 19)
|-
|1303
|701.7652
|(no Schismic 19)
|-
|566
|701.7668
|(no Schismic 19)
|-
|961
|701.7690
|(no Schismic 19)
|-
|395
|701.7722
|(no Schismic 19)
|-
|1014
|701.7751
|(no Schismic 19)
|-
|619
|701.7771
|(no Schismic 19)
|-
|843
|701.7794
|(no Schismic 19)
|-
|1067
|701.7807
|(no Schismic 19)
|-
|224
|701.7857
|(no Schismic 19)
|-
|725
|701.7931
|(no Schismic 19)
|-
|501
|701.7964
|(no Schismic 19)
|-
|778
|701.7995
|(no Schismic 19)
|-
|277
|701.8051
|(no Schismic 19)
|-
|607
|701.8122
|(no Schismic 19)
|-
|330
|701.8182
|(no Schismic 19)
|-
|383
|701.8277
|(no Schismic 19)
|-
|436
|701.8349
|(no Schismic 19)
|-
|489
|701.8405
|(no Schismic 19)
|-
|542
|701.8450
|(no Schismic 19)
|-
|53
|701.8868
|Garibaldi
|-
|253
|701.9763
|(no Schismic 19)
|-
|200
|702.0000
|(no Schismic 19)
|-
|147
|702.0408
|Garibaldi
|-
|94
|702.1277
|Garibaldi
|-
|135
|702.2222
|Garibaldi
|-
|41
|702.4390
|Garibaldi
|-
|29
|703.4483
|Garibaldi
|-
|17
|705.8824
|
|}

== Footnotes ==
[a] Despite ending in -isma, ''schisma'' is a [[Temperament naming#Comma declension categories|3rd-declension]] comma name, therefore its temperaments are ''Schismatic'' and ''Schismic'' (which, as it is a 2.3.5 comma, refer to the same temperament).
{{Navbox regtemp}}

159edo

2026-03-29T00:04:34Z

Aura:

159edo

2026-03-29T00:00:58Z

Aura:

159edo

2026-03-28T23:54:09Z

Aura: Split the two big charts into two smaller ones each, and added the chart for the neutral thirds

159edo

2026-03-28T23:37:39Z

Aura: Two of the three charts of thirds have now been added

53edo

2026-03-28T22:48:35Z

Aura:

'''53edo''', or 53 equal divisions of the octave, is the equal tuning featuring steps of (1200/53) ~= 22.64 cents, 53 of which stack to the perfect octave [[2/1]]. 53edo is an excellent approximation to Pythagorean tuning (stacking pure 3/2 fifths). Theoretical interest in this tuning system goes back to antiquity.

== Theory ==
Unless one has a set of accidentals for the syntonic comma (see the Notation section) one is left in the unenviable position of having to label a Ptolemaic major third the same way as the Pythagorean diminished fourth, for example. Apart from that issue, 53edo is very useful for 5-limit music.

==== Edostep interpretations ====
53edo's edostep has the following interpretations in the 2.3.5.7.13 subgroup:

* 65/64, the difference between the 13-limit tendoneutral third 16/13 and the classical major third 5/4
* 81/80 (the syntonic comma), the difference between 5/4 and the diatonic major third
* The Pythagorean comma, the difference between the Pythagorean diatonic and chromatic semitones
* 91/90, the difference between the 13-limit ultramajor third 13/10 and the septimal supermajor third 9/7
* 64/63, the difference between the diatonic major third and 9/7
* 512/507, the difference between the 13-limit neutral thirds

53edo tempers out the following commas:

* The schisma (the difference between 5/4 and the Pythagorean diminished fourth)
* The vulture comma (the difference between four 320/243 intervals and the tritave)
* The amiton (the difference between a stack of five 10/9 intervals and 27/16)
* The kleisma (the difference between a stack of three 25/24 intervals and 9/8)
* The semicomma (the difference between a stack of three 75/64 intervals and 8/5)
* 225/224 (the difference between 15/14 and 16/15)
* 385/384 (the difference between 77/64 and 6/5)
* 121/120 (the difference between 12/11 and 11/10)
* 625/624 (the difference between 25/24 and 26/25)
* 676/675 (the difference between a stack of two 15/13 intervals and the perfect fourth)

==== JI approximation ====
53edo is most usefully seen as a 2.3.5.7.13 tuning, but the 2.3.5.13 restriction is more accurate and shared with a number of its multiples, such as [[159edo]]. Because it is not a Meantone system, there are actually multiple potential diatonic scales to use for 5-limit harmony, one of which is the Zarlino diatonic scale (LMsLMLs), tuned in 53edo as 9-8-5-9-8-9-5, though this particular scale is arguably best used for Lydian or Locrian modes. There's also the Didymic diatonic scale, tuned in 53edo as 9-8-5-9-9-8-5, which is better suited for Ionian mode and Major tonality in general. However, 53edo also features a MOS diatonic of 9-9-4-9-9-9-4, which is basically the Pythagorean diatonic scale.

{{Harmonics in ED|53|31|0}}
{| class="wikitable"
|+Thirds in 53edo
!Quality
|Inframinor
|Subminor
|'''Farminor'''
|Nearminor
|Supraminor
|Submajor
|Nearmajor
|'''Farmajor'''
|Supermajor
|Ultramajor
|-
!Cents
|249
|272
|'''294'''
|317
|340
|362
|385
|'''408'''
|430
|453
|-
!Just interpretation
|15/13
|7/6, 75/64
|'''32/27'''
|6/5
|39/32
|16/13
|5/4
|'''81/64'''
|9/7, 32/25
|13/10
|-
!Steps
|11
|12
|'''13'''
|14
|15
|16
|17
|'''18'''
|19
|20
|}
Diatonic thirds are bolded.

=== Chords ===
53edo has four different flavors of minor and major intervals as well as supraminor and submajor intervals. Its inframinor and ultramajor thirds approximate 15/13 and 13/10 respectively. At the same time, 53edo's subminor and supermajor intervals approximate 7/6 and 9/7. Then there's the novaminor and novamajor thirds, which are extremely close approximations of Pythagorean minor and major thirds and can be referred to as such. There are also the pentaminor and pentamajor thirds, which are very close approximations of the Ptolemaic minor and major thirds and can also be referred to as such. Finally, the supraminor and submajor thirds approximate 39/32 and 16/13. For fourth-bounded triads, there's only really five options. The first two, which involve the approximations of 9/8 and 32/27, have a marked propensity to cause crowding, and thus are dissonant. Then there's the next two, the latal triads, which involve the approximations of 8/7 and 7/6, and which, due to their tuning are markedly less dissonant, but still dissonant. Finally, the last option, which splits the perfect fourth cleanly in half, is an ambisonance- that is, an interval that is halfway between the extremes of consonance and dissonance.

== Notation ==
This section provides some of the options for notating 53edo.

=== Pythagorean notation ===
In 53edo, the space between each of the notes that is separated by 2 steps in 12edo is instead 9 steps; notes separated by a single step in 12edo have to be distinguished from each other as the Pythagorean diatonic semitone is 4 steps while the Pythagorean chromatic semitone is 5 steps. Furthermore, the Pythagorean comma is a single step in 53edo, unlike in 12edo where it's tempered out. It is important to understand the usage of enharmonic equivalence here; unlike in systems such as 31edo where each note has an easily derivable "canonical" notation, it is important to understand the multiple faces of each of 53edo's pitches (which some might consider as a downside of using the Pythagorean system).

{| class="wikitable"
|+
!
!
|-
| colspan="2" |D
|-
|^^Ebb
|^D
|-
|vvEb
|^^D
|-
|vEb
|vvD#
|-
|Eb
|vD#
|-
|^Eb
|D#
|-
|^^Eb
|^D#
|-
|vvE
|^^D#
|-
|vE
|vvDx
|-
| colspan="2" |E
|}

==== Ups and Downs ====
Ups and downs naturally reflect 53edo's structure, as 5/4 is downmajor, 81/64 is major, and 9/7 is upmajor.

==== Syntonic-Rastmic Subchroma notation ====
Syntonic-Rastmic Subchroma notation, or SRS notation for short, uses '''synsharp''' and '''synflat''' as accidentals to cover the syntonic comma. However, while SRS notation is a 2.3.5.11 notation, only the 2.3.5 portion of the notation for 53edo is shared with multiples like 159edo.

==== Accidentals ====
53edo's accidentals, as mentioned and demonstrated previously, consist of sharps and flats, as well as either up and down accidentals, or, alternatively, synsharps and synflats and their derivatives.
{{Navbox EDO}}
{{Cat|Edos}}

== Multiples ==

===106edo===
106edo has inconsistent 11th and 17th harmonics, and also loses some ability to be consistent that 53edo has due to inconsistencies in the 7-odd-limit.

{{Harmonics in ED|106|31|0}}

===159edo===
''Main article: [[159edo]]''

159edo corrects 53edo's approximate 11th and 17th harmonics to near-just qualities, and also improves the 7th harmonic to a lesser extent, resulting in it being consistent to the 17-odd-limit. If you go further than that, you're forced to choose between the 17the harmonic on one hand and both the 19th and 29th harmonics on the other, but you do get a good 23rd harmonic regardless. In addition, you also gain access to a pitch hue palette that approximates those of simpler systems such as [[10edo]], [[12edo]], [[13edo]], [[14edo]], [[17edo]], [[19edo]], [[22edo]], [[24edo]] and [[31edo]] among others, opening up additional compositional techniques.

{{Harmonics in ED|159|31|0}}

53edo

2026-03-28T22:48:03Z

Aura:

'''53edo''', or 53 equal divisions of the octave, is the equal tuning featuring steps of (1200/53) ~= 22.64 cents, 53 of which stack to the perfect octave [[2/1]]. 53edo is an excellent approximation to Pythagorean tuning (stacking pure 3/2 fifths). Theoretical interest in this tuning system goes back to antiquity.

== Theory ==
Unless one has a set of accidentals for the syntonic comma (see the Notation section) one is left in the unenviable position of having to label a Ptolemaic major third the same way as the Pythagorean diminished fourth, for example. Apart from that issue, 53edo is very useful for 5-limit music.

==== Edostep interpretations ====
53edo's edostep has the following interpretations in the 2.3.5.7.13 subgroup:

* 65/64, the difference between the 13-limit tendoneutral third 16/13 and the classical major third 5/4
* 81/80 (the syntonic comma), the difference between 5/4 and the diatonic major third
* The Pythagorean comma, the difference between the Pythagorean diatonic and chromatic semitones
* 91/90, the difference between the 13-limit ultramajor third 13/10 and the septimal supermajor third 9/7
* 64/63, the difference between the diatonic major third and 9/7
* 512/507, the difference between the 13-limit neutral thirds

53edo tempers out the following commas:

* The schisma (the difference between 5/4 and the Pythagorean diminished fourth)
* The vulture comma (the difference between four 320/243 intervals and the tritave)
* The amiton (the difference between a stack of five 10/9 intervals and 27/16)
* The kleisma (the difference between a stack of three 25/24 intervals and 9/8)
* The semicomma (the difference between a stack of three 75/64 intervals and 8/5)
* 225/224 (the difference between 15/14 and 16/15)
* 385/384 (the difference between 77/64 and 6/5)
* 121/120 (the difference between 12/11 and 11/10)
* 625/624 (the difference between 25/24 and 26/25)
* 676/675 (the difference between a stack of two 15/13 intervals and the perfect fourth)

==== JI approximation ====
53edo is most usefully seen as a 2.3.5.7.13 tuning, but the 2.3.5.13 restriction is more accurate and shared with a number of its multiples, such as [[159edo]]. Because it is not a Meantone system, there are actually multiple potential diatonic scales to use for 5-limit harmony, one of which is the Zarlino diatonic scale (LMsLMLs), tuned in 53edo as 9-8-5-9-8-9-5, though this particular scale is arguably best used for Lydian or Locrian modes. There's also the Didymic diatonic scale, tuned in 53edo as 9-8-5-9-9-8-5, which is better suited for Ionian mode and Major tonality in general. However, 53edo also features a MOS diatonic of 9-9-4-9-9-9-4, which is basically the Pythagorean diatonic scale.

{{Harmonics in ED|53|31|0}}
{| class="wikitable"
|+Thirds in 53edo
!Quality
|Inframinor
|Subminor
|'''Farminor'''
|Nearminor
|Supraminor
|Submajor
|Nearmajor
|'''Farmajor'''
|Supermajor
|Ultramajor
|-
!Cents
|249
|272
|'''294'''
|317
|340
|362
|385
|'''408'''
|430
|453
|-
!Just interpretation
|15/13
|7/6, 75/64
|'''32/27'''
|6/5
|39/32
|16/13
|5/4
|'''81/64'''
|9/7, 32/25
|13/10
|-
!Steps
|11
|12
|'''13'''
|14
|15
|16
|17
|'''18'''
|19
|20
|}
Diatonic thirds are bolded.

=== Chords ===
53edo has four different flavors of minor and major intervals as well as supraminor and submajor intervals. Its inframinor and ultramajor thirds approximate 15/13 and 13/10 respectively. At the same time, 53edo's subminor and supermajor intervals approximate 7/6 and 9/7. Then there's the novaminor and novamajor thirds, which are extremely close approximations of Pythagorean minor and major thirds and can be referred to as such. There are also the pentaminor and pentamajor thirds, which are very close approximations of the Ptolemaic minor and major thirds and can also be referred to as such. Finally, the supraminor and submajor thirds approximate 39/32 and 16/13. For fourth-bounded triads, there's only really five options. The first two, which involve the approximations of 9/8 and 32/27, have a marked propensity to cause crowding, and thus are dissonant. Then there's the next two, the latal triads, which involve the approximations of 8/7 and 7/6, and which, due to their tuning are markedly less dissonant, but still dissonant. Finally, the last option, which splits the perfect fourth cleanly in half, is an ambisonance- that is, an interval that is halfway between the extremes of consonance and dissonance.

== Notation ==
This section provides some of the options for notating 53edo.

=== Pythagorean notation ===
In 53edo, the space between each of the notes that is separated by 2 steps in 12edo is instead 9 steps; notes separated by a single step in 12edo have to be distinguished from each other as the Pythagorean diatonic semitone is 4 steps while the Pythagorean chromatic semitone is 5 steps. Furthermore, the Pythagorean comma is a single step in 53edo, unlike in 12edo where it's tempered out. It is important to understand the usage of enharmonic equivalence here; unlike in systems such as 31edo where each note has an easily derivable "canonical" notation, it is important to understand the multiple faces of each of 53edo's pitches (which some might consider as a downside of using the Pythagorean system).

{| class="wikitable"
|+
!
!
|-
| colspan="2" |D
|-
|^^Ebb
|^D
|-
|vvEb
|^^D
|-
|vEb
|vvD#
|-
|Eb
|vD#
|-
|^Eb
|D#
|-
|^^Eb
|^D#
|-
|vvE
|^^D#
|-
|vE
|vvDx
|-
| colspan="2" |E
|}

==== Ups and Downs ====
Ups and downs naturally reflect 53edo's structure, as 5/4 is downmajor, 81/64 is major, and 9/7 is upmajor.

==== Syntonic-Rastmic Subchroma notation ====
Syntonic-Rastmic Subchroma notation, or SRS notation for short, uses '''synsharp''' and '''synflat''' as accidentals to cover the syntonic comma. However, while SRS notation is a 2.3.5.11 notation, only the 2.3.5 portion of the notation for 53edo is shared with multiples like 159edo.

==== Accidentals ====
53edo's accidentals, as mentioned and demonstrated previously, consist of sharps and flats, as well as either up and down accidentals, or, alternatively, synsharps and synflats and their derivatives.
{{Navbox EDO}}
{{Cat|Edos}}

== Multiples ==

===106edo===
106edo has inconsistent 11th and 17th harmonics, and also loses some ability to be consistent due to inconsistencies in the 7-odd-limit.

{{Harmonics in ED|106|31|0}}

===159edo===
''Main article: [[159edo]]''

159edo corrects 53edo's approximate 11th and 17th harmonics to near-just qualities, and also improves the 7th harmonic to a lesser extent, resulting in it being consistent to the 17-odd-limit. If you go further than that, you're forced to choose between the 17the harmonic on one hand and both the 19th and 29th harmonics on the other, but you do get a good 23rd harmonic regardless. In addition, you also gain access to a pitch hue palette that approximates those of simpler systems such as [[10edo]], [[12edo]], [[13edo]], [[14edo]], [[17edo]], [[19edo]], [[22edo]], [[24edo]] and [[31edo]] among others, opening up additional compositional techniques.

{{Harmonics in ED|159|31|0}}

159edo

2026-03-28T22:39:07Z

Aura:

159edo

2026-03-28T22:35:05Z

Aura:

159edo

2026-03-28T22:29:22Z

Aura:

159edo

2026-03-28T21:00:05Z

Aura: