Xenharmonic Reference - User contributions [en]

7edo

2026-07-28T02:57:50Z

2^67-1:

User:2^67-1/7afdo

2026-07-15T08:48:17Z

2^67-1: Created page with "'''7afdo''' is a scale which is defined by taking the harmonic series from harmonics 7 to 14. Due to it not having a 3/2 over the tonic it does not lend itself well to common-practice-style harmony unless rotated. The smallest edo which makes the notes distinct is 7edo, and the smallest which makes the intervals distinct is 87edo. == Modes == === 8:9:10:11:12:13:14:16 === This is the just-intonation version of the acoustic scale. This is 8afdo without the 15t..."

'''7afdo''' is a scale which is defined by taking the harmonic series from harmonics 7 to 14. Due to it not having a 3/2 over the tonic it does not lend itself well to common-practice-style harmony unless rotated.

The smallest edo which makes the notes distinct is [[7edo]], and the smallest which makes the intervals distinct is [[87edo]].

== Modes ==

=== 8:9:10:11:12:13:14:16 ===

This is the just-intonation version of the acoustic scale. This is 8afdo without the 15th harmonic. As a result, it can sound very Mixolydian.

=== 12:13:14:16:18:20:22:24 ===

This scale sounds like a warped Dorian scale and has a 4/3 and 3/2 above the tonic, allowing one to think of it as a tetrachordal scale with two different tetrachords.

User:2^67-1

2026-07-15T08:41:25Z

2^67-1: Created page with "=About me= My favourite EDOs are 7, 10, 14, and 28. My favourite EDT is 11. Banned from the XA due to some controversies. Whatever you do, please do not DM me and ask me about them. Was part of the Hemipyth cult (yes, yes, that Pergele nonsense) but then resigned. For one, I support the exploration of nonoctave equal-step tunings. =Subpages= {{Special:Prefixindex|prefix={{FULLPAGENAME}}/|hideredirects=1|stripprefix=1}}"

=About me=

My favourite EDOs are [[7edo|7]], [[10edo|10]], [[14edo|14]], and [[28edo|28]]. My favourite EDT is [[11edt|11]].

Banned from the XA due to some controversies. Whatever you do, please do not DM me and ask me about them.

Was part of the Hemipyth cult (yes, yes, that Pergele nonsense) but then resigned.

For one, I support the exploration of nonoctave equal-step tunings.

=Subpages=

{{Special:Prefixindex|prefix={{FULLPAGENAME}}/|hideredirects=1|stripprefix=1}}

7edo

2026-06-17T04:09:20Z

2^67-1: /* Polysomatic tuning */

'''7edo''' is the basic equiheptatonic, where all the steps are tuned to be precisely equal. It features steps of (1200/7) ~= 171.4 cents.

== Theory ==

=== Edostep interpretations ===
7edo's edostep has the following interpretations in the 2.3.5 subgroup:

* 9/8 (the diatonic major second)
* 10/9 (the interval separating 9/8 and 5/4)
* 16/15 (the interval separating 5/4 and 4/3)

=== JI approximation ===
7edo is, very crudely, a 2.3.5 system, and strength in 2.3.5 is generally what carries into other equiheptatonic scales. It can also be viewed in various other subgroups, most notably 2.3.13 and 2.3.11, and equiheptatonic temperaments can be found that represent those subgroups as well. The diatonic scale in 7edo is equivalent to every note in the tuning system; sharps and flats are not meaningful and all intervals are perfect.
{{Harmonics in ED|7|31}}
{| class="wikitable"
|+Thirds in 7edo
!Quality
|'''Neutral'''
|-
!Cents
|'''343'''
|-
!Just interpretation
|'''11/9'''
|}
Diatonic thirds are bolded.

=== Chords ===
7edo features, for tertian triadic harmony, only a neutral chord [0 2 4] and the (rather discordant) sus chords [0 1 4] and [0 3 4]. Regardless, due to its triads and due to representing all seven degrees of the diatonic scale, it is the smallest edo where Western functional harmony works.

=== Scales ===
7edo is the first edo to distinguish the modes of the [[pentic]] scale. However, it is still small enough that it is well-temperable into scales (specifically, those of the 7-form discussed elsewhere in this article). In real world musical cultures which use near-equal 7-note scales, perfect 7edo is almost never used.

=== Derivation ===
7edo is derived by equalizing an equiheptatonic scale.

== Notation ==
In 7edo, pretty much all reasonable notation schemes collapse to ABCDEFG on A=440Hz. Accidentals are not used.

== Polysomatic tuning ==
Polysomatic tuning, coined by Cole Parker, refers to an octave stretch of 7edo (close to 11edt and 34ed30, about 6.929edo, step size 173.19c, octave 1212.33 cents) that has the property of approximating the first six harmonics of a standard harmonic instrument and of an unsupported bar ([http://hyperphysics.phy-astr.gsu.edu/hbase/Music/barres.html more context]), such as a glockenspiel, within 25% relative error (and in fact approximates them within 25 cents absolute error, except for the 6th frequency of an unsupported bar).
{| class="wikitable"
|+
! rowspan="2" |Frequency #
! colspan="5" |Unsupported bar
! colspan="5" |Harmonic instrument
|-
!Decimal
!Cents
!Polysomatic tuning
!Deviation
!Steps
!Decimal
!Cents
!Polysomatic tuning
!Deviation
!Steps
|-
|1
|1
|0
|0
|0
|0
|1
|0
|0
|0
|0
|-
|2
|2.75625
|1755.25
|1731.91
| -23.34
|10
|2
|1200.00
|1212.33
|12.33
|7
|-
|3
|5.40225
|2920.27
|2944.24
|23.97
|17
|3
|1901.96
|1905.10
|3.14
|11
|-
|4
|8.93025
|3790.44
|3810.19
|19.75
|22
|4
|2400.00
|2424.67
|24.67
|14
|-
|5
|13.34025
|4485.26
|4502.95
|17.70
|26
|5
|2786.31
|2771.05
| -15.26
|16
|-
|6
|18.63225
|5063.68
|5022.53
| -41.15
|29
|6
|3101.96
|3117.43
|15.48
|18
|}
Within the octave, polysomatic tuning also slightly improves the intervals 5/4 and 3/2, but makes 4/3 less accurate.

== Whitewood temperament ==
7edo may be interpreted as ''Whitewood'' temperament, which tempers out the Pythagorean [[chromatic semitone]]. The most obvious rank-2 extension is to add a free generator corresponding to 7/4, resulting in a system containing multiple copies of 7edo separated by the interval 7/4. This extension is supported by [[21edo]], which, along with 14edo, supports the [[Diatonic|omnidiatonic]] ternary diatonic scale.

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

MOS

2026-02-27T10:51:18Z

2^67-1:

[[File:MOS.png|thumb|401x401px|The comparison of non-MOS and MOS diatonic scales.]]
A '''MOS''' (or '''mos''', or '''moment of symmetry''' scale) is a member of a family of scales which generalize properties of the diatonic and pentatonic scales within 12 equal temperament. Microtonal MOS scales are often seen as some of the easier microtonal scales to work with, because the number of distinct intervals within the scale is quite low.

As in any scale, we can define "n-step interval" to refer to one of many intervals that arise from ascending by n steps of the scale. Formally, a MOS is a scale where, for all n, there are at most two distinct sizes of n-step intervals. Any multiple of the period (which is usually an octave or a fraction thereof) has only one size.[https://en.xen.wiki/w/MOS_scale] MOS scales are also known as '''MV2''' (or '''maximum variety 2''') scales.

MOS scales are often referred to as MOSes, thus MOS can be used as either an adjective or a noun.

MOSses can be named according to their number of large and small steps (for example, 5L 2s for the diatonic MOS), because there is exactly one step pattern that fits the MOS criteria with any given number of small and large steps. This form of name does not specify the tuning of the MOS or which scale degree is defined as the tonic.

== Examples ==
The most widely used MOS scale is the 12edo [[diatonic]] scale, which has five equal large steps (major seconds) and two equal small steps (minor seconds) within the octave. It can thus be notated 5L 2s. In contrast, while the melodic minor scale (LsLLLLs) has only two step sizes, it is still not MOS since it has three different sizes of fifths: perfect, diminished, and augmented.

A MOS exists for any whole number of large and small steps, for example [[Mosh|3L 4s]] (mosh), which functions as a "neutral" version of the diatonic scale, and [[Onyx|1L 6s]] (onyx), which has 1 large step and thus a very wide range of tunings.

The [[equave]] of a MOS is denoted using angle brackets: for example, 3L2s{{angbr|3/2}} denotes the 3L 2s MOS pattern but using 3/2 as the interval of equivalence rather than 2/1.

== Periods and generators ==
Every MOS scale can be ''generated'' by stacking a certain interval called the [[generator]] a number of times, then moving each note by multiples of the period (which is usually the octave) to fit within the span of one period.[https://www.anaphoria.com/wilsonintroMOS.html] The latter step is called period reduction. For example, the diatonic scale is generated by stacking 6 fifths (or equivalently, 6 fourths) and octave-reducing to get a 7 note scale. Another example, [[Pentic|2L 3s]], is generated by stacking 4 fifths to get 5 notes. However, stacking 5 fifths to get a hexatonic scale such as C D E F G A C does not produce a MOS, because there are more than 2 sizes of some interval classes.

The amount of stacking that produces a MOS scale depends only on the size of the generator relative to the size to the period. For a just fifth and a just octave, the valid scale sizes are 2, 3, 5, 7, 12, 17, 29, 41, 53... However for a quarter-comma [[meantone]] fifth, the valid sizes are 2, 3, 5, 7, 12, 19, 31, 50...

A 2/1-equivalent MOS scale aLbs always has 1\gcd(a, b) as its period. For example, 5L2s has period 1\1; 5L5s has period 1\5; 2L8s has period 1\2.

== Hardness ==
One way to specify the tuning of a given MOS pattern (with a given equave) is ''hardness'', which refers to the logarithmic ratio between the size of the L step versus the size of the s step. A tuning of a given MOS that has a higher hardness is ''harder'', and one with a lower hardness is ''softer''.

The hardness of a MOS scale is mostly associated with its melodic shape. Softer tunings (with more similar step sizes) may sound melodically smoother, softer, or more mellow. In contrast, harder tunings of the same MOS (with steps of very different sizes) may sound jagged, dramatic, or sparkly.

Hardness is usually given as L/s, and can fall anywhere between 1 and positive infinity. A hardness value of 1 is called ''equalized'' (since L = s), and positive infinity is called ''collapsed'' (since s = 0). We call hardness 2/1 the ''basic'' tuning of the MOS; the basic tuning is the smallest equal tuning that meaningfully supports the MOS scale. More generally, the basic tuning of a MOS aLbs is always (2a+b)-edo.[https://sevish.com/2021/getting-hard-with-scales/]

Examples for MOS diatonic:
* 12edo diatonic is 2221221, so it has hardness 2/1.
* 17edo diatonic is 3331331, so it has hardness 3/1.
* 19edo diatonic is 3332332, so it has hardness 3/2.
* The equalized tuning is 7edo (1111111).
* The collapsed tuning is 5edo (1110110).

== Table of MOS scales ==
The following table lists common MOS scales grouped by scale size, along with their step patterns and associated [[Temperament|temperaments]]. All temperaments mentioned in the table be found in the [[List of regular temperaments]]. All names in the first column do not specify anything about tuning or which scale degree is the tonic. Common MOSes are highlighted.
{| class="wikitable"
|+ Table of some 2/1-equivalent MOSes.
|-
!Name
!aLbs
!Brightest mode
!Equalized (softest) gen.
!Collapsed (hardest) gen.
!Period<br />(1\1 (1200c) unless otherwise stated)
!Description
|-
! colspan="7" |5-note MOSses
|-
! class="thl" |pentic
! class="thl" |2L3s
|LsLss
|2\5 (480c)
|1\2 (600c)
|
|Called "pentatonic" in 12edo music theory. Five-note subset of both MOS diatonic and antidiatonic.
|-
!|antipentic
!|3L2s
|LLsLs
|2\5 (480c)
|1\3 (400c)
|
|Five-note subset of both oneirotonic and checkertonic.
|-
!|manual
!|4L1s
|LLLLs
|1\5 (240c)
|1\4 (300c)
|
|Five-note subset of both semiquartal and gramitonic.
|-
! colspan="7" |6-note MOSses
|-
!|machinoid
!|5L1s
|LLLLLs
|1\6 (200c)
|1\5 (240c)
|
|Some temperament interpretations: Machine[6], Gorgo[6], Slendric[6].
|-
! colspan="7" |7-note MOSses
|-
! class="thl" |onyx
! class="thl" |1L6s
|Lssssss
|1\7 (171.4c)
|0\6 (0c)
|
|One temperament interpretation is Porcupine[7].
|-
!|antidiatonic
!|2L5s
|LssLsss
|3\7 (685.7c)
|1\2 (600c)
|
|When using a very flat fifth, reverses the interval qualities of diatonic. One temperament interpretation is Mabilic[7].
|-
! class="thl" |mosh
! class="thl" |3L4s
|LsLsLss
|2\7 (342.9c)
|1\3 (400c)
|
|Neutral thirds generate this MOS.
|-
!|smitonic
!|4L3s
|LLsLsLs
|2\7 (342.9c)
|1\4 (300c)
|
|So named because the generator is a sharp minor third. Inthar finds that it sounds like a brighter, stretched version of the diatonic scale. In fact, it can be seen as a "warped" diatonic scale, because it can be found by replacing one large step of diatonic with a small step. One temperament interpretation is Orgone[7].
|-
! class="thl" |(MOS) diatonic
! class="thl" |5L2s
|LLLsLLs
|4\7 (685.7c)
|3\5 (720c)
|
|
|-
!|arch(a)eotonic
!|6L1s
|LLLLLLs
|1\7 (171.4c)
|1\6 (200c)
|
|{{Adv|Some temperament interpretations: Tetracot[7], Didacus[7].}}
|-
! colspan="7" |8-note MOSses
|-
!|checkertonic
!|3L5s
|LsLssLss
|3\8 (450c)
|1\3 (400c)
|
|Somewhat like a stretched tcherepnin scale. Associated with Squares (when relatively hard) and Sensi (hardness 3/2 or softer). In softer tunings, it is the most consonant 8-tone MOS.
|-
! class="thl" |tetrawood
! class="thl" |4L4s
|LsLsLsLs
|1\8 (150c)
|1\4 (300c)
|1\4 (300c)
|Exists in 12edo; often called the "diminished" or "octatonic" scale in 12edo theory.[https://en.wikipedia.org/wiki/Octatonic_scale]
|-
!|oneirotonic
!|5L3s
|LLsLLsLs
|3\8 (450c)
|2\5 (480c)
|
|Sounds like a darker, compressed version of the diatonic scale according to Inthar.
|-
!|ekic
!|6L2s
|LLLsLLLs
|1\8 (150c)
|1\6 (200c)
|1\2 (600c)
| Adv |The 22edo (step ratio L:s = 3:2) tuning is associated with Hedgehog temperament. Lumi Pakkanen has described the sound of this MOS as "dreamy, yet oppressing", especially when tuned to step ratio 3:1.[https://en.xen.wiki/w/6L_2s]
|-
! class="thl" |pine
! class="thl" |7L1s
|LLLLLLLs
|1\8 (150c)
|1\7 (171.4c)
|
|
|-
! colspan="7" |9-note MOSses
|-
!|tcherepnin
!|3L6s
|LssLssLss
|1\9 (133.3c)
|1\3 (400c)
|1\3 (400c)
|Exists in 12edo; associated with Augmented temperament, which collapses the circle of just (5/4) major thirds into one augmented chord formed from 400c major thirds.
|-
!|gramitonic
!|4L5s
|LsLsLsLss
|2\9 (266.7c)
|1\4 (300c)
|
|So named because the generator is a grave minor third. Associated with Orwell temperament. One of the more consonant 9-note MOSses, especially when tuned with a step ratio 4:3 < L:s < 3:2.
|-
! class="thl" |semiquartal
! class="thl" |5L4s
|LLsLsLsLs
|2\9 (266.7c)
|1\5 (240c)
|
|So named because the generator is half a fourth. One of the more consonant 9-note MOSses.
|-
!|armotonic
!|7L2s
|LLLLsLLLs
|5\9 (666.7c)
|4\7 (685.7c)
|
|Contains antidiatonic as a subset. One temperament interpretation is Mabilic[9], or it can be seen in lower complexity as Mavila[9], where three just perfect fourths stack to a just (5/2) major tenth.
|-
!|subneutralic
!|8L1s
|LLLLLLLLs
|1\9 (133.3c)
|1\8 (150c)
|
|Has tunings that split the perfect fifth into 5 equal parts, e.g. in 17edo.
|-
! colspan="7" |10-note MOSses
|-
!|jaric
!|2L8s
|LssssLssss
|1\10 (120c)
|1\2 (600c)
|1\2 (600c)
|{{Adv|So named because of Pajara[10] and Injera[10] interpretations. Interpreted as Diaschismic[10] in 34edo and 46edo.}}
|-
! class="thl" |pentawood
! class="thl" |5L5s
|LsLsLsLsLs
|1\10 (120c)
|0\5 (0c)
|1\5 (240c)
|One temperament interpretation is Blackwood[10]. Often called the Blackwood scale because composer Easley Blackwood was one of the first to use it. Contains either a major or minor triad on every note, and thus can be seen as having similar properties to MOSdiatonic in 5n-edos.
|-
!|dicoid
!|7L3s
|LLLsLLsLLs
|3\10 (360c)
|2\7 (342.9c)
|
|The 10-note MOS generated by neutral thirds. So named because of the exotemperament [[Dichotic]].
|-
!|taric
!|8L2s
|LLLLsLLLLs
|1\10 (120c)
|1\8 (150c)
|1\2 (600c)
|Generated by an oneirotonic generator (3\5 to 5\8). Named after Hindi for 18 (''aṭhārah''), because 18edo is the basic tuning.
|-
!|sinatonic
!|9L1s
|LLLLLLLLLs
|1\10 (120c)
|1\9 (133.3c)
|
|So named because of the "sinaic" generator (named after ibn Sina), which is 1/4 of a perfect fourth.
|-
! colspan="7" |Larger MOSses
|-
!|slentonic
!|5L6s
|sLsLsLsLsLs
|2\11 (218.2c)
|1\5 (240c)
|
|The 11-note MOS of Slendric.
|-
!|p-chro smitonic
!|4L7s
|LsLssLssLss
|3\11 (327.3c)
|1\4 (300c)
|
|The 11-note MOS associated with Kleismic/Cata and Orgone, both accurate temperaments. Older material may call this scale "kleistonic."
|-
! class="thl" |p-chromatic
! class="thl" |5L7s
|LsLsLssLsLss
|7\12 (700c)
|3\5 (720c)
|
|The chromatic scale generated by sharp-of-12edo fifths. Superpyth[12] is a particularly interesting interpretation.
|-
!|hexawood
!|6L6s
|LsLsLsLsLsLs
|1\12 (100c)
|1\6 (200c)
|1\6 (200c)
|A "straddle-fifth chromatic scale", as it can be constructed by stacking alternating flat and sharp fifths as long as they stack to 1\6, in e.g. 18edo.
|-
! class="thl" |m-chromatic
! class="thl" |7L5s
|LLsLsLLsLsLs
|7\12 (700c)
|4\7 (685.7c)
|
|The chromatic scale generated by flat-of-12edo fifths. Used in many 17th-century keyboards and still used in some church organs.
|-
!|telluric
!|10L2s
|LLLLLsLLLLLs
|1\12 (100c)
|1\10 (120c)
|1\2 (600c)
|Commonly interpreted as Pajara[12] or Diaschismic[12].
|-
!|heptawood
!|7L7s
|LsLsLsLsLsLsLs
|1\14 (85.7c)
|0\7 (0c)
|1\7 (171.4c)
|Two offset rings of 7edo fifths; the 7edo analogue of the blackwood MOS.
|-
!|
!|11L4s
|LLLsLLLsLLLsLLs
|4\15 (320c)
|3\11 (327.3c)
|
|The first MOS unambiguously interpreted as Orgone.
|-
!|
!|15L4s
|LLLLsLLLLsLLLLsLLLs
|7\19 (315.8c)
|4\15 (320c)
|
|The first MOS unambiguously interpreted as Kleismic.
|}

{{Cat|
Scale construction
Core knowledge
}}

Neutral temperaments

2026-02-27T10:48:57Z

2^67-1:

'''Dichotic''' is an exotemperament that can be defined to temper out [[25/24]], the dicot comma, [[45/44]], and [[64/63]]. As a result it also tempers out [[55/54]].

== Interval chart ==

This interval chart uses Partch's 11-limit tonality diamond and maps it onto [[7edo]] and [[10edo]]. This also gives an idea of what intervals are possible in dichotic temperament. '''Intervals in bold''' are in the 7-note MOS (symmetric mode) and ''intervals in italics'' are in the 10-note MOS (both near-symmetric modes). Note that the interval that can be mapped to 11/8 and 10/7 and the interval mapped to 7/5 and 16/11 cannot coexist in the same 10-note MOS.

{| class="wikitable"
! !! 0\7 !! 1\7 !! 2\7 !! 3\7 !! 4\7 !! 5\7 !! 6\7 !! 7\7
|-
! scope="row"| 0\10
| '''1/1''' || || || || || || ||
|-
! scope="row"| 1\10
| || '''12/11, 10/9''' || || || || || ||
|-
! scope="row"| 2\10
| || ''11/10, 9/8, 8/7'' || 7/6 || || || || ||
|-
! scope="row"| 3\10
| || || '''6/5, 11/9, 5/4''' || 14/11 || || || ||
|-
! scope="row"| 4\10
| || || 9/7 || '''4/3''' || || || ||
|-
! scope="row"| 5\10
| || || || ''11/8, 10/7'' || ''7/5, 16/11'' || || ||
|-
! scope="row"| 6\10
| || || || || '''3/2''' || 14/9 || ||
|-
! scope="row"| 7\10
| || || || || 11/7 || '''8/5, 18/11, 5/3''' || ||
|-
! scope="row"| 8\10
| || || || || || 12/7 || ''7/4, 16/9, 20/11'' ||
|-
! scope="row"| 9\10
| || || || || || || '''9/5, 11/6''' ||
|-
! scope="row"| 10\10
| || || || || || || || '''2/1'''
|}

== Theory ==

Dichotic is generated by a neutral 3rd, or one half of a perfect fifth. A notation system based on dichotic temperament is still in the works.

Neutral temperaments

2026-02-27T10:47:02Z

2^67-1: Created page with "'''Dichotic''' is an exotemperament that can be defined to temper out 25/24, the dicot comma, 45/44, and 64/63. As a result it also tempers out 55/54. == Interval chart == This interval chart uses Partch's 11-limit tonality diamond and maps it onto 7edo and 10edo. This also gives an idea of what intervals are possible in dichotic temperament. '''Intervals in bold''' are in the 7-note MOS (symmetric mode) and ''intervals in italics'' are in the 1..."

Erac

2026-01-07T01:30:54Z

2^67-1: /* Erac-derived temperaments */

'''Eracs''', short for error accidentals, are symbols that indicate how much a tempered interval is flat or sharp relative to others in a subgroup. They act as variables representing small pitch differences and have no set size. They are the standard notation for subgroups involving [[Straddle primes]].

Eracs provide a more complete picture of error cancelation than the standard notation of non-prime intervals. For example, 11edo almost perfectly misses primes 3 and 5 present 22edo, which still allows them to cancel out for an accurate 5/3 and 15. 11edo's subgroup might be 2.5/3.15.7.11 in standard notation, or 2.x3.x5.7.11 in erac notation. This becomes even more important in edos like 23 or 29, where several low primes are critically inaccurate and representing all of the error cancelation in standard notation creates a very long subgroup.

Example erac subgroups for edos can be found in [[EDO#List of Edos]].

== Symbols ==

{| class="wikitable"
|+Erac Meanings (intervals represented by an underscore)
|-
|<_
|Flat by a set arbitrary amount.
|-
|>_
|Sharp by a set arbitrary amount.
|-
|x_
|Critically flat/sharp. Shorthand for <_.>_.
|-
|<x_ and >x_
|Shorthand for <<_.>_ and <_.>>_ respectively.
|-
|{ and }
|Partial eracs, indicating error that may be ignored.
|-
|~_
|Approximate, tempered equivalent. A widely accepted symbol used in this context to indicate that no eracs apply.
|}

Eracs are placed before their respective numbers by default (although they may be placed after as well) This is for readability and to remove ambiguity with the denominator, as eracs on the denominator have an inverse effect on the size of a tempered ratio. For example, >5/3 is sharp or 5/3, but 5/>3 is flat.

== Erac temperaments ==
Temperament optimization algorithms can be built around eracs to create optimal error cancelation, such as [https://spoogly.website/tools/gto.html GTO]. Erac temperaments have the potential to replace exotemperaments by properly representing the large error in approximating certain primes, providing less misleading mappings with systematic accuracy.

== Erac-derived temperaments ==

These were a concept derived by [[User:2^67-1|Cole]] to be able to quickly create temperaments.

His procedure:

* For a flat harmonic and a sharp harmonic:
** take the flat harmonic and take it to the power of how many eracs the sharp harmonic has
** take the sharp harmonic and take it to the power of how many eracs the flat harmonic has
** multiply them together
** take it to the power of how many steps there are in an EDO
** octave-reduce (if the interval is greater than 600c octave reduced, take the inverse)

{{Cat|Atypical ratios
Notation}}

Erac

2026-01-06T01:39:45Z

2^67-1:

'''Eracs''', short for error accidentals, are symbols that indicate how much a tempered interval is flat or sharp relative to others in a subgroup. They act as variables representing small pitch differences and have no set size. They are the standard notation for subgroups involving [[Straddle primes]].

Eracs provide a more complete picture of error cancelation than the standard notation of non-prime intervals. For example, 11edo almost perfectly misses primes 3 and 5 present 22edo, which still allows them to cancel out for an accurate 5/3 and 15. 11edo's subgroup might be 2.5/3.15.7.11 in standard notation, or 2.x3.x5.7.11 in erac notation. This becomes even more important in edos like 23 or 29, where several low primes are critically inaccurate and representing all of the error cancelation in standard notation creates a very long subgroup.

Example erac subgroups for edos can be found in [[EDO#List of Edos]].

== Symbols ==

{| class="wikitable"
|+Erac Meanings (intervals represented by an underscore)
|-
|<_
|Flat by a set arbitrary amount.
|-
|>_
|Sharp by a set arbitrary amount.
|-
|x_
|Critically flat/sharp. Shorthand for <_.>_.
|-
|<x_ and >x_
|Shorthand for <<_.>_ and <_.>>_ respectively.
|-
|{ and }
|Partial eracs, indicating error that may be ignored.
|-
|~_
|Approximate, tempered equivalent. A widely accepted symbol used in this context to indicate that no eracs apply.
|}

Eracs are placed before their respective numbers by default (although they may be placed after as well) This is for readability and to remove ambiguity with the denominator, as eracs on the denominator have an inverse effect on the size of a tempered ratio. For example, >5/3 is sharp or 5/3, but 5/>3 is flat.

== Erac temperaments ==
Temperament optimization algorithms can be built around eracs to create optimal error cancelation, such as [https://spoogly.website/tools/gto.html GTO]. Erac temperaments have the potential to replace exotemperaments by properly representing the large error in approximating certain primes, providing less misleading mappings with systematic accuracy.

== Erac-derived temperaments ==

These were a concept derived by [[User:2^67-1|Cole]] to be able to quickly create temperaments.

His procedure:

* For a flat harmonic and a sharp harmonic:
** take the flat harmonic and take it to the power of how many eracs the sharp harmonic has
** take the sharp harmonic and take it to the power of how many eracs the flat harmonic has
** multiply them togethertake it to the power of how many steps there are in an EDO
** octave-reduce (if the interval is greater than 600c octave reduced, take the inverse)

{{Cat|Atypical ratios
Notation}}