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

Aura at 19:57, 31 March 2026

2026-03-31T19:57:53Z

← Older revision		Revision as of 19:57, 31 March 2026
Line 1,307:		Line 1,307:

	== Trines ==		== 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.		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.		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.

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

2026-03-31T19:55:14Z

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

← Older revision		Revision as of 19:55, 31 March 2026
Line 1,307:		Line 1,307:

	== Trines ==		== 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.		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"		{\| class="wikitable"

Aura at 06:55, 31 March 2026

2026-03-31T06:55:33Z

← Older revision		Revision as of 06:55, 31 March 2026
Line 8:		Line 8:
	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.		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.		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.		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.

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

2026-03-31T06:26:59Z

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

← Older revision		Revision as of 06:26, 31 March 2026
Line 1,334:		Line 1,334:
	\| 0, 65, 0		\| 0, 65, 0
	\| 64:85:128		\| 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~~		\| This trine is the first of two that are often used in the extended harmony of t<IV chords
	\|-		\|-
	\| Utonal Archagall		\| Utonal Archagall
Line 1,340:		Line 1,340:
	\| 0, 94, 0		\| 0, 94, 0
	\| 1/(64:85:128)		\| 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~~		\| This trine is the second of two that are often used in the extended harmony of t<IV chords
	\|-		\|-
	\| Bass-Up Marvelous		\| Bass-Up Marvelous
Line 1,346:		Line 1,346:
	\| 0, 92, 0		\| 0, 92, 0
	\| 75:112:150		\| 75:112:150
	\| This ~~dissonant~~ trine is the first of two that are formed from stacking identical approximations of the LCJI neutral third		\| This trine is the first of two that are formed from stacking identical approximations of the LCJI neutral third
	\|-		\|-
	\| Treble-Down Marvelous		\| Treble-Down Marvelous
Line 1,352:		Line 1,352:
	\| 0, 67, 0		\| 0, 67, 0
	\| 1/(75:112:150)		\| 1/(75:112:150)
	\| This ~~dissonant~~ trine is the second of two that are formed from stacking identical approximations of the LCJI neutral third		\| This trine is the second of two that are formed from stacking identical approximations of the LCJI neutral third
	\|-		\|-
	\| Narrow Supernaiadic		\| Narrow Supernaiadic

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

2026-03-31T06:21:53Z

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

← Older revision		Revision as of 06:21, 31 March 2026
Line 2:		Line 2:

	== Intervals and Notation ==		== 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 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.

	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.		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"		{\| class="wikitable"
Line 1,300:		Line 1,306:
	\|}		\|}

	== Trines ~~and Tetrachords~~ ==		== Trines ==
	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~~.		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.

	~~First, the trines shall be covered, for they serve as one of the fundamental units framing scales, melodies and harmonies.~~
	{\| class="wikitable"		{\| class="wikitable"
	\|+Table of 159edo Trines		\|+Table of 159edo Trines
Line 1,565:		Line 1,570:
	\| This trine is very likely to be used as a partial basis for suspended chords		\| 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.

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

2026-03-31T00:24:59Z

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

← Older revision		Revision as of 00:24, 31 March 2026
Line 1,566:		Line 1,566:
	\|}		\|}

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

	~~{\| 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~~.

Aura at 23:45, 30 March 2026

2026-03-30T23:45:45Z

← Older revision		Revision as of 23:45, 30 March 2026
Line 1,582:		Line 1,582:
	\|Pythagorean Whole Tone (2)		\|Pythagorean Whole Tone (2)
	\|		\|
	\|Pythagorean ~~Diatonic Semitone~~		\|Pythagorean Limma
	\|		\|
	\|-		\|-
Line 1,588:		Line 1,588:
	\|Pythagorean Whole Tone		\|Pythagorean Whole Tone
	\|Ptolemaic Whole Tone		\|Ptolemaic Whole Tone
	\|Ptolemaic ~~Diatonic Semitone~~		\|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.

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

2026-03-30T23:41:12Z

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

Show changes

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

2026-03-30T22:53:49Z

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

Show changes

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

2026-03-30T22:41:39Z

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

New page

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
|}