159edo/Interval names and harmonies

159edo contains all the intervals of 53edo, however, as some of the interpretations differ due 159edo having different mappings for certain primes, those differences show up in how harmonies are constructed. It should be noted that since 159edo does a better job of representing the 2.3.11 subgroup than 24edo, some of the chords listed on the page for 24edo interval names and harmonies carry over to this page, even though the exact sets of enharmonics differ between the two systems.

Table of 159edo intervals

Step	Cents	5 limit	7 limit	11 limit	13 limit	17 limit	Interval Names			Notes
0	0	1/1					P1	Perfect Unison	D	This interval... Is the basic representation of a given chord's root Is the basic representation of the Tonic Is one of four perfect consonances in this system
1	7.5471698		225/224	243/242	196/195, 351/350	256/255	R1	Wide Prime	D/	This interval... Approximates the rastma, and thus... Is useful for defining 11-limit subchromatic alterations in the Western-Classical-based functional harmony of this system Approximates the marvel comma, and thus... Can function as both a type of subchroma and a type of reverse diesis in this system Is useful for slight dissonances that convey something less than satisfactory Can only be approached in melodic lines indirectly with one or more intervening notes Can add to the bandwidth of a sound
2	15.0943396		?	121/120, 100/99	144/143	120/119	rK1	Narrow Superprime	D↑\	This interval... Approximates the ptolemisma and the biyatisma Is useful for slight dissonances that create noticeable tension Can only be approached in melodic lines indirectly with one or more intervening notes
3	22.6415094	81/80	?	?	78/77	85/84	K1	Lesser Superprime	D↑	This interval... Approximates the syntonic comma, and as such... Is especially useful as a basis for defining 5-limit subchromatic alterations in the Western-Classical-based functional harmony of this system Approximates the Pythagorean comma, and thus... Can be considered a type of reverse diesis Is a dissonance to be avoided in Western-Classical-based harmony unless deliberately used for expressive purposes Is useful in melody as... An appoggiatura An acciaccatura Part of a series of quick passing tones
4	30.1886792		64/63	56/55, 55/54	?	52/51	S1, kU1	Greater Superprime, Narrow Inframinor Second	Edb<, Dt<↓	This interval... Approximates the septimal comma Approximates the telepathma Is a dissonance to be avoided in Western-Classical-based harmony unless... Used for hidden subchromatic voice-leading in the middle voices Used in a contrapuntal passage in which both of the notes separated by this interval in one voice work well against the notes in all other voices Deliberately used for expressive purposes Is useful in melody as... An appoggiatura An acciaccatura Part of a series of quick passing tones The destination for a glissando
5	37.7358491		?	45/44	?	51/50	um2, RkU1	Inframinor Second, Wide Superprime	Edb>, Dt>↓	This interval... Approximates the Undecimal Fifth-Tone Approximates a complex 11-limit paradiatonic quartertone that is the namesake of 24edo's own Inframinor Second Is the closest approximation of 31edo's own Superprime found in this system, and thus... Is capable of being used in progressions reminiscent of that system's spiral progressions Is a dissonance to be avoided in Western-Classical-based harmony unless... Used for hidden voice-leading in the middle voices Used for tonality-flux-based chord progressions Used in a contrapuntal passage in which both of the notes separated by this interval in one voice work well against the notes in all other voices Deliberately used for expressive purposes Is useful in melody as... A non-chord passing tone The destination for a glissando
6	45.2830189	?	?	?	40/39	192/187	kkm2, Rum2, rU1	Wide Inframinor Second, Narrow Ultraprime	Eb↓↓, Dt<\	This interval... Approximates the Tridecimal Minor Diesis Is one half of a Pythagorean Minor Second in this system, and thus... It functions like an Ultraprime in that... It has the potential to move directly back down to the Tonic through a parachromatic motion It has the potential to move away from the Tonic towards either a Contralead or Supertonic harmony through a diatonic or paradiatonic motion It cannot occur as the distance between any two notes in a single chord in Western-Classical-based polypedal harmony due to its dissonance It functions like an Inframinor Second in that... It can be used in Western-Classical-based harmony as part of a contrapuntal passage in which both of the notes separated by this interval in one voice work well against the notes in all other voices It can be used in Western-Classical-based harmony for hidden voice-leading in the middle voices Is one of the more important intervals for use in tonality-flux-based chord progressions
7	52.8301887		?	33/32	?	34/33	U1, rKum2	Ultraprime, Narrow Subminor Second	Dt<, Edb<↑	This interval... Approximates the Al-Farabi Quartertone, and as such... It functions as the default parachromatic quartertone in Western-Classical-based Paradiatonic functional harmony, and thus... Can be used more overtly in both melodic and harmonic voice-leading in general, though doing so in Western-Classical-based music requires a proper set-up Cannot occur as the distance between any two notes in a single chord in Western-Classical-based polypedal harmony due to its dissonance Has the potential to move directly back down to the Tonic through a parachromatic motion Has the potential to move away from the Tonic towards either a Contralead or Supertonic harmony through a diatonic or paradiatonic motion Is one fifth of this system's approximation of the Septimal Subminor Third Is the closest approximation of 22edo's Lesser Minor Second in this system, and thus... Can be used in Superpyth-based melodic and harmonic gestures reminiscent of those found in that system, albeit with caveats, since such moves on their own don't work the exact same way in this system Is the closest approximation of 24edo's own Ultraprime in this system, and thus... Follows similar interval arithmetic logic in relation to Pythagorean intervals, albeit with caveats, since the rastma is not tempered out Is one of the more important intervals for use in tonality-flux-based chord progressions
8	60.3773585		28/27	?	?	88/85	sm2, Kum2, uA1	Lesser Subminor Second, Wide Ultraprime, Infra-Augmented Prime	Dt>, Eb↓\	This interval... Approximates the Septimal Subminor Second, and thus... Is the narrowest interval that can be used in Western-Classical-based harmony and Neo-Medieval harmony as a proper leading tone Compared to other options, it has a markedly more tense feel Can be used as an unexpected option for a chromatic-type semitone in Western-Classical-based harmony Is the closest approximation of 19edo's Augmented Prime found in this system, and thus... Is capable of being used for similar modulatory moves, albeit with caveats, since such moves on their own don't work the exact same way in this system Is one third of this system's approximation of the Ptolemaic Major Second Can be used for tonality-flux-based chord progressions
9	67.9245283	25/24	?	?	26/25, 27/26	?	km2, RuA1, kkA1	Greater Subminor Second, Diptolemaic Augmented Prime	Eb↓, Dt<↑\, D#↓↓	This interval... Approximates the Classic Chroma, and thus... It frequently acts as a chromatic semitone in Western-Classical-based harmony Approximates the Large Tridecimal Third-Tone and the Small Tridecimal Third-Tone, and thus... It demonstrates third-tone functionality- especially in relation to this system's approximation of the Pythagorean Major Second- due to the combination commas tempered out in this system Is the closest approximation of 17edo's Minor Second found in this system, and thus... Is capable of being used for similar modulatory moves, albeit with caveats, since such moves on their own don't work the exact same way in this system
10	75.4716981		?	?	?	160/153	Rkm2, rKuA1	Wide Subminor Second, Lesser Sub-Augmented Prime	Eb↓/, Dt<↑	This interval... Approximates multiple complex 17-limit intervals relative to the Tonic and can be used... In Western-Classical-based harmony as part of the simul cadence due to it providing a smooth option for both voice-leading and chord construction As an unexpected option for a chromatic-type semitone in Western-Classical-based harmony Is the closest approximation of 31edo's Subminor Second found in this system, and thus... Is capable of being used in certain similar melodic and harmonic gestures, albeit with caveats, since such moves on their own don't work the exact same way in this system Is the closest approximation of 16edo's Subminor Second found in this system, and thus... Can be used in melodic and harmonic gestures reminiscent of those found in that system, albeit with caveats, since such moves on their own don't work the exact same way in this system
11	83.0188679		21/20	22/21	?	?	rm2, KuA1	Narrow Minor Second, Greater Sub-Augmented Prime	Eb\, Dt>↑	This interval... Approximates the Septimal Minor Semitone, and thus... It serves as a type of leading tone when resolving septimal harmony constructions to classic harmony constructions Approximates the Small Undecimal Semitone, and thus... It serves as a type of small chromatic semitone in undecimal harmony constructions Is one sixth of this system's approximation of the Perfect Fourth
12	90.5660377	256/243, 135/128	?	?	?	?	m2, kA1	Pythagorean Minor Second, Ptolemaic Augmented Prime	Eb, D#↓	This interval... Approximates the Pythagorean Limma or Pythagorean Minor Second, and as such... It serves as a Diatonic semitone in both Western-Classical-based functional harmony and Neo-Medieval harmony in general, and thus... Can be used readily in both melodic and harmonic voice-leading in general Can occur as the distance between two notes in a single chord in Western-Classical-based polypedal harmony, though this causes dissonance, and thus requires resolution Has the potential to move directly back down to the Tonic as a reverse leading-tone, though in non-meantone systems like this one, such a gesture using this interval has a slightly more tense feel Can serve as a possible interval between the Tonic and the root of a Neapolitan chord Approximates the Major Chroma or Ptolemaic Augmented Prime, and as such... It serves as a chromatic semitone in the 5-limit Diatonic settings that are common to Western-Classical-based harmony, and thus... It separates Pythagorean Major intervals from Ptolemaic Minor Intervals from Pythagorean Major intervals, and likewise separates Ptolemaic Major intervals from Pythagorean Minor intervals As a further consequence of the schisma being tempered out in this system, this interval is one half of this system's approximation of the Ptolemaic Major Second Is the closest approximation of 13edo's own Minor Second in this system, and thus... Can be used in Warped Diatonic gestures reminiscent of those found in that system, albeit with caveats, since such moves on their own don't work the exact same way in this system
13	98.1132075		?	128/121	55/52	18/17	Rm2, RkA1	Artomean Minor Second, Artomean Augmented Prime	Eb/, D#↓/	This interval... Approximates the Small Septendecimal Semitone, and thus... Can be used as an unexpected option for a chromatic-type semitone in Western-Classical-based harmony Approximates the Axirabian Limma, and thus... Can be used as a type of Diatonic semitone in undecimal harmony Is one of two in this system that are essential in executing the frameshift cadence Is the closest approximation of the 12edo Minor Second found in this system, and thus... Can be used in melodic and harmonic gestures reminiscent of those found in that system, albeit with caveats, since such moves on their own don't work the exact same way in this system
14	105.6603774		?	?	?	17/16	rKm2, rA1	Tendomean Minor Second, Tendomean Augmented Prime	D#\, Eb↑\	This interval... Approximates the Large Septendecimal Semitone or Octave-Reduced Seventeenth Harmonic, and thus... Can be used as an unexpected option for a Diatonic-type semitone in Western-Classical-based harmony Approximates the Parapotome and thus... Can be used as a type of chromatic semitone in undecimal harmony Is the closest approximation of 22edo's Greater Minor Second in this system, and thus... Can be used in Faux-Classical-based melodic and harmonic gestures reminiscent of those found in that system, albeit with caveats, since the biyatisma is not tempered out Can be used in Warped Diatonic gestures reminiscent of those found in that system, albeit with caveats, since such moves on their own don't work the exact same way in this system
15	113.2075472	16/15	?	?	?	?	Km2, A1	Ptolemaic Minor Second, Pythagorean Augmented Prime	D#, Eb↑	This interval... Approximates the Classic Minor Second or Ptolemaic Minor Second, and thus... It serves as the the traditional leading tone in 5-limit Western-Classical-based functional harmony and thus... Is one of the staples of both melodic and harmonic voice-leading Can occur as the distance between two notes in a single chord in Western-Classical-based polypedal harmony, though this causes crowding, and thus requires resolution Has the potential to move directly back down to the Tonic as a reverse leading-tone, though in non-meantone systems like this one, such a gesture using this interval has a slightly more lax and natural feel Can serve as a possible interval between the Tonic and the root of a Neapolitan chord Approximates the Apotome or Pythagorean Augmented Prime, and thus... Is generally the interval that defines the default value of sharps and flats in this system, and is thus very helpful as a reference interval Is one of two in this system that are essential in executing the frameshift cadence Is the closest approximation of 31edo's own Minor Second found in this system, and thus... Can be used in melodic and harmonic gestures reminiscent of those found in that system, albeit with caveats, since such moves on their own don't work the exact same way in this system
16	120.7547170		15/14	275/256	?	?	RKm2, kn2, RA1	Wide Minor Second, Artoretromean Augmented Prime	Ed<↓, Eb↑/, D#/	In addition to being the approximation of the Septimal Major Semitone, this interval is also one third of a Lesser Submajor Third in this system, and is thus used accordingly.
17	128.3018868		?	?	14/13	128/119	kN2, rKA1	Lesser Supraminor Second, Tendoretromean Augmented Prime	Ed>↓, D#↑\	In addition to its properties as a type of supraminor second, this interval is also one third of a Ptolemaic Major Third in this system and is thus used accordingly.
18	135.8490566	27/25	?	?	13/12	?	KKm2, rn2, KA1	Greater Supraminor Second, Diptolemaic Limma, Retroptolemaic Augmented Prime	Ed<\, Eb↑↑, D#↑	This interval is not only both two thirds of Pythagorean Major Second and the approximation of the Large Limma or Diptolemaic Limma in this system, but also a type of supraminor second, and is thus used accordingly.
19	143.3962264		?	88/81	?	?	n2, SA1, kUA1	Artoneutral Second, Lesser Super-Augmented Prime	Ed<, Dt#<↓	As one of two neutral seconds in this system, this interval is notable for being half of the approximation of the Neo-Gothic Minor Third, though it is also sometimes used in much the same way as 24edo's own Neutral Second.
20	150.9433962		?	12/11	?	?	N2, RkUA1	Tendoneutral Second, Greater Super-Augmented Prime	Ed>, Dt#>↓	As one of two neutral seconds in this system, this interval is the one that most closely resembles the low-complexity JI neutral second, and thus, it is frequently used in much the same way as 24edo's own Neutral Second.
21	158.4905660	?	?	?	128/117	561/512, 1024/935	kkM2, RN2, rUA1	Lesser Submajor Second, Diretroptolemaic Augmented Prime	Ed>/, E↓↓, Dt#>↓/, D#↑↑	In addition to being a type of Submajor Second and the closest approximation of the 31edo Middle Second found in this system, two of these add up to the approximation of the Ptolemaic Minor Third.
22	166.0377358		?	11/10	?	?	Kn2, UA1	Greater Submajor Second, Ultra-Augmented Prime	Ed<↑, Dt#<, Fb↓/	In addition to its properties as the interval that most closely resembles the Undecimal Submajor Second, this interval serves as both the Ultra-Augmented Prime and as one third of a Perfect Fourth, and is used accordingly.
23	173.5849057		567/512	243/220	?	425/384	rkM2, KN2	Narrow Major Second	Ed>↑, E↓\, Dt#>, Fb\	While this interval is large enough to act as a type of whole tone, it is worth noting that two of these add up to the approximation of the low-complexity JI Neutral Third in this system.
24	181.1320755	10/9	?	256/231	?	?	kM2	Ptolemaic Major Second	E↓, Fb	As the approximation of the Ptolemaic Major Second, this interval is used accordingly, though it is worth noting that in this system, two of these add up to the approximation of the thirteenth subharmonic; furthermore, it is also one the intervals in this system that are essential in executing any sort of variation on Jacob Collier's "Four Magical chords" from his rendition of "In the Bleak Midwinter".
25	188.6792458		?	?	143/128	512/459	RkM2	Artomean Major Second	E↓/, Fb/	This interval has surprising utility in modulating to keys that are not found on the same circle of fifths owing to both its size and its ease of access through octave-reducing stacks of approximated low-complexity JI intervals.
26	196.2264151		28/25	121/108	?	?	rM2	Tendomean Major Second	E\, Fb↑\	In addition to being the closest approximation of the 31edo Major Second found in this system, it is one of two intervals that come the closest to approximating the 12edo Major Second found in this system.
27	203.7735849	9/8	?	?	?	?	M2	Pythagorean Major Second	E, Fb↑	This interval is the standard-issue whole tone in this system as it is one of two intervals that come the closest to approximating the 12edo Major Second, and the only one of the two that actually approximates the Pythagorean Major Second; furthermore, it is the whole tone that is used as a reference interval in diatonic-and-chromatic-style interval logic in this system as it pertains to both semitones and quartertones.
28	211.3207547		?	?	44/39	289/256	RM2	Wide Major Second	E/, Fd<↓	This interval is interesting on the basis that it is formed by stacking two instances of the octave-reduced approximation of the seventeenth harmonic.
29	218.8679245		?	?	?	17/15	rKM2	Narrow Supermajor Second	E↑\, Fd>↓	This interval is interesting not only because it is utilized in approximations of the 17-odd-limit, but also because it is the whole tone found in this system's Superpyth scale, and is of such quality that two of these add up to this system's approximation of the Septimal Supermajor Third.
30	226.4150943	256/225	?	154/135	?	?	KM2	Lesser Supermajor Second	E↑, Fd<\, Fb↑↑, Dx	This interval can be interpreted as a type of second on the basis of it approximating the sum of the syntonic comma and the Pythagorean Major Second; it also appears in approximations of 5-limit Neapolitan scales as the interval formed from stacking two Ptolemaic Minor Seconds, making it double as a type of diminished third, and is likely the smallest interval in this system that can be used in chords without causing crowding.
31	233.9622642		8/7	55/48	?	?	SM2, kUM2	Greater Supermajor Second, Narrow Inframinor Third	Fd<, Et<↓, E↑/	As the approximation of the octave-reduced seventh subharmonic- that is, the Septimal Supermajor Second- this interval is used accordingly; in fact, since three of these add up to a Perfect Fifth in this system, there are multiple ways this interval can be used in chords to great effect.
32	241.5094340		?	1024/891	?	?	um3, RkUM2	Inframinor Third, Wide Supermajor Second	Fd>, Et>↓	The 11-limit ratio this interval approximates is the namesake of 24edo's own Inframinor Third; however, in a higher-fidelity system such as this, one will notice that this is a syntactic third that sounds more like a second.
33	249.0566038	?	?	?	15/13	?	kkm3, KKM2, Rum3, rUM2	Wide Inframinor Third, Narrow Ultramajor Second, Semifourth	Fd>/, Et<\, F↓↓, E↑↑	This interval is particularly likely to be used as a cross between an Ultramajor Second and an Inframinor Third; furthermore, as the name "Semifourth" suggests, this interval is one half of a Perfect Fourth, and used in exactly the same way as 24edo's own Semifourth, right down to the low-complexity 13-limit interpretation.
34	256.6037736		?	297/256	?	?	UM2, rKum3	Ultramajor Second, Narrow Subminor Third	Et<, Fd<↑	The 11-limit ratio this interval approximates is the namesake of 24edo's own Ultramajor Second; however, in a higher-fidelity system such as this, one will notice that this is a syntactic second that sounds more like a third.
35	264.1509434		7/6	64/55	?	?	sm3, Kum3	Lesser Subminor Third, Wide Ultramajor Second	Et>, Fd>↑, F↓\	As the approximation of the Septimal Subminor Third, those who are not already familiar with septimal harmony will find this interval useful in forming not only strident-sounding triads framed by the Perfect Fifth, but also other, ambisonant triads framed by the Perfect Fourth; in addition, three of these add up to the Pythagorean Minor Sixth.
36	271.6981132	75/64	?	?	?	?	km3	Greater Subminor Third	F↓, Et>/, E#↓↓, Gbb	This interval is useful for evoking the feel of 31edo due to approximating that system's Subminor Third, and even approximates the result of subtracting a syntonic comma from a Pythagorean Minor Third; however, it most frequently appears in approximations of 5-limit Harmonic scales as the interval between the Ptolemaic Minor Sixth and the Ptolemaic Major Seventh, making it double as a type of augmented second.
37	279.2452830		?	?	?	20/17	Rkm3	Wide Subminor Third	F↓/, Et<↑	This interval is utilized in approximations of the 17-odd-limit, courtesy of acting as the fourth complement to the Narrow Supermajor Second; it is also good for evoking the feel of 17edo due to approximating that system's Minor Third.
38	286.7924528		?	33/28	13/11	85/72	rm3	Narrow Minor Third	F\, Et>↑	This interval is of particular interest because it is the approximation of the Neo-Gothic Minor Third and is used accordingly; what's more, this interval and the approximation of the Neo-Gothic Major Third add up to make the Perfect Fifth in this system.
39	294.3396226	32/27	?	?	?	?	m3	Pythagorean Minor Third	F	This interval approximates the Pythagorean Minor Third, and since this system does not temper out the syntonic comma, this interval- in contrast to the Ptolemaic Minor Third- is very useful as an interpretation of the dissonant Minor Third from Medieval music's florid organum, and can thus be used in creating a subtle instability in certain Diatonic harmonies.
40	301.8867925		25/21	144/121	?	?	Rm3	Artomean Minor Third	F/	This interval is the closest approximation of the 12edo Minor Third found in this system, and, conveniently enough, it is easily accessed by stacking instances of this system's approximation of the low-complexity JI neutral second.
41	309.4339622		?	?	512/429	153/128	rKm3	Tendomean Minor Third	F↑\	In addition to being the closest approximation of the 31edo Minor Third found in this system, this interval is also half of this system's approximation of the Greater Septimal Tritone and is thus used accordingly as part of a triad.
42	316.9811321	6/5	?	77/64	?	?	Km3	Ptolemaic Minor Third	F↑, E#	As the approximation of the Ptolemaic Minor Third- that is, the traditional 5-limit minor third- this interval is one of four imperfect consonances in this system, and, unsurprisingly, is thus used accordingly; however, one should also note that this interval can be reached by stacking three of this system's approximation of the octave-reduced seventeenth harmonic.
43	324.5283019		135/112	?	?	512/425	RKm3, kn3	Wide Minor Third	Ft<↓, F↑/, Gdb<	The main thing of note concerning this interval is that two of these add up to this system's approximation of the Paraminor Fifth, thus facilitating the formation of strange-sounding triads
44	332.0754717		?	40/33, 121/100	?	144/119, 165/136	kN3, ud4	Lesser Supraminor Third, Infra-Diminished Fourth	Ft>↓, Gdb>	This interval is mainly of interest due to the fact that it's exactly twice the size of it's fourth complement- the approximation of the Undecimal Submajor Second- and its interesting properties as a type of supraminor third.
45	339.6226415	?	?	?	39/32	17/14	KKm3, rn3, Rud4	Greater Supraminor Third, Diretroptolemaic Diminished Fourth	Ft<\, F↑↑, Gdb<↑\, Gb↓↓	This interval is of interest because not only does it have 13-limit interpretations, but it also has usage as a 17-odd-limit interval, and all while being easily reached by stacking three Ptolemaic Minor Seconds.
46	347.1698113		?	11/9	?	?	n3, rKud4	Artoneutral Third, Lesser Sub-Diminished Fourth	Ft<, Gdb<↑	As one of two neutral thirds in this system, this interval is the one that most closely resembles the low-complexity JI neutral third, and thus, it is frequently used in much the same way as 24edo's own Neutral Third; on top of that, it can be stacked in interesting ways in this system.
47	354.7169811		?	27/22	?	?	N3, sd4, Kud4	Tendoneutral Third, Greater Sub-Diminished Fourth	Ft>, Gdb>↑	As one of two neutral seconds in this system, this interval is notable for being one half of a possible generator for this system's superpyth scale.
48	362.2641509	?	?	?	16/13	21/17	kkM3, RN3, kd4	Lesser Submajor Third, Retroptolemaic Diminished Fourth	Ft>/, F#↓↓, Gb↓	As both the approximation of the octave-reduced thirteenth subharmonic, and ostensibly one of the easiest 13-limit thirds to utilize in chords framed by some type of sharp wolf fifth, this interval is used accordingly.
49	369.8113208		?	?	?	68/55	Kn3, Rkd4	Greater Submajor Third, Artoretromean Diminished Fourth	Ft<↑, Gb↓/	In addition to its properties as a type of submajor third, this interval is also one third of a Pythagorean Major Seventh in this system and is thus used accordingly.
50	377.3584906		56/45	1024/825	?	?	rkM3, KN3, rd4	Narrow Major Third, Tendoretromean Diminished Fourth	Ft>↑, F#↓\, Gb\	The main thing of note concerning this interval is that two of these add up to this system's approximation of the Paramajor Fifth, thus facilitating the formation of strange-sounding triads.
51	384.9056604	5/4	?	96/77	?	?	kM3, d4	Ptolemaic Major Third, Pythagorean Diminished Fourth	Gb, F#↓	This interval is none other than the approximation of the octave-reduced fifth harmonic- that is, the traditional 5-limit major third- and thus, it one of four imperfect consonances in this system, and, unsurprisingly, is used accordingly; however, this interval is also the approximation of the Pythagorean Diminished Fourth in this system, which sometimes leads to interesting enharmonic substitutions when building chords for purposes of voice-leading.
52	392.4528302		?	?	?	64/51	RkM3, Rd4	Artomean Major Third, Artomean Diminished Fourth	Gb/, F#↓/	As this interval is situated between the Ptolemaic Major Third on one hand and the familiar major third of 12edo on the other, this interval can easily be used in modulatory maneuvers similar to those performed by Jacob Collier.
53	400		63/50	121/96	?	?	rM3, rKd4	Tendomean Major Third, Tendomean Diminished Fourth	F#\, Gb↑\	As none other than the familiar major third of 12edo, this interval is useful for creating the familiar augmented triads of 12edo, performing modulatory maneuvers based around said triads, and evoking the feel of 12edo in other ways.
54	407.5471698	81/64	?	?	?	?	M3, Kd4	Pythagorean Major Third, Ptolemaic Diminished Fourth	F#, Gb↑	This interval approximates the Pythagorean Major Third, and, since this system does not temper out the syntonic comma, this interval- in contrast to the Ptolemaic Major Third- is very useful as an interpretation of the dissonant Major Third from Medieval music's florid organum, and can thus be used in creating a subtle instability in certain Diatonic harmonies, though it's also useful in building oddly-charming augmented triads.
55	415.0943396		?	14/11	33/26	108/85	RM3, kUd4	Wide Major Third, Lesser Super-Diminished Fourth	F#/, Gd<↓, Gb↑/	This interval is of particular interest because it is the approximation of the Neo-Gothic Major Third and is used accordingly; what's more, this interval has additional applications in Paradiatonic harmony, particularly when such harmony is found in what is otherwise the traditional Diatonic context of a Major key.
56	422.6415094		?	?	143/112	51/40	rKM3, RkUd4	Narrow Supermajor Third, Greater Super-Diminished Fourth	F#↑\, Gd>↓	This interval is useful for evoking the feel of 31edo due to approximating that system's Supermajor Third, and is even better for evoking the feel of 17edo due to approximating that system's Major Third.
57	430.1886792	32/25	?	?	?	?	KM3, rUd4, KKd4	Lesser Supermajor Third, Diptolemaic Diminished Fourth	F#↑, Gd<\, Gb↑↑	This interval is easily very useful due to it being a consistent approximation of the Classic Diminished Fourth; despite its dissonance- or perhaps even because of said dissonance- this interval is even useful when it comes to building chords.
58	437.7358491		9/7	165/128	?	?	SM3, kUM3, rm4, Ud4	Greater Supermajor Third, Ultra-Diminished Fourth	Gd<, F#↑/	This interval is the approximation of the Septimal Supermajor Third and is directly on this system's Superpyth scale as well; those who are not already familiar with septimal harmony will find this interval useful in forming not only strident-sounding triads framed by the Perfect Fifth, but also different types of augmented and superaugmented triad.
59	445.2830189		?	128/99	?	22/17	m4, RkUM3	Paraminor Fourth, Wide Supermajor Third	Gd>, Ft#>↓	Although this interval is not found on the Paradiatonic scale, it is nevertheless important for usage in Parachromatic gestures and in various types of harmony based on such gestures; it is the namesake of 24edo's own Paraminor Fourth interval, and, just like that interval, it tends to want to be followed up by either the Unison, the Perfect Fourth, or, its Paramajor counterpart- the latter having additional follow-up options.
60	452.8301887	?	?	?	13/10	?
61	460.3773585		?	176/135	?	?
62	467.9245283		21/16	55/42, 72/55	?	17/13
63	475.4716981	320/243, 675/512	?	?	?	?
64	483.0188679		?	33/25	?	45/34
65	490.5660377		?	?	?	85/64
66	498.1132075	4/3	?	?	?	?
67	505.6603774		75/56	162/121	?	?
68	513.2075472		?	121/90	?	?
69	520.7547170	27/20	?	?	104/77	?
70	528.3018868		?	110/81	?	?
71	535.8490566		?	15/11	?	?
72	543.3962264	?	?	?	160/117	256/187
73	550.9433962		?	11/8	?	?
74	558.4905660		112/81	?	?	?
75	566.0377358	25/18	?	?	18/13	?
76	573.5849057		?	?	?	357/256
77	581.1320755		7/5	?	?	?
78	588.6792458	1024/729, 45/32	?	?	?	?
79	596.2264151		?	?	?	24/17
80	603.7735849		?	?	?	17/12
81	611.3207547	729/512, 64/45	?	?	?	?
82	618.8679245		10/7	?	?	?
83	626.4150943		?	?	?	512/357
84	633.9622642	36/25	?	?	13/9	?
85	641.5094340		81/56	?	?	?
86	649.0566038		?	16/11	?	?
87	656.6037736	?	?	?	117/80	187/128
88	664.1509434		?	22/15	?	?
89	671.6981132		?	81/55	?	?
90	679.2452830	40/27	?	?	77/52	?
91	686.7924528		?	180/121	?	?
92	694.3396226		112/75	121/81	?	?
93	701.8867925	3/2	?	?	?	?
94	709.4339622		?	?	?	128/85
95	716.9811321		?	50/33	?	68/45
96	724.5283019	243/160, 1024/675	?	?	?	?
97	732.0754717		32/21	84/55, 55/36	?	26/17
98	739.6226415		?	135/88	?	?
99	747.1698113	?	?	?	20/13	?
100	754.7169811		?	99/64	?	17/11
101	762.2641509		14/9	256/165	?	?
102	769.8113208	25/16	?	?	?	?
103	777.3584906		?	?	224/143	80/51
104	784.9056604		?	11/7	52/33	85/54
105	792.4528302	128/81	?	?	?	?
106	800		100/63	192/121	?	?
107	807.5471698		?	?	?	51/32
108	815.0943396	8/5	?	77/48	?	?
109	822.6415094		45/28	825/512	?	?
110	830.1886792		?	?	?	55/34
111	837.7358491	?	?	?	13/8	34/21
112	845.2830189		?	44/27	?	?
113	852.8301887		?	18/11	?	?
114	860.3773585	?	?	?	64/39	28/17
115	867.9245283		?	33/20, 200/121	?	119/72, 272/165
116	875.4716981		224/135	?	?	425/256
117	883.0188679	5/3	?	128/77	?	?
118	890.5660377		?	?	429/256	256/153
119	898.1132075		42/25	121/72	?	?
120	905.6603774	27/16	?	?	?	?
121	913.2075472		?	56/33	22/13	144/85
122	920.7547170		?	?	?	17/10
123	928.3018868	128/75	?	?	?	?
124	935.8490566		12/7	55/32	?	?
125	943.3962264		?	512/297	?	?
126	950.9433962	?	?	?	26/15	?
127	958.4905660		?	891/512	?	?
128	966.0377358		7/4	96/55	?	?
129	973.5849057	225/128	?	135/77	?	?
130	981.1320755		?	?	?	30/17
131	988.6792458		?	?	39/22	512/289
132	996.2264151	16/9	?	?	?	?
133	1003.7735849		25/14	216/121	?	?
134	1011.3207547		?	?	256/143	459/256
135	1018.8679245	9/5	?	231/128	?	?
136	1026.4150943		1024/567	440/243	?	768/425
137	1033.9622642		?	20/11	?	?
138	1041.5094340	?	?	?	117/64	1024/561, 935/512
139	1049.0566038		?	11/6	?	?
140	1056.6037736		?	81/44	?	?
141	1064.1509434	50/27	?	?	24/13	?
142	1071.6981132		?	?	13/7	119/64
143	1079.2452830		28/15	512/275	?	?
144	1086.7924528	15/8	?	?	?	?
145	1094.3396226		?	?	?	32/17
146	1101.8867925		?	121/64	104/55	17/9
147	1109.4339622	243/128, 256/135	?	?	?	?
148	1116.9811321		40/21	21/11	?	?
149	1124.5283019		?	?	?	153/80
150	1132.0754717	48/25	?	?	25/13, 52/27	?
151	1139.6226415		27/14	?	?	85/44
152	1147.1698113		?	64/33	?	33/17
153	1154.7169811	?	?	?	39/20	187/96
154	1162.2641509		?	88/45	?	100/51
155	1169.8113208		63/32	55/28, 108/55	?	51/26
156	1177.3584906	160/81	?	?	77/39	168/85
157	1184.9056604		?	240/121, 99/50	143/72	119/60
158	1192.4528302		448/225	484/243	195/98, 700/351	255/128
159	1200	2/1					P8	Perfect Octave	D	This interval... Is the reduplication of a chord's root Is the reduplication of the Tonic Is one of four perfect consonances in this system