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, and thus... Can function as both a type of parachroma and a type of diesis in this system Approximates the telepathma, and thus... Can be considered a type of parachroma 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 quartertone motion Has the potential to move away from the Tonic towards either a Contralead or Supertonic harmony through a type of Diatonic or Paradiatonic semitone 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 or Diptolemaic 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 of 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... 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 It serves as a Diatonic semitone in both Western-Classical-based functional harmony and Neo-Medieval harmony in general, and thus... Has the potential to move directly back down to the Tonic as a Contralead, 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, and likewise separates Ptolemaic Major intervals from Pythagorean Minor intervals Is one half of this system's approximation of the Classic Major Second as a consequence of the schisma being tempered out in this system 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 as such... 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 It readily serves as the traditional leading tone in 5-limit Western-Classical-based functional harmony and thus... Has the potential to move directly back down to the Tonic as a Contralead, though in non-meantone systems like this one, such a gesture using this interval has a slightly more lax and natural feel Has close affinities with the Serviant due to being located at roughly a Ptolemaic Major Third away from it 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#/	This interval... Approximates the Septimal Major Semitone, and thus... It functions as both a type of chromatic semitone and a type of Diatonic semitone in septimal harmony Is one third of this system's approximation of the Octave-Reduced Thirteenth Subharmonic Is the closest approximation of 10edo's Minor Second found 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
17	128.3018868		?	?	14/13	128/119	kN2, rKA1	Lesser Supraminor Second, Tendoretromean Augmented Prime	Ed>↓, D#↑\	This interval... Approximates the Tridecimal Supraminor Second and a similar 11-limit interval that acts as the Supraminor counterpart to the Undecimal Submajor Second, and thus... It can be thought of as something along the lines of a "wide semitone" in voice-leading It demonstrates trienthird functionality- namely in relation to this system's approximation of the Classic Major Third- due to the combination of commas tempered out in this system Approximates a complex yet uprooted 17-limit interval relative to the Tonic and can be used... As an unexpected option for a Diatonic-type semitone in Western-Classical-based harmony Is the closest approximation of 19edo's Minor Second found in this system, and thus... Can be used in melodic and harmonic gestures reminiscent of those found in that system, since such moves on their own don't work the exact same way in this system
18	135.8490566	27/25	?	?	13/12	?	KKm2, rn2, KA1	Greater Supraminor Second, Diptolemaic Limma, Retroptolemaic Augmented Prime	Ed<\, Eb↑↑, D#↑	This interval... Approximates the Large Limma, and thus... It frequently acts as a Diatonic semitone in Western-Classical-based harmony Approximates the Tridecimal Neutral Second, and thus... It demonstrates two-third-tone functionality- especially in relation to this system's approximation of the Pythagorean Major Second- due to the combination of commas tempered out in this system It demonstrates trienthird functionality- namely in relation to this system's approximation of the Pythagorean Major Third- due to the combination of commas tempered out in this system Is found in 53edo as that system's Supraminor Second, and can thus be used to create identical-sounding melodic and harmonic gestures in this system
19	143.3962264		?	88/81	?	?	n2, SA11	Artoneutral Second, Lesser Super-Augmented Prime	Ed<, Dt#<↓	This interval... Approximates the Alpharabian Artoneutral Second or 2nd Undecimal Neutral Second, and as such... It can be thought of as something along the lines of a "wide semitone" in voice-leading 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 It serves as the smaller and more dissonant of two Neutral Seconds in Western-Classical-based Paradiatonic functional harmony, and thus... Has the potential to move to the Pythagorean Minor Third through a Paradiatonic "narrow whole tone" motion Has the potential to move to the Lesser Subminor Second through a type of Chromatic semitone motion Is one half of this system's approximation of the Neo-Gothic Minor Third Is one third of this system's approximation of the Classic Diminished Fourth Is the closest approximation of 17edo's Neutral Second found in this system, and thus... Is capable of being used for similar modulatory moves, albeit with caveats, albeit with caveats, since such moves on their own don't work the exact same way in this system
20	150.9433962		?	12/11	?	?	N2, RkUA1	Tendoneutral Second, Greater Super-Augmented Prime	Ed>, Dt#>↓	This interval... Approximates the Alpharabian Tendoneutral Second, which is the traditional, low complexity Undecimal Neutral Second, and as such... It can be thought of as something along the lines of a "narrow whole tone" in 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 It serves as the larger and more consonant of two Neutral Seconds in Western-Classical-based Paradiatonic functional harmony, and thus... Has the potential to move to the Pythagorean Minor Third through a Paradiatonic "wide semitone" motion Has the potential to move to the Lesser Subminor Second through a type of Chromatic semitone motion Is one fifth of this system's approximation of the Just Paramajor Fifth Is the closest approximation of 24edo's own Neutral Second 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 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
21	158.4905660	?	?	?	128/117	561/512, 1024/935	kkM2, RN2, rUA1	Lesser Submajor Second, Diretroptolemaic Augmented Prime	Ed>/, E↓↓, Dt#>↓/, D#↑↑	This interval... Is one half of this system's approximation of the Classic Minor Third Is the closest approximation of 31edo's own Middle 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 Is found in 53edo as that system's Submajor Second, and can thus be used to create identical-sounding melodic and harmonic gestures in this system
22	166.0377358		?	11/10	?	?	Kn2, UA1	Greater Submajor Second, Ultra-Augmented Prime	Ed<↑, Dt#<, Fb↓/	This interval... Approximates the Undecimal Submajor Second and a similar 13-limit interval that acts as the Submajor counterpart to the Tridecimal Supraminor Second, and thus... It can be thought of as something along the lines of a "narrow whole tone" in voice-leading Approximates a complex 11-limit Parachromatic interval formed by stacking an Al-Farabi Quartertone on top of an Apotome, and thus... It can be thought of as a type of sesquichroma when acting in this capacity Is one third of this system's approximation of the Perfect Fourth Is the closest approximation of 22edo's Lesser Major 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 such moves on their own don't work the exact same way in this system
23	173.5849057		567/512	243/220	?	425/384	rkM2, KN2	Narrow Major Second	Ed>↑, E↓\, Dt#>, Fb\	This interval... Is one half of the approximation of the traditional, low complexity Undecimal Neutral Third in this system Is one third of the approximation of the Classic Acute Fourth in this system Is the closest approximation of the 7edo 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
24	181.1320755	10/9	?	256/231	?	?	kM2	Ptolemaic Major Second	E↓, Fb	This interval... Approximates the Classic Major Second or Ptolemaic Major Second, and as such... 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 crowding, and thus requires resolution Is 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" It readily serves as a Diatonic whole tone in Western-Classical-based functional harmony, since... It has close affinities with the Serviant due to being located at roughly a Ptolemaic Minor Third away from it Is one half of this system's approximation of the Octave-Reduced Thirteenth Subharmonic Is one fifth of this system's approximation of the Pythagorean Major Sixth Is the closest approximation of 13edo's own Major 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
25	188.6792458		?	?	143/128	512/459	RkM2	Artomean Major Second	E↓/, Fb/	This interval... Approximates the Grossmic Whole Tone, and thus... Is useful for modulating to keys that are not found on the same circle of fifths Is one third of this system's approximation of the Classic Augmented Fourth Is the closest approximation of 19edo's Major Second found in this system, and thus... Can be used in melodic and harmonic gestures reminiscent of those found in that system, since such moves on their own don't work the exact same way in this system
26	196.2264151		28/25	121/108	?	?	rM2	Tendomean Major Second	E\, Fb↑\	This interval... Approximates the Middle Major Second Is one of two intervals that serve as the closest approximation of the 12edo Major 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 Is the closest approximation of 31edo's Major Second found in this system, and thus... Can be used in melodic and harmonic gestures reminiscent of those found in that system, since such moves on their own don't work the exact same way in this system
27	203.7735849	9/8	?	?	?	?	M2	Pythagorean Major Second	E, Fb↑	This interval... Approximates the Pythagorean Major Second, and as such... 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 It readily serves as a Diatonic whole tone in both Western-Classical-based functional harmony and Neo-Medieval harmony in general, since... It functions as a Double Dominant due to being the result of stacking two Perfect Fifths and octave-reducing 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, and thus... It sees usage in Paradiatonic and Parachromatic harmonies in addition to the more obvious Diatonic-related uses Is one fourth of this system's approximation of the Classic Minor Sixth as a consequence of the schisma being tempered out in this system Is reachable through stacking three of this system's approximation of the Septimal Subfourth and octave-reducing Is one of two intervals that serve as the closest approximation of the 12edo Major 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
28	211.3207547		?	?	44/39	289/256	RM2	Wide Major Second	E/, Fd<↓	This interval... Approximates the Tridecimal Major Second, and thus... It is very likely to be treated as a type of whole tone when working in Neo-Medieval harmony Is reachable through stacking two of this system's approximation of the Octave-Reduced Seventeenth Harmonic Is the closest approximation of 17edo's Major 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
29	218.8679245		?	?	?	17/15	rKM2	Narrow Supermajor Second	E↑\, Fd>↓	This interval... Approximates the Septendecimal Whole Tone, and thus... Can be used as an unexpected option for a diminished third in Western-Classical-based harmony Approximates a complex 11-limit interval formed by stacking an Parapotome on top of a Classic Minor Second, and thus... It can be thought of as a type of whole tone when acting in this capacity Is one half of this system's approximation of the Septimal Supermajor Third Is reachable through stacking two of this system's approximation of the Septendecimal Fifth and octave-reducing Is the closest approximation of 22edo's Greater Major 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
30	226.4150943	256/225	?	154/135	?	?	KM2	Lesser Supermajor Second	E↑, Fd<\, Fb↑↑, Dx	This interval... Approximates the Neapolitan Diminished Third, and thus... It readily appears in approximations of 5-limit Neapolitan scales as the interval formed from stacking two Ptolemaic Minor Seconds Approximates a complex 5-limit interval formed by stacking a syntonic comma on top of a Pythagorean Major Second, and thus... It can be thought of as a type of second when acting in this capacity Is likely the smallest interval in this system that can be used in chords without causing crowding Is the closest approximation of 16edo's Supermajor 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
31	233.9622642		8/7	55/48	?	?	SM2, kUM2	Greater Supermajor Second, Narrow Inframinor Third	Fd<, Et<↓, E↑/	This interval... Approximates the Septimal Supermajor Second or Octave-Reduced Seventh Subharmonic, and as such... Can readily occur as the distance between two notes in a single chord in Western-Classical-based polypedal harmony, as per this system's version of the Dinner Party Rules, and it should be noted that... Since three of these add up to this system's approximation of the Perfect Fifth, there are multiple ways it can be used in chords to great effect This causes ambisonance, so chords that utilize it are prone to decomposition It readily serves as one of the key Paradiatonic intervals in Western-Classical-based Paradiatonic functional harmony, since... It functions as a Contravaricant due to its semiambitonal properties relative to the Diatonic scale Is one half of this system's approximation of the Septimal Subfourth Is the closest approximation of 31edo's Supermajor 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
32	241.5094340		?	1024/891	?	?	um3, RkUM2	Inframinor Third, Wide Supermajor Second	Fd>, Et>↓	This interval... Approximates a complex 11-limit Paradiatonic interval that functions as a syntactic third that sounds more like a second, and as such... It has the potential to move back down to the Supertonic through a diatonic or paradiatonic motion It has the potential to move up towards a Mediant harmony through a parachromatic motion Is one fourth of this system's approximation of the Octave-Reduced Seventh Harmonic Is the closest approximation of 10edo's Major Second slash Minor Third found 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
33	249.0566038	?	?	?	15/13	?	kkm3, KKM2, Rum3, rUM2	Wide Inframinor Third, Narrow Ultramajor Second, Semifourth	Fd>/, Et<\, F↓↓, E↑↑	This interval... Approximates the Tridecimal Semifourth, and thus... It can be used both in triads framed by a Perfect Fourth and in triads Framed by a Perfect Fifth Is one half of a Perfect Fourth in this system Is the closest approximation of 19edo's Semifourth found in this system, and thus... Can be used in melodic and harmonic gestures reminiscent of those found in that system, since such moves on their own don't work the exact same way in this system Is the closest approximation of 24edo's Semifourth, 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
34	256.6037736		?	297/256	?	?	UM2, rKum3	Ultramajor Second, Narrow Subminor Third	Et<, Fd<↑	This interval... Approximates a complex 11-limit Paradiatonic interval that functions as a syntactic second that sounds more like a third, and as such... It has the potential to move back down to the Supertonic through a parachromatic motion It has the potential to move up towards a Mediant harmony through a diatonic or paradiatonic motion Is one third of this system's approximation of the Classic Augmented Fifth Is reachable through stacking two of this system's approximation of the Tridecimal Supraminor Second
35	264.1509434		7/6	64/55	?	?	sm3, Kum3	Lesser Subminor Third, Wide Ultramajor Second	Et>, Fd>↑, F↓\	This interval... Approximates the Septimal Subminor Third, and as such... Can readily occur as the distance between two notes in a single chord in Western-Classical-based polypedal harmony, as per this system's version of the Dinner Party Rules, and it should be noted that... This causes ambisonance, so chords that utilize it are prone to decomposition It readily serves as one of the key Paradiatonic intervals in Western-Classical-based Paradiatonic functional harmony, since... It functions as a Contravaricant due to its semiambitonal properties relative to the Diatonic scale It is useful in forming not only strident-sounding triads framed by the Perfect Fifth, but also other, ambisonant triads framed by the Perfect Fourth Is one third of this system's approximation of the Pythagorean Minor Sixth Is very useful for essentially tempered chords such as Keenanismic chords
36	271.6981132	75/64	?	?	?	?	km3	Greater Subminor Third	F↓, Et>/, E#↓↓, Gbb	This interval... Approximates the classic augmented second, and as such... It most frequently appears in approximations of 5-limit Harmonic scales as the interval between the Ptolemaic Minor Sixth and the Ptolemaic Major Seventh Approximates a complex 5-limit interval formed by subtracting a syntonic comma from a Pythagorean Minor Third, and thus... It can be thought of as a type of third when acting in this capacity Is the closest approximation of 22edo's Lesser Minor Third 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 31edo's Subminor Third 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
37	279.2452830		?	?	?	20/17	Rkm3	Wide Subminor Third	F↓/, Et<↑	This interval... Approximates the Septendecimal Minor Third Is the closest approximation of 13edo's Minor Third found 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 Is the closest approximation of 17edo's Minor Third 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
38	286.7924528		?	33/28	13/11	85/72	rm3	Narrow Minor Third	F\, Et>↑	This interval... Approximates the Neo-Gothic Minor Third, and thus... Can be used in Western-Classical-based harmony and Neo-Medieval harmony very easily Has additional applications in Paradiatonic harmony, particularly... When it is found in what is otherwise the traditional Diatonic context of a Minor key Is one third of this system's approximation of the Greater Tridecimal Neutral Sixth Is very useful for essentially tempered chords such as gentle chords, ainic chords and nicolic chords
39	294.3396226	32/27	?	?	?	?	m3	Pythagorean Minor Third	F	This interval... Approximates the Pythagorean Minor Third, and as such... Is one of the staples of both melodic and harmonic motion in general Readily occurs as the distance between two notes in a single chord in Western-Classical-based polypedal harmony It differs from the Ptolemaic Minor Third in that... It is very useful as an interpretation of the dissonant Minor Third from Medieval music's florid organum It can be used in creating a subtle instability in certain Diatonic harmonies Is one third of this system's approximation of the Classic Major Sixth as a consequence of the schisma being tempered out in this system Is reachable through stacking three of this system's approximation of the Axirabian Limma
40	301.8867925		25/21	144/121	?	?	Rm3	Artomean Minor Third	F/	This interval... Approximates the Quasi-Tempered Minor Third, and as such... It is the closest approximation of 12edo's Minor Third 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 Is one half of this system's approximation of the Larger Septendecimal Tritone Is reachable through stacking two of this system's approximation of the low-complexity JI neutral second
41	309.4339622		?	?	512/429	153/128	rKm3	Tendomean Minor Third	F↑\	This interval... Approximates the Septendecimal Tendomean Minor Third Is one half of this system's approximation of the Greater Septimal Tritone, and thus... Is used accordingly as part of a diminished triad Is reachable through stacking two of this system's approximation of the Just Paramajor Fifth and octave-reducing Is the closest approximation of 31edo's Minor Third 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 Can easily be used in modulatory maneuvers similar to those performed by Jacob Collier
42	316.9811321	6/5	?	77/64	?	?	Km3	Ptolemaic Minor Third	F↑, E#	This interval... Approximates the Classic Minor Third, and as such... Is one of the staples of both melodic and harmonic motion in general, and in particular... It is a staple interval in Western-Classical-based Diatonic scales Readily occurs as the distance between two notes in a single chord in Western-Classical-based polypedal harmony It differs from the Pythagorean Minor Third in that... It is one of four imperfect consonances in this system, rendering it extremely suitable for use in a Tonic triad in Western-Classical-based polypedal harmony Is reachable through stacking three of this system's approximation of the Octave-Reduced Seventeenth Harmonic Is the closest approximation of 19edo's Minor Third found in this system, and thus... Can be used in melodic and harmonic gestures reminiscent of those found in that system, since such moves on their own don't work the exact same way in this system
43	324.5283019		135/112	?	?	512/425	RKm3, kn3	Wide Minor Third	Ft<↓, F↑/, Gdb<	This interval... Approximates the Marvelous Minor Third, and as such... Is the widest interval that can reasonably be used in Western-Classical-based harmony and Neo-Medieval harmony as a type of Minor Third Is one half of this system's approximation of the Just Paraminor Fifth Is one third of this system's approximation of the Neapolitan Augmented Sixth as a consequence of the hemimage comma being tempered out in this system Is the closest approximation of 22edo's Greater Minor Third 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 such moves on their own don't work the exact same way in this system
44	332.0754717		?	40/33, 121/100	?	144/119, 165/136	kN3, ud4	Lesser Supraminor Third, Infra-Diminished Fourth	Ft>↓, Gdb>	This interval... Approximates the Undecimal Supraminor Third, and thus... It functions as the fourth complement to this system's approximation of the Undecimal Submajor Second Is one half of this system's approximation of the Undecimal Diminished Fifth Is one third of this system's approximation of the Pythagorean Minor Seventh
45	339.6226415	?	?	?	39/32	17/14	KKm3, rn3, Rud4	Greater Supraminor Third, Diretroptolemaic Diminished Fourth	Ft<\, F↑↑, Gdb<↑\, Gb↓↓	This interval... Approximates the Lesser Tridecimal Neutral Third, and thus... It serves as the fifth complement of the Octave-Reduced Thirteenth Subharmonic Approximates the Septendecimal Supraminor Third, and thus... Is very useful for essentially tempered chords such as tannic chords Is reachable through stacking three of this system's approximation of the Classic Minor Second Is the closest approximation of the 7edo Third 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 Is found in 53edo as that system's Supraminor Third, and can thus be used to create identical-sounding melodic and harmonic gestures in this system
46	347.1698113		?	11/9	?	?	n3, rKud4	Artoneutral Third, Lesser Sub-Diminished Fourth	Ft<, Gdb<↑	This interval... Approximates the Alpharabian Artoneutral Third, which is the traditional, low complexity Undecimal Neutral Third, and as such... 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 It is a staple interval of the Western-Classical based Paradiatonic scale in this system It serves as the smaller and more consonant of two Neutral Thirds in Western-Classical-based Paradiatonic functional harmony, and thus... Has the potential to move to the Perfect Fourth through a Paradiatonic "narrow whole tone" motion Has the potential to move to the Lesser Subminor Third through a type of Chromatic semitone motion Is the closest approximation of 24edo's own Neutral Third 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 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 Is the closest approximation of 31edo's own Middle Third in this system, and thus... Can be used in melodic and harmonic gestures reminiscent of those found in that system, albeit with caveats, since the rastma is not tempered out
47	354.7169811		?	27/22	?	?	N3, sd4, Kud4	Tendoneutral Third, Greater Sub-Diminished Fourth	Ft>, Gdb>↑	This interval... Approximates the Alpharabian Tendoneutral Third or 2nd Undecimal Neutral Second, and as such... 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 It serves as the larger and more dissonant of two Neutral Thirds in Western-Classical-based Paradiatonic functional harmony, and thus... Has the potential to move to the Perfect Fourth through a Paradiatonic "wide semitone" motion Has the potential to move to the Lesser Subminor Third through a type of Chromatic semitone motion Is one half of this system's approximation of the Septendecimal Fifth, which is a a possible generator for this system's Superpyth scale Is the closest approximation of 17edo's Neutral Third found in this system, and thus... Is capable of being used for similar modulatory moves, albeit with caveats, albeit with caveats, since such moves on their own don't work the exact same way in this system
48	362.2641509	?	?	?	16/13	21/17	kkM3, RN3, kd4	Lesser Submajor Third, Retroptolemaic Diminished Fourth	Ft>/, F#↓↓, Gb↓	This interval Approximates the Greater Tridecimal Neutral Third or Octave-Reduced Thirteenth Subharmonic, and as such... Is ostensibly one of the easiest 13-limit thirds to utilize in chords framed by either a Grave Fifth or an Acute Fifth Is one third of this system's approximation of the Classic Major Seventh Is the closest approximation of 10edo's Major Third found 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 Is found in 53edo as that system's Submajor Third, and can thus be used to create identical-sounding melodic and harmonic gestures in this system
49	369.8113208		?	?	26/21	68/55	Kn3, Rkd4	Greater Submajor Third, Artoretromean Diminished Fourth	Ft<↑, Gb↓/	This interval... Approximates the Tridecimal Submajor Third and a similar 11-limit interval that acts as the Submajor counterpart to the Undecimal Supraminor Third, and thus... Serves as the fourth complement to this system's approximation of the Tridecimal Supraminor Second Is one third of this system's approximation of the Pythagorean Major Seventh Is the closest approximation of 13edo's Minor Third found 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
50	377.3584906		56/45	1024/825	?	?	rkM3, KN3, rd4	Narrow Major Third, Tendoretromean Diminished Fourth	Ft>↑, F#↓\, Gb\	This interval... Approximates the Marvelous Major Third, and as such... Is the narrowest interval that can reasonably be used in Western-Classical-based harmony and Neo-Medieval harmony as a type of Major Third Is one half of this system's approximation of the Just Paramajor Fifth Is the closest approximation of 16edo's Major Third 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 19edo's Major Third found in this system, and thus... Can be used in melodic and harmonic gestures reminiscent of those found in that system, since such moves on their own don't work the exact same way in this system
51	384.9056604	5/4	?	96/77	?	?	kM3, d4	Ptolemaic Major Third, Pythagorean Diminished Fourth	Gb, F#↓	This interval... Approximates the Classic Major Third or Octave-Reduced Fifth Harmonic, and as such... Is one of the staples of both melodic and harmonic motion in general, and in particular... It is a staple interval in Western-Classical-based Diatonic scales Readily occurs as the distance between two notes in a single chord in Western-Classical-based polypedal harmony It differs from the Pythagorean Major Third in that... It is one of four imperfect consonances in this system, rendering it extremely suitable for use in a Tonic triad in Western-Classical-based polypedal harmony Approximates the Pythagorean Diminished Fourth, and as such... Serves as an enharmonic substitution for the Classic Major Third when building chords for purposes of voice-leading, in which case it counts as an unresolved interval Is the closest approximation of 22edo's Lesser Major Third 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 such moves on their own don't work the exact same way in this system Is the closest approximation of 31edo's Major Third 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
52	392.4528302		?	?	?	64/51	RkM3, Rd4	Artomean Major Third, Artomean Diminished Fourth	Gb/, F#↓/	This interval... Approximates the Septendecimal Artomean Major Third Is reachable through stacking two of this system's approximation of the Middle Major Second Can easily be used in modulatory maneuvers similar to those performed by Jacob Collier
53	400		63/50	121/96	?	34/27	rM3, rKd4	Tendomean Major Third, Tendomean Diminished Fourth	F#\, Gb↑\	This interval... Approximates the Quasi-Tempered Major Third Is none other than the familiar major third of 12edo, and thus... It is useful for evoking the feel of 12edo in various ways, such as... Creating the familiar augmented triads of 12edo Performing modulatory maneuvers based around the aforementioned triads Is very useful for essentially tempered chords such as palingenetic chords, and, oddly... It just so happens that stacking this interval with this system's approximation of the Quasi-Tempered Minor Third makes triads of just this sort in the 27-odd-limit
54	407.5471698	81/64	?	?	?	?	M3, Kd4	Pythagorean Major Third, Ptolemaic Diminished Fourth	F#, Gb↑	This interval... Approximates the Pythagorean Major Third, and as such... Is a viable option in both melodic and harmonic motion in general, and in particular... It is a useful interval in Western-Classical-based Diatonic scales Can easily occur as the distance between two notes in a single chord in Western-Classical-based polypedal harmony It differs from the Ptolemaic Minor Third in that... It is very useful as an interpretation of the dissonant Major Third from Medieval music's florid organum It can be used in creating a subtle instability in certain Diatonic harmonies Is one half of this system's approximation of the Classic Minor Sixth as a consequence of the schisma being tempered out in this system, which... Leads to this interval being useful in forming oddly charming augmented triads Moving up by seven of these with two octave-reductions is an unexpected alternative means for exploiting the frameshift comma, though this system's approximation of the Axirabian Limma remains essential to the process even in this case
55	415.0943396		?	14/11	33/26	108/85	RM3, kUd4	Wide Major Third, Lesser Super-Diminished Fourth	F#/, Gd<↓, Gb↑/	This interval... Approximates the Neo-Gothic Major Third, and thus... Can be used in Western-Classical-based harmony and Neo-Medieval harmony very easily Has additional applications in Paradiatonic harmony, particularly... When it is found in what is otherwise the traditional Diatonic context of a Major key Is one half of this system's approximation of the Tridecimal Supraminor Sixth, and thus... Can be used to make augmented triads framed by this system's closest approximation of acoustic phi Is reachable through stacking five of this system's approximation of the Septimal Minor Semitone
56	422.6415094		?	?	143/112	51/40	rKM3, RkUd4	Narrow Supermajor Third, Greater Super-Diminished Fourth	F#↑\, Gd>↓	This interval... Approximates the Septendecimal Major Third Is the closest approximation of 17edo's Major Third 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 Is the closest approximation of 31edo's Supermajor Third 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
57	430.1886792	32/25	?	?	?	?	KM3, rUd4, KKd4	Lesser Supermajor Third, Diptolemaic Diminished Fourth	F#↑, Gd<\, Gb↑↑	This interval... Approximates the Classic Diminished Fourth, and thus... It is easily very useful when it comes to building chords despite- or perhaps even because of- its dissonance Approximates a complex 5-limit interval formed by adding a syntonic comma to a Pythagorean Major Third, and thus... It can be thought of as a type of third when acting in this capacity Is one half of this system's approximation of the Greater Tridecimal Neutral Sixth
58	437.7358491		9/7	165/128	?	?	SM3, kUM3, rm4, Ud4	Greater Supermajor Third, Ultra-Diminished Fourth	Gd<, F#↑/	This interval... Approximates the Septimal Supermajor Third, and as such... 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 It is useful in forming not only strident-sounding triads framed by the Perfect Fifth, but also different types of augmented and superaugmented triad Is one half of this system's approximation of the Marvelous Minor Sixth as a consequence of the hemimage comma being tempered out in this system Is the closest approximation of 22edo's Greater Major Third 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
59	445.2830189		?	128/99	?	22/17	m4, RkUM3	Paraminor Fourth, Wide Supermajor Third	Gd>, Ft#>↓	This interval... Approximates the Just Paraminor Fourth, and as such... Although it is not found on the Paradiatonic scale, it is nevertheless readily serves as one of the key Parachromatic intervals in Western-Classical-based Parachromatic functional harmony, since... It functions as a Misoserviant due to its dissonance and its properties relative to the Diatonic scale It has the potential to move back down to the Tonic harmony through a Paradiatonic motion It has the potential to move up towards a Serviant harmony through a Parachromatic quatertone-type motion It has the potential to move up towards an Intersubiant harmony through a Parachromatic semitone-type motion, with this move granting additional follow-up options Follows after the pattern of inframinor and ultramajor intervals sounding like members of the adjacent interval classes- specifically, the paraminor fourth sounds more like a third than a fourth Is the closest approximation of 19edo's Diminished Fourth 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
60	452.8301887	?	?	?	13/10	?	Rm4, KKM3, rUM3	Wide Paraminor Fourth, Narrow Ultramajor Third	Gd>/, F#↑↑, G↓↓	This interval... Approximates the Tridecimal Semisixth Is the closest approximation of 24edo's Paraminor Fourth, 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 very useful for essentially tempered chords such as island chords
61	460.3773585		?	176/135	?	?	UM3, rKm4	Ultramajor Third, Narrow Grave Fourth	Gd<↑, Ft#<	This interval... Approximates a complex 11-limit Paradiatonic interval that functions as a syntactic third that sounds more like a fourth, and as such... It has the potential to move up to the Intersubiant harmony through Paradiatonic motion It has the potential to move back down to a Mediant harmony through a type of Chromatic or Parachromatic semitone motion Is reachable through stacking four of this system's approximation of the Neo-Gothic Major Third and octave-reducing Is the closest approximation of 13edo's Minor Fourth found 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
62	467.9245283		21/16	55/42, 72/55	?	17/13	s4, Km4	Lesser Grave Fourth, Wide Ultramajor Third	Gd>↑, G↓\	This Interval... Approximates the Septimal Subfourth, and thus... Is really useful for forming suspensions on account of its dissonance Is one half of this system's approximation of the Septimal Supermajor Sixth Is the closest approximation of 31edo's own Subfourth 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
63	475.4716981	320/243, 675/512	?	?	?	?	k4	Greater Grave Fourth	G↓	This interval... Approximates a complex 5-limit interval formed by subtracting a syntonic comma from a Perfect Fourth Is one half of this system's approximation of the Tridecimal Semitwelfth Is reachable through stacking nine of this system's approximation of the Al-Farabi Quartertone
64	483.0188679		?	33/25	?	45/34	Rk4	Wide Grave Fourth	G↓/	This interval... Is one half of this system's approximation of the Octave-Reduced Seventh Harmonic Is the closest approximation of 10edo's Perfect Fourth found 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 Is one of two intervals that can generate a Diatonic MOS with a more extreme hardness than that seen in Ultrapyth temperament
65	490.5660377		?	?	?	85/64	r4	Narrow Fourth	G\	This interval... Approximates the Septendecimal Fifth, and thus... Is one of two intervals that can generate a Diatonic MOS with a sound and feel akin to that seen in Superpyth temperament Is the closest approximation of 17edo's Perfect Fourth 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 Is the closest approximation of 22edo's Perfect Fourth in this system, and thus... Can be used in both Superpyth-based and Faux-Classical-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
66	498.1132075	4/3	?	?	?	?	P4	Perfect Fourth	G	This interval... Approximates the Perfect Fourth or Octave-Reduced Third Subharmonic, and as such... Is one of the staples of both melodic and harmonic motion in general, and in particular... It is a staple interval in Western-Classical-based Diatonic scales in this system, as... It is one of two intervals that can generate this system's approximation of the Pythagorean Diatonic MOS Can easily occur as the distance between two notes in a single chord in Western-Classical-based polypedal harmony Is the basic representation of the Serviant, and thus... It is the basic interval for framing a standard tetrachord in this system Is one of four perfect consonances in this system Is the closest approximation of the 12edo Perfect Fourth 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 Inherits a sizable portion of its functionality from its 53edo counterpart, including... A chain of 53 Perfect Fourths closing at the octave due to Mercator's comma being tempered out Virtually all of its functionality in the realm of Western-Classical-based Diatonic scales and Diatonic functional harmony New elements to its functionality include... New approaches enabled by this system supporting temperaments such as sextilififths A sizable chunk of its functionality in the realm of Western-Classical-based Paradiatonic functional harmony
67	505.6603774		75/56	162/121	?	?	R4	Wide Fourth	G/	This interval... Approximates the Marvelous Fourth, and thus... Is one of two intervals that can generate a Diatonic MOS with a sound and feel akin to that seen in Flattone temperament Is the closest approximation of 19edo's Perfect Fourth 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 Is the closest approximation of 31edo's Perfect Fourth 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
68	513.2075472		?	121/90	?	?	rK4	Narrow Acute Fourth	G↑\	This interval... Approximates a complex 11-limit interval, which, in this system... Is one of two intervals that can generate a Diatonic MOS with a softness so extreme as to be quasi-equalized Is reachable through stacking four of this system's approximation of the Tridecimal Supraminor Second Is the closest approximation of the 7edo Fourth 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
69	520.7547170	27/20	?	?	104/77	?	K4	Lesser Acute Fourth	G↑	This interval... Approximates the Classic Acute Fourth, and as such... Is one of two xenharmonic intervals that appear in the otherwise traditional settings of Western-Classical-based Diatonic scales Specifically, it is found between the Major Third and Major Sixth in the Lydian Mode of the familiar Ptolemaic Sequence, and is ideally in the exact same position for both Ionian and Mixolydian modes, though this technically results in there being Diatonic scales of different varieties- namely the Bilawal and Myxian scale types... It is useful for overtly making the VImin chord of these modes do what a traditional deceptive cadence normally does Is one of two intervals that can generate an Antidiatonic MOS with a sound and feel that blends in easily with traditional Western-Classical-based Diatonic harmonies Is reachable through stacking twelve of this system's approximation of the 2nd Undecimal Neutral Second and octave-reducing Is very useful for essentially tempered chords such as marveltwin chords and ibnsinmic chords in the 27-odd-limit
70	528.3018868		256/189	110/81	?	?	S4, kM4	Greater Acute Fourth	Gt<↓, G↑/, Adb<	This interval... Is reachable through stacking two of this system's approximation of the Septimal Subminor Third Is reachable through stacking five of this system's approximation of the Large Septendecimal Semitone Is the closest approximation of 16edo's Major Fourth 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
71	535.8490566		?	15/11	?	?	RkM4, ud5	Wide Acute Fourth, Infra-Diminished Fifth	Gt>↓, Adb>	This interval... Approximates the Undecimal Grave Infra-Augmented Fourth, and thus... Is extremely useful as an imperfect dissonance in Western-Classical-based Paradiatonic functional harmony Has interesting functions in undecimal harmony in which it can act as both an acute fourth and an infra-augmented fourth Approximates a complex 11-limit Parachromatic interval formed by taking both an Apotome and an Al-Farabi Quartertone away from a Perfect Fifth, and thus... It can be thought of as a type of sesquiflat-fifth when acting in this capacity Is one half of this system's approximation of the Tridecimal Submajor Seventh
72	543.3962264	?	?	?	160/117	256/187	rM4, Rud5	Narrow Paramajor Fourth, Diretroptolemaic Diminished Fifth	Gt<\, G↑↑, Ab↓↓	This interval... Is reachable through stacking three of this system's approximation of the Classic Major Second....... Is the closest approximation of 22edo's Diminished Fifth 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 31edo's Superfourth 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
73	550.9433962		?	11/8	?	?	M4, rKud5	Paramajor Fourth, Lesser Sub-Diminished Fifth	Gt<, Adb<↑	This interval... Approximates the Just Paramajor Fourth, and as such... Is one of the key intervals on the Paradiatonic scale, and one of the key Paradiatonic intervals in Western-Classical-based Parachromatic functional harmony, since... It functions as an Intersubiant due to its ambisonance and its properties relative to the Diatonic scale It has the potential to move up towards to the Dominant harmony through a Paradiatonic motion It has the potential to move back down to a Serviant harmony through a Parachromatic quatertone-type motion It has the potential to move up towards an Interregnant harmony through a Paradiatonic semitone-type motion, with this move granting additional follow-up options Is reachable through stacking eight of this system's approximation of the Septendecimal Whole Tone and octave-reducing Is the closest approximation of 13edo's Major Fourth found 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 Is the closest approximation of 24edo's own Paramajor Fourth found 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 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
74	558.4905660		112/81	?	?	?	RM4, uA4, Kud5	Infra-Augmented Fourth, Greater Sub-Diminished Fifth	Gt>, Adb>↑	This interval... Approximates a complex 7-limit interval formed by stacking a Septimal Subminor Second on top of a Perfect Fourth, and thus... It can be thought of as a type of fifth when acting in this capacity Is one half of this system's approximation of the Undecimal Major Seventh Is reachable through stacking two of this system's approximation of the Septendecimal Minor Third
75	566.0377358	25/18	?	?	18/13	?	kkA4, RuA4, kd5	Diptolemaic Augmented Fourth, Retroptolemaic Diminished Fifth	G#↓↓, Ab↓	This interval... Approximates the Classic Augmented Fourth, and thus... It frequently acts as an Augmented Fourth in Western-Classical-based harmony Approximates the Tridecimal Augmented Fourth, and thus... It acts as an Augmented Fourth in Western-Classical-based Paradiatonic harmony Is the closest approximation of 17edo's Diminished Fifth 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 Is the closest approximation of 19edo's Augmented Fourth 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
76	573.5849057		?	?	?	357/256	rKuA4, Rkd5	Lesser Sub-Augmented Fourth, Artoretromean Diminished Fifth	Gt<↑, Ab↓/	This interval... Approximates a complex 11-limit interval formed by stacking a Syntonic Comma on top of a Paramajor Fourth, and thus... It can be thought of as a type of fourth when acting in this capacity Is one half of this system's approximation of the Undecimal Infraoctave Is reachable through stacking two of this system's approximation of the Neo-Gothic Minor Third
77	581.1320755		7/5	?	?	?	KuA4, rd5	Greater Sub-Augmented Fourth, Tendoretromean Diminished Fifth	Gt>↑, Ab\	This interval... Approximates the Lesser Septimal Tritone and thus... It occurs frequently in septimal harmony, especially in harmonic seventh chords Is reachable through stacking eleven of this system's approximation of the Al-Farabi Quartertone Is the closest approximation of 31edo's Augmented Fourth 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
78	588.6792458	1024/729, 45/32	?	?	?	?	kA4, d5	Ptolemaic Augmented Fourth, Pythagorean Diminished Fifth	Ab, G#↓	This interval... Approximates the Smaller Diatonic Tritone, and as such... Is one of only two intervals in the tritonic region that is allowed to be built directly on top of the Tonic in Western-Classical-based harmony It functions as an Augmented Fourth in Western-Classical-based functional harmony by default, and is the signature interval of 5-limit Lydian Mode Approximates the Pythagorean Narrow Tritone, and as such... It functions as a Diminished Fifth when acting in this capacity, though this usually only occurs due to voice-leading or the stacking of Pythagorean intervals Is reachable through stacking three of this system's approximation of the Middle Major Second
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		189/128	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	?	27/17
107	807.5471698		?	?	?	51/32
108	815.0943396	8/5	?	77/48	?	?
109	822.6415094		45/28	825/512	?	?
110	830.1886792		?	?	21/13	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