159edo/Interval names and harmonies

< 159edo

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. Furthermore, just as with 24edo can be thought of as essentially having two fields of 12edo separated by a quartertone, 159edo can be thought of as having three fields of 53edo, each separated from the others by a third of a 53edo step on either side. This even lends to 159edo having its own variation on the Dinner Party Rules- represented here by the Harmonic Compatibility Rating and Melodic Compatibility Rating columns where 5 is a full-blown friend relative to the root and −5 if a full-blown enemy relative to the root. Note that the Harmonic Compatibility and Melodic Compatibility ratings are based on octave-equivalence, and many of the ratings are speculative at this point, adjustments are to be expected in the future.

Interval chart

Table of 159edo intervals

Step	Cents	Interval names			Compatibility rating		Notes
					Harmonic	Melodic
0	0	P1	Perfect Unison	D	5	5	This interval… Is the perfect unison, and thus… 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 Is the only interval shared by all tuning systems
1	7.5471698	R1	Wide Prime	D/	0	0	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 retrodiesis 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	rK1	Narrow Superprime	D↑\	−5	−5	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	K1	Lesser Superprime	D↑	−5	−2	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 retrodiesis 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	S1, kU1	Greater Superprime, Narrow Inframinor Second	Edb<, Dt<↓	−5	2	This interval… Approximates the Archytas 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	um2, RkU1	Inframinor Second, Wide Superprime	Edb>, Dt>↓	−5	5	This interval… Approximates the Undecimal Fifth-Tone Approximates the Alpharabian Inframinor Second, which 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	kkm2, Rum2, rU1	Wide Inframinor Second, Narrow Ultraprime	Eb↓↓, Dt<\	−5	5	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	U1, rKum2	Ultraprime, Narrow Subminor Second	Dt<, Edb<↑	−5	5	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	sm2, Kum2, uA1	Lesser Subminor Second, Wide Ultraprime, Infra-Augmented Prime	Dt>, Eb↓\	−4	5	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 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 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	km2, RuA1, kkA1	Greater Subminor Second, Diptolemaic Augmented Prime	Eb↓, D#↓↓	−4	5	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	Rkm2, rKuA1	Wide Subminor Second, Lesser Sub-Augmented Prime	Eb↓/, Dt<↑	−4	5	This interval… Approximates multiple complex 17-limit intervals relative to the Tonic and can be used… 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	rm2, KuA1	Narrow Minor Second, Greater Sub-Augmented Prime	Eb\, Dt>↑	−3	5	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	m2, kA1	Pythagorean Minor Second, Ptolemaic Augmented Prime	Eb, D#↓	−3	5	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	Rm2, RkA1	Artomean Minor Second, Artomean Augmented Prime	Eb/, D#↓/	−3	5	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	rKm2, rA1	Tendomean Minor Second, Tendomean Augmented Prime	D#\, Eb↑\	−2	5	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	Km2, A1	Ptolemaic Minor Second, Pythagorean Augmented Prime	D#, Eb↑	−2	5	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	RKm2, kn2, RA1	Wide Minor Second, Artoretromean Augmented Prime	Ed<↓, Eb↑/, D#/	−2	5	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	kN2, rKA1	Lesser Supraminor Second, Tendoretromean Augmented Prime	Ed>↓, D#↑\	−3	4	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	KKm2, rn2, KA1	Greater Supraminor Second, Diptolemaic Limma, Retroptolemaic Augmented Prime	Ed<\, Eb↑↑, D#↑	−4	3	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	n2, SA1	Artoneutral Second, Lesser Super-Augmented Prime	Ed<, Dt#<↓	−4	2	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	N2, RkUA1	Tendoneutral Second, Greater Super-Augmented Prime	Ed>, Dt#>↓	−4	2	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	kkM2, RN2, rUA1	Lesser Submajor Second, Retrodiptolemaic Augmented Prime	Ed>/, E↓↓, Dt#>↓/, D#↑↑	−4	3	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	Kn2, UA1	Greater Submajor Second, Ultra-Augmented Prime	Ed<↑, Dt#<, Fb↓/	−3	4	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	rkM2, KN2	Narrow Major Second	Ed>↑, E↓\, Dt#>, Fb\	−3	5	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	kM2	Ptolemaic Major Second	E↓, Fb	−2	5	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	RkM2	Artomean Major Second	E↓/, Fb/	−2	5	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	rM2	Tendomean Major Second	E\, Fb↑\	−2	5	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	M2	Pythagorean Major Second	E, Fb↑	−1	5	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	RM2	Wide Major Second	E/, Fd<↓	−1	5	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	rKM2	Narrow Supermajor Second	E↑\, Fd>↓	−1	5	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 a 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	KM2	Lesser Supermajor Second	E↑, Fd<\, Fb↑↑, Dx	−1	5	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 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	SM2, kUM2	Greater Supermajor Second, Narrow Inframinor Third	Fd<, Et<↓, E↑/	0	5	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	um3, RkUM2	Inframinor Third, Wide Supermajor Second	Fd>, Et>↓	0	4	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	kkm3, KKM2, Rum3, rUM2	Wide Inframinor Third, Narrow Ultramajor Second, Semifourth	Fd>/, Et<\, F↓↓, E↑↑	0	4	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	UM2, rKum3	Ultramajor Second, Narrow Subminor Third	Et<, Fd<↑	0	3	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	sm3, Kum3	Lesser Subminor Third, Wide Ultramajor Second	Et>, Fd>↑, F↓\	0	3	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	km3	Greater Subminor Third	F↓, Et>/, E#↓↓, Gbb	−1	3	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	Rkm3	Wide Subminor Third	F↓/, Et<↑	−1	3	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	rm3	Narrow Minor Third	F\, Et>↑	0	4	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, ainismic chords and nicolic chords
39	294.3396226	m3	Pythagorean Minor Third	F	−1	5	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	Rm3	Artomean Minor Third	F/	0	5	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	rKm3	Tendomean Minor Third	F↑\	1	5	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	Km3	Ptolemaic Minor Third	F↑, E#	3	5	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	RKm3, kn3	Wide Minor Third	Ft<↓, F↑/, Gdb<	1	4	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	kN3, ud4	Lesser Supraminor Third, Infra-Diminished Fourth	Ft>↓, Gdb>	0	3	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 Acute Ultra-Diminished Fifth Is one third of this system's approximation of the Pythagorean Minor Seventh
45	339.6226415	KKm3, rn3, Rud4	Greater Supraminor Third, Retrodiptolemaic Diminished Fourth	Ft<\, F↑↑, Gdb<↑\, Gb↓↓	−1	3	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	n3, rKud4	Artoneutral Third, Lesser Sub-Diminished Fourth	Ft<, Gdb<↑	0	3	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	N3, sd4, Kud4	Tendoneutral Third, Greater Sub-Diminished Fourth	Ft>, Gdb>↑	−1	3	This interval… Approximates the Alpharabian Tendoneutral Third or 2nd 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 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	kkM3, RN3, kd4	Lesser Submajor Third, Retroptolemaic Diminished Fourth	Ft>/, F#↓↓, Gb↓	0	3	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	Kn3, Rkd4	Greater Submajor Third, Artoretromean Diminished Fourth	Ft<↑, Gb↓/	1	3	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	rkM3, KN3, rd4	Narrow Major Third, Tendoretromean Diminished Fourth	Ft>↑, F#↓\, Gb\	2	4	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	kM3, d4	Ptolemaic Major Third, Pythagorean Diminished Fourth	Gb, F#↓	4	5	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	RkM3, Rd4	Artomean Major Third, Artomean Diminished Fourth	Gb/, F#↓/	2	5	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	rM3, rKd4	Tendomean Major Third, Tendomean Diminished Fourth	F#\, Gb↑\	0	5	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	M3, Kd4	Pythagorean Major Third, Ptolemaic Diminished Fourth	F#, Gb↑	−1	5	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 Major 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	RM3, kUd4	Wide Major Third, Lesser Super-Diminished Fourth	F#/, Gd<↓, Gb↑/	0	4	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	rKM3, RkUd4	Narrow Supermajor Third, Greater Super-Diminished Fourth	F#↑\, Gd>↓	−1	3	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	KM3, rUd4, KKd4	Lesser Supermajor Third, Diptolemaic Diminished Fourth	F#↑, Gd<\, Gb↑↑	−1	2	This interval… Approximates the Classic Diminished Fourth or Diptolemaic 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	SM3, kUM3, rm4, Ud4	Greater Supermajor Third, Ultra-Diminished Fourth	Gd<, F#↑/	0	1	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	m4, RkUM3	Paraminor Fourth, Wide Supermajor Third	Gd>, Ft#>↓	−1	0	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	Rm4, KKM3, rUM3	Wide Paraminor Fourth, Narrow Ultramajor Third	Gd>/, F#↑↑, G↓↓	−1	−1	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	UM3, rKm4	Ultramajor Third, Narrow Grave Fourth	Gd<↑, Ft#<	−2	−2	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	s4, Km4	Lesser Grave Fourth, Wide Ultramajor Third	Gd>↑, G↓\	−3	−1	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	k4	Greater Grave Fourth	G↓	−2	0	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	Rk4	Wide Grave Fourth	G↓/	−1	1	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	r4	Narrow Fourth	G\	1	3	This interval… Approximates the Septendecimal Fourth, 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	P4	Perfect Fourth	G	5	5	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 a frequent destination for motion away from the Tonic harmony either upwards or downwards 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	R4	Wide Fourth	G/	1	4	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	rK4	Narrow Acute Fourth	G↑\	−2	3	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	K4	Lesser Acute Fourth	G↑	−3	3	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	S4, kM4	Greater Acute Fourth	Gt<↓, G↑/, Adb<	−2	2	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	RkM4, ud5	Wide Acute Fourth, Infra-Diminished Fifth	Gt>↓, Adb>	−1	2	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	rM4, Rud5	Narrow Paramajor Fourth, Retrodiptolemaic Diminished Fifth	Gt<\, G↑↑, Ab↓↓	−1	3	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	M4, rKud5	Paramajor Fourth, Lesser Sub-Diminished Fifth	Gt<, Adb<↑	0	3	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, a motion which… When used as the roots of two successive chords, is known as a simul cadence 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	RM4, uA4, Kud5	Infra-Augmented Fourth, Greater Sub-Diminished Fifth	Gt>, Adb>↑	−1	2	This interval… Approximates the Septimal Subdiminished Fifth, and thus… It can be thought of as a type of fifth when acting in this capacity Approximates a complex 11-limit Parachromatic interval that results from subtracting an Al-Farabi Quartertone from a Pythagorean Augmented Fourth, and as such… 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 Major Seventh Is reachable through stacking two of this system's approximation of the Septendecimal Minor Third
75	566.0377358	kkA4, RuA4, kd5	Diptolemaic Augmented Fourth, Retroptolemaic Diminished Fifth	G#↓↓, Ab↓	−2	2	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	rKuA4, Rkd5	Lesser Sub-Augmented Fourth, Artoretromean Diminished Fifth	Gt<↑, Ab↓/	−1	2	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	KuA4, rd5	Greater Sub-Augmented Fourth, Tendoretromean Diminished Fifth	Gt>↑, Ab\	0	3	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	kA4, d5	Ptolemaic Augmented Fourth, Pythagorean Diminished Fifth	Ab, G#↓	−3	3	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	RkA4, Rd5	Artomean Augmented Fourth, Artomean Diminished Fifth	G#↓/, Ab/	−5	3	This interval… Approximates the Smaller Septendecimal Tritone, and thus… It can be thought of as a type of augmented fourth when acting in this capacity Is reachable through stacking fourteen of this system's approximation of the Tridecimal Supraminor Second and octave-reducing Is one of two intervals that serve as the closest approximation of the Semioctave found in this system, and thus… Can be used in melodic and harmonic gestures reminiscent of those found in 10edo, 12edo and 22edo, 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 ainismic chords
80	603.7735849	rKd5, rA4	Tendomean Diminished Fifth, Tendomean Augmented Fourth	Ab↑\, G#\	−5	3	This interval… Approximates the Larger Septendecimal Tritone, and thus… It can be thought of as a type of diminished fifth when acting in this capacity Is reachable through stacking four of this system's approximation of the Low-Complexity JI Neutral Second Is one of two intervals that serve as the closest approximation of the Semioctave found in this system, and thus… Can be used in melodic and harmonic gestures reminiscent of those found in 10edo, 12edo and 22edo, 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 ainismic chords
81	611.3207547	Kd5, A4	Ptolemaic Diminished Fifth, Pythagorean Augmented Fourth	Ab↑, G#	−3	3	This interval… Approximates the Larger 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 a Diminished Fifth in Western-Classical-based functional harmony by default, and is the signature interval of 5-limit Locrian Mode Approximates the Pythagorean Wide Tritone, and as such… It functions as an Augmented Fourth 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 Larger Septendecimal Tritone and octave-reducing
82	618.8679245	kUd5, RA4	Lesser Super-Diminished Fifth, Artoretromean Augmented Fourth	Ad<↓, G#/	0	3	This interval… Approximates the Greater Septimal Tritone and thus… It occurs frequently in septimal harmony, especially in inversions of harmonic seventh chords Is reachable through stacking four of this system's approximation of the Just Paramajor Fifth and octave-reducing Is the closest approximation of 31edo'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
83	626.4150943	RkUd5, rKA4	Greater Super-Diminished Fifth, Tendoretromean Augmented Fourth	Ad>↓, G#↑\	−1	2	This interval… Approximates a complex 11-limit interval formed by subtracting a Syntonic Comma from a Paraminor Fifth, 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 Ultraoctave Is reachable through stacking eleven of this system's approximation of the Undecimal Submajor Second
84	633.9622642	KKd5, rUDd5, KA4	Diptolemaic Diminished Fifth, Retroptolemaic Augmented Fourth	Ab↑↑, G#↑	−2	2	This interval… Approximates the Classic Diminished Fifth, and thus… It frequently acts as a Diminished Fifth in Western-Classical-based harmony Approximates the Tridecimal Diminished Fifth, and thus… It acts as a Diminished Fifth in Western-Classical-based Paradiatonic harmony Is the closest approximation of 17edo'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 Is the closest approximation of 19edo'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
85	641.5094340	rm5, Ud5, kUA4	Ultra-Diminished Fifth, Lesser Super-Augmented Fourth	Ad<, Gt#<↓	−1	2	This interval… Approximates the Septimal Superaugmented Fourth, and thus… It can be thought of as a type of fourth when acting in this capacity Approximates a complex 11-limit Parachromatic interval that results from stacking an Al-Farabi Quartertone on top of a Pythagorean Diminished Fifth, and as such… It can be thought of as a type of fifth when acting in this capacity Is reachable through stacking five of this system's approximation of the Tridecimal Supraminor Second
86	649.0566038	m5, RkUA4	Paraminor Fifth, Greater Super-Augmented Fourth	Ad>, Gt#>↓	0	3	This interval… Approximates the Just Paraminor Fifth, 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 Interregnant due to its ambisonance and its properties relative to the Diatonic scale It has the potential to move back down to a Serviant harmony through a Paradiatonic motion It has the potential to move up towards to the Dominant harmony through a Parachromatic quatertone-type motion It has the potential to move back down to an Intersubiant harmony through a Paradiatonic semitone-type motion, with this move granting additional follow-up options Is reachable through stacking five of this system's approximation of the Tridecimal Submajor Third and octave-reducing Is the closest approximation of 13edo's Minor Fifth 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 Paraminor Fifth 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
87	656.6037736	Rm5, rUA4	Wide Paraminor Fifth, Retrodiptolemaic Augmented Fourth	Ad<\, G#↑, Ab↑↑	−1	3	This interval… Is reachable through stacking three of this system's approximation of the Septendecimal Whole Tone Is the closest approximation of 22edo's Augmented Fourth 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 Subfifth 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
88	664.1509434	rKm5, UA4	Narrow Grave Fifth, Ultra-Augmented Fourth	Ad<↑, Gt#<	−1	2	This interval… Approximates the Undecimal Acute Ultra-Diminished Fifth, 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 grave fifth and an ultra-augmented fourth Approximates a complex 11-limit Parachromatic interval formed by stacking both an Apotome and an Al-Farabi Quartertone on top of a Perfect Fourth, and thus… It can be thought of as a type of sesquisharp-fourth when acting in this capacity Is reachable through stacking thirteen of this system's approximation of the Alpharabian Artoneutral Second and octave-reducing
89	671.6981132	s5, Km5	Lesser Grave Fifth	Ad>↑, A↓\, Gt#>	−2	2	This interval… Is reachable through stacking four of this system's approximation of the Werckismic Subfourth and octave-reducing Is the closest approximation of 16edo's Minor Fifth 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
90	679.2452830	k5	Greater Grave Fifth	A↓	−3	3	This interval… Approximates the Classic Grave Fifth, 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 Minor Third and Minor Seventh in the Locrian Mode of the familiar Ptolemaic Sequence, and is ideally in the exact same position for, Phrygian, Aeolian and Dorian modes, though this technically results in there being Diatonic scales of different varieties- namely the Contrazarlino, Contrabilawal and Contramyxian 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 ten of this system's approximation of the Large Tridecimal Third-Tone Is very useful for essentially tempered chords such as marveltwin chords and ibnsinmic chords in the 27-odd-limit
91	686.7924528	Rk5	Wide Grave Fifth	A↓/	−2	3	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 seven of this system's approximation of the Small Septendecimal Semitone due to the combination of commas tempered out in this system Is the closest approximation of the 7edo 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
92	694.3396226	r5	Narrow Fifth	A\	1	4	This interval… Approximates the Marvelous Fifth, 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 reachable through stacking two of this system's approximation of the Low-Complexity JI Neutral Third Is the closest approximation of 19edo's Perfect 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 31edo's Perfect 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
93	701.8867925	P5	Perfect Fifth	A	5	5	This interval… Approximates the Perfect Fifth or Octave-Reduced Third 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 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 Dominant, and thus… It is the basic interval for framing a standard triad in this system It can easily move to the Tonic harmony either upwards or downwards Is one of four perfect consonances in this system Is the closest approximation of the 12edo Perfect 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 Inherits a sizable portion of its functionality from its 53edo counterpart, including… A chain of 53 Perfect Fifths 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 gamelan and gentle A sizable chunk of its functionality in the realm of Western-Classical-based Paradiatonic functional harmony
94	709.4339622	R5	Wide Fifth	A/	1	3	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 reachable through stacking two of this system's approximation of the 2nd Undecimal Neutral Third Is the closest approximation of 17edo's Perfect 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 22edo's Perfect Fifth 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
95	716.9811321	rK5	Narrow Acute Fifth	A↑\	−1	1	This interval… Is reachable through stacking five of this system's approximation of the 2nd Undecimal Neutral Second Is the closest approximation of 10edo's Perfect Fifth 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
96	724.5283019	K5	Lesser Acute Fifth	A↑	−2	0	This interval… Approximates a complex 5-limit interval formed by stacking a syntonic comma on top of a Perfect Fifth Is reachable through stacking two of this system's approximation of the Octave-Reduced Thirteenth Subharmonic Is reachable through stacking three of this system's approximation of the Septimal Superaugmented Fourth and octave-reducing
97	732.0754717	S5, kM5	Greater Acute Fifth, Narrow Inframinor Sixth	At<↓, A↑/	−3	−1	This Interval… Approximates the Septimal Superfifth, and thus… Is really useful for forming unexpected chords in place of a normal fifth Is the closest approximation of 31edo's own Superfifth 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
98	739.6226415	um6, RkM5	Inframinor Sixth, Wide Acute Fifth	At>↓, Bdb>	−2	−2	This interval… Approximates a complex 11-limit Paradiatonic interval that functions as a syntactic sixth that sounds more like a fifth, and as such… It has the potential to move back down to the Interregnant harmony through Paradiatonic motion It has the potential to move up to a Contramediant harmony through a type of Chromatic or Parachromatic semitone motion Is the closest approximation of 13edo's Major Fifth 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
99	747.1698113	Rm4, KKM3, rUM3	Narrow Paramajor Fifth, Wide Inframinor Sixth	At<\, Bb↓↓, A↑↑	−1	−1	This interval… Approximates the Tridecimal Semitenth Is the closest approximation of 24edo's Paramajor Fifth, 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
100	754.7169811	M5, rKum6	Paramajor Fifth, Narrow Subminor Sixth	At<, Bdb<↑	−1	0	This interval… Approximates the Just Paramajor Fifth, 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 Misodominant 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 back down towards a Dominant harmony through a Parachromatic quatertone-type motion It has the potential to move back down towards an Interregnant 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 paramajor fifth sounds more like a sixth than a fifth Is the closest approximation of 19edo's Augmented Fifth 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
101	762.2641509	sm6, Kum6, RM5, uA5	Lesser Subminor Sixth, Infra-Augmented Fifth	At>, Bb↓\	0	1	This interval… Approximates the Septimal Subminor Sixth, 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 Is one half of this system's approximation of the Marvelous Minor Tenth as a consequence of the hemimage comma being tempered out in this system Is the closest approximation of 22edo's Lesser Minor Sixth 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
102	769.8113208	km6, RuA5, kkA5	Greater Subminor Sixth, Diptolemaic Augmented Fifth	Bb↓, At>/, A#↓↓	−1	2	This interval… Approximates the Classic Augmented Fifth or Diptolemaic Augmented Fifth, and thus… It functions as an Augmented Fifth in Western-Classical-based functional harmony by default, and is the signature interval of certain 5-limit Non-Diatonic modes such as Lydian Augmented Can be used in Western-Classical-based harmony as an extension to the simul cadence due to its relationship to multiple notes It is easily very useful when it comes to building chords despite- or perhaps even because of- its dissonance, specifically… It is the basic interval for framing a 5-limit augmented triad, though it can also be used for certain other triads Approximates a complex 5-limit interval formed by subtracting a syntonic comma from a Pythagorean Minor Sixth, and thus… It can be thought of as a type of sixth when acting in this capacity Is one half of this system's approximation of the Lesser Tridecimal Neutral Tenth as a consequence of the tunbarsma being tempered out in this system
103	777.3584906	Rkm6, rKuA5	Wide Subminor Sixth, Lesser Sub-Augmented Fifth	Bb↓/, At<↑	−1	3	This interval… Approximates the Septendecimal Minor Sixth Is the closest approximation of 17edo's Minor Sixth 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 Subminor Sixth 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
104	784.9056604	rm6, KuA5	Narrow Minor Sixth, Greater Sub-Augmented Fifth	Bb\, At>↑, A#↓\	0	4	This interval… Approximates the Neo-Gothic Minor Sixth, 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 half of this system's approximation of the Tridecimal Submajor Tenth Is reachable through stacking eight of this system's approximation of the Axirabian Limma
105	792.4528302	m6, kA5	Pythagorean Minor Sixth, Ptolemaic Augmented Fifth	Bb, A#↓	−1	5	This interval… Approximates the Pythagorean Minor Sixth, 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 Sixth in that… It is very useful as an interpretation of the dissonant Minor Sixth 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 Major Tenth
106	800	Rm6, RkA5	Artomean Minor Sixth, Artomean Augmented Fifth	Bb/, A#↓/	0	5	This interval… Approximates the Quasi-Tempered Minor Sixth Is none other than the familiar Minor Sixth of 12edo, and thus… It is useful for evoking the feel of 12edo in various ways, such as… Framing the familiar augmented triads of 12edo Performing modulatory maneuvers based around the aforementioned triads
107	807.5471698	rKm6, rA5	Tendomean Minor Sixth, Tendomean Augmented Fifth	A#\, Bb↑\	2	5	This interval… Approximates the Septendecimal Tendomean Minor Sixth Can easily be used in modulatory maneuvers similar to those performed by Jacob Collier
108	815.0943396	Km6, A5	Ptolemaic Minor Sixth, Pythagorean Augmented Fifth	A#, Bb↑	4	5	This interval… Approximates the Classic Minor Sixth or Octave-Reduced Fifth 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 Readily occurs as the distance between two notes in a single chord in Western-Classical-based polypedal harmony It differs from the Pythagorean Minor Sixth in that… It is one of four imperfect consonances in this system, rendering it extremely suitable for use in a spaced out Tonic chord in Western-Classical-based polypedal harmony Approximates the Pythagorean Augmented Fifth, and as such… Serves as the frame for oddly charming augmented chords Is the closest approximation of 22edo's Greater Minor Sixth 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 Minor Sixth 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
109	822.6415094	RKm6, kn6, RA5	Wide Minor Sixth, Artoretromean Augmented Fifth	Bd<↓, Bb↑/, A#/	2	4	This interval… Approximates the Marvelous Minor Sixth, 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 Sixth Is the closest approximation of 16edo's Minor Sixth 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 Minor Sixth 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
110	830.1886792	kN6, rKA5	Lesser Supraminor Sixth, Tendoretromean Augmented Fifth	Bd>↓, A#↑\	1	3	This interval… Approximates the Tridecimal Supraminor Sixth and a similar 11-limit interval that acts as the Supraminor counterpart to the Undecimal Submajor Sixth Is this system's closest approximation of acoustic phi Is the closest approximation of 13edo's Major Sixth 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
111	837.7358491	KKm6, rn6, KA5	Greater Supraminor Sixth, Retroptolemaic Augmented Fifth	Bd<\, Bb↑↑, A#↑	0	3	This interval Approximates the Lesser Tridecimal Neutral Sixth or Octave-Reduced Thirteenth Harmonic, and as such… Is rather common in Paradiatonic melodies and harmonies, in particular… It shows up as a key interval in the minor simul chord. Is the closest approximation of 10edo's Minor Sixth 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 Supraminor Third, and can thus be used to create identical-sounding melodic and harmonic gestures in this system
112	845.2830189	n6, SA5, kUA5	Artoneutral Sixth, Lesser Super-Augmented Fifth	Bd<, At#<↓	−1	3	This interval… Approximates the Alpharabian Artoneutral Sixth or 2nd Undecimal Neutral Sixth, 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 smaller and more dissonant of two Neutral Sixths in Western-Classical-based Paradiatonic functional harmony, and thus… Has the potential to move to the Perfect Fifth through a Paradiatonic "wide semitone" motion Has the potential to move to the Greater Supermajor Sixth through a type of Chromatic semitone motion Is the closest approximation of 17edo's Neutral Sixth 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
113	852.8301887	N6, RkUA5	Tendoneutral Sixth, Greater Super-Augmented Fifth	Bd>, At#>↓	0	3	This interval… Approximates the Alpharabian Tendoneutral Sixth, which is the traditional, low complexity Undecimal Neutral Sixth, 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 larger and more consonant of two Neutral Sixths in Western-Classical-based Paradiatonic functional harmony, and thus… Has the potential to move to the Perfect Fifth through a Paradiatonic "narrow whole tone" motion Has the potential to move to the Greater Supermajor Sixth through a type of Chromatic semitone motion Is the closest approximation of 24edo's own Neutral Sixth 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 Sixth 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
114	860.3773585	kkM6, RN6, rUA5	Lesser Submajor Sixth, Retrodiptolemaic Augmented Fifth	Bd>/, B↓↓, At#>↓/, A#↑↑	−1	3	This interval… Approximates the Greater Tridecimal Neutral Sixth Approximates the Septendecimal Submajor Sixth, and thus… Is very useful for essentially tempered chords such as tannic chords Is reachable through stacking six of this system's approximation of the 2nd Undecimal Neutral Second Is the closest approximation of the 7edo Sixth 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 Sixth, and can thus be used to create identical-sounding melodic and harmonic gestures in this system
115	867.9245283	Kn6, UA5	Greater Submajor Sixth, Ultra-Augmented Fifth	Bd<↑, At#<	0	3	This interval… Approximates the Undecimal Submajor Sixth Is one half of this system's approximation of the Undecimal Grave Infra-Augmented Eleventh
116	875.4716981	rkM6, KN6	Narrow Major Sixth	Bd>↑, B↓\, At#>	1	4	This interval… Approximates the Marvelous Major Sixth, 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 Sixth Is the closest approximation of 22edo's Lesser Major Sixth 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
117	883.0188679	kM6	Ptolemaic Major Sixth	B↓, Cb	3	5	This interval… Approximates the Classic Major Sixth, 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 Sixth in that… It is one of four imperfect consonances in this system, rendering it extremely suitable for use in a spaced out Tonic chord in Western-Classical-based polypedal harmony Is surprisingly ill-suited for use as the interval between the Tonic and the Major Sixth scale degree above it in a fixed-pitch Bass-Up Diatonic system due to it creating a wolf fifth in a less-than-ideal location within the main scale Is reachable through stacking six of this system's approximation of the low complexity Undecimal Neutral Third and octave-reducing Is the closest approximation of 19edo's Major Sixth 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
118	890.5660377	RkM6	Artomean Major Sixth	B↓/	1	5	This interval… Approximates the Septendecimal Artomean Major Sixth Is the closest approximation of 31edo's Major Sixth 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
119	898.1132075	rM6	Tendomean Major Sixth	B\	0	5	This interval… Approximates the Quasi-Tempered Major Sixth, and as such… It is the closest approximation of 12edo's Major Sixth 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 reachable through stacking seven of this system's approximation of the Tridecimal Supraminor Second
120	905.6603774	M6	Pythagorean Major Sixth	B	−1	5	This interval… Approximates the Pythagorean Major Sixth, 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 It is the optimal choice for use as the interval between the Tonic and the Major Sixth scale degree above it in a fixed-pitch Bass-Up Diatonic system due to it creating a wolf fifth in one of the two possible ideal locations within the main scale Is reachable through stacking six of this system's approximation of the low complexity Undecimal Neutral Second Is reachable through stacking eight of this system's approximation of the Apotome as a consequence of Mercator's comma being tempered out in this system
121	913.2075472	RM6	Wide Major Sixth	B/, Cd<↓	0	4	This interval… Approximates the Neo-Gothic Major Sixth, 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 very useful for essentially tempered chords such as gentle chords, ainismic chords and nicolic chords
122	920.7547170	rKM6	Narrow Supermajor Sixth	B↑\, Cd>↓	−1	3	This interval… Approximates the Septendecimal Major Sixth Is the closest approximation of 13edo's Major Sixth 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 Major Sixth 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
123	928.3018868	KM6	Lesser Supermajor Sixth	B↑, Cd<\, Cb↑↑, A##	−1	3	This interval… Approximates the Classic Diminished Seventh, and as such… It can be thought of as a type of seventh when acting in this capacity Approximates a complex 5-limit interval formed by stacking a syntonic comma on top of a Pythagorean Major Sixth, and thus… It can be thought of as a type of sixth when acting in this capacity Is the closest approximation of 22edo's Greater Major Sixth 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 Supermajor Sixth 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
124	935.8490566	SM6, kUM6	Greater Supermajor Second, Narrow Inframinor Seventh	Cb<, Bt<↓, B↑/	0	3	This interval… Approximates the Septimal Supermajor Sixth, 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 Varicant due to its semiambitonal properties relative to the Diatonic scale It is useful in forming ambisonant triads framed by the Perfect Twelfth Is very useful for essentially tempered chords such as Keenanismic chords
125	943.3962264	um7, RkUM6	Inframinor Seventh, Wide Supermajor Sixth	Cd>, Bt>↓	0	3	This interval… Approximates a complex 11-limit Paradiatonic interval that functions as a syntactic seventh that sounds more like a sixth, and as such… It has the potential to move up towards the Subtonic through a parachromatic motion It has the potential to move back down to a Contramediant harmony through a diatonic or paradiatonic motion Is reachable through stacking five of this system's approximation of the Grossmic Whole Tone due to the combination of commas tempered out in this system
126	950.9433962	KKM6, kkm7, rUM6, Rum7	Narrow Ultramajor Sixth, Wide Inframinor Seventh, Semitwelfth	Bt<\, Cd>/, B↑↑, C↓↓	0	4	This interval… Approximates the Tridecimal Semitwelfth, and thus… It can be in triads framed by a Perfect Twelfth Is one half of a Perfect Twelfth in this system Is the closest approximation of 19edo's Semitwelfth 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 Semitwelfth, 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
127	958.4905660	UM6, rKum7	Ultramajor Sixth, Narrow Subminor Seventh	Bt<, Cd<↑	0	4	This interval… Approximates a complex 11-limit Paradiatonic interval that functions as a syntactic sixth that sounds more like a seventh, and as such… It has the potential to move up towards the Subtonic through a diatonic or paradiatonic motion It has the potential to move back down to a Submediant harmony through a parachromatic motion Is the closest approximation of 10edo's Minor Seventh slash Major Sixth 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
128	966.0377358	sm7, Kum7	Lesser Subminor Seventh, Wide Ultramajor Sixth	Bt>, Cd>↑, C↓\	0	5	This interval… Approximates the Septimal Subminor Seventh or Octave-Reduced Seventh Harmonic, 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 Varicant due to its semiambitonal properties relative to the Diatonic scale It is useful in forming ambisonant triads framed by the Perfect Twelfth Is the closest approximation of 31edo's Subminor Seventh 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
129	973.5849057	km7	Greater Subminor Seventh	C↓, Bt>/, B#↓↓, Dbb	−1	5	This interval… Approximates the Neapolitan Augmented Sixth, and thus… It readily appears in approximations of 5-limit Neapolitan scales Approximates a complex 5-limit interval formed by subtracting a syntonic comma from a Pythagorean Minor Seventh, and thus… It can be thought of as a type of seventh when acting in this capacity Is the closest approximation of 16edo's Subminor Seventh 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
130	981.1320755	Rkm7	Wide Subminor Seventh	C↓/, Bt<↑	−1	5	This interval… Approximates the Septendecimal Minor Seventh, and thus… Can be used as an unexpected option for an augmented sixth in Western-Classical-based harmony Approximates a complex 11-limit interval formed by subtracting a Parapotome from a Classic Major Seventh, and thus… It can be thought of as a type of seventh when acting in this capacity Is reachable through stacking five of this system's approximation of the Middle Major Second Is the closest approximation of 22edo's Lesser Minor Seventh 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
131	988.6792458	rm7	Narrow Minor Seventh	C\, Bt>↑	−1	5	This interval… Approximates the Tridecimal Minor Seventh, and thus… It is very likely to be treated as a type of minor seventh when working in Neo-Medieval harmony Is the closest approximation of 17edo's Minor Seventh 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
132	996.2264151	m7	Pythagorean Minor Seventh	C, B#↓	−1	5	This interval… Approximates the Pythagorean Minor Seventh, 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 crowds the octave, and is thus prone to decomposition It readily serves as a Diatonic Minor Seventh in both Western-Classical-based functional harmony and Neo-Medieval harmony in general, since… It functions as a Double Serviant due to being the result of stacking two Perfect Fourths Is reachable through stacking three of this system's approximation of the Undecimal Supraminor Third Is one of two intervals that serve as the closest approximation of the 12edo Minor Seventh 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
133	1003.7735849	Rm7	Artomean Minor Seventh	C/, B#↓/	−2	3	This interval… Approximates the Middle Minor Seventh Is one of two intervals that serve as the closest approximation of the 12edo Minor Seventh 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 Minor Seventh 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
134	1011.3207547	rKm7	Tendomean Minor Seventh	C↑\, B#\	−2	5	This interval… Approximates the Grossmic Minor Seventh, and thus… Is useful for modulating to keys that are not found on the same circle of fifths Is the closest approximation of 19edo's Minor Seventh 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
135	1018.8679245	kM2	Ptolemaic Minor Seventh	C↑, B#	−2	5	This interval… Approximates the Classic Minor Seventh or Ptolemaic Minor Seventh, 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 crowds the octave, and is thus prone to decomposition It readily serves as a Diatonic Minor Seventh in Western-Classical-based functional harmony, since… It has close affinities with the Dominant due to being located at roughly a Ptolemaic Minor Third away from it Is reachable through stacking five of this system's approximation of the Pythagorean Major Second Is the closest approximation of 13edo's own Minor Seventh 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
136	1026.4150943	RKm7, kn7	Wide Minor Seventh	Ct<↓, C↑/, Ddb<, B#/	−3	5	This interval… Is reachable through stacking eight of this system's approximation of the Tridecimal Supraminor Second Is the closest approximation of the 7edo Seventh 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
137	1033.9622642	kN7, ud8	Lesser Supraminor Seventh, Infra-Diminished Octave	Ct>↓, Ddb>, B#↑\	−3	4	This interval… Approximates the Undecimal Supraminor Seventh and a similar 13-limit interval that acts as the Supraminor counterpart to the Tridecimal Submajor Seventh It can be thought of as a type of seventh in voice-leading Approximates a complex 11-limit Parachromatic interval formed by subtracting an Al-Farabi Quartertone and an Apotome from an Octave, and thus… It can be thought of as the inverse of a type of sesquichroma when acting in this capacity Is the closest approximation of 22edo's Greater Minor Seventh 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
138	1041.5094340	KKm7, rn7, Rud8	Greater Supraminor Seventh, Retrodiptolemaic Diminished Octave	Ct<\, C↑↑, Ddb<↑\, Db↓↓	−4	3	This interval… Is the closest approximation of 31edo's own Middle Seventh 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 Seventh, and can thus be used to create identical-sounding melodic and harmonic gestures in this system
139	1049.0566038	n7, rKud8	Artoneutral Seventh, Lesser Sub-Diminished Octave	Ct<, Ddb<↑	−4	2	This interval… Approximates the Alpharabian Artoneutral Seventh, which is the traditional, low complexity Undecimal Neutral Seventh, and as such… Can occur as the distance between two notes in a single chord in Western-Classical-based polypedal harmony, though this crowds the octave, and is thus prone to decomposition It serves as the smaller and more consonant of two Neutral Sevenths in Western-Classical-based Paradiatonic functional harmony, and thus… Has the potential to move to the Pythagorean Major Sixth through a Paradiatonic "wide semitone" motion Has the potential to move to the Greater Supermajor Seventh through a type of Chromatic semitone motion Is the closest approximation of 24edo's own Neutral Seventh 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
140	1056.6037736	N7, sd8	Tendoneutral Seventh, Greater Sub-Diminished Octave	Ct>, Ddb>↑	−4	2	This interval… Approximates the Alpharabian Tendoneutral Seventh or 2nd Undecimal Neutral Seventh, 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 Sevenths in Western-Classical-based Paradiatonic functional harmony, and thus… Has the potential to move to the Pythagorean Major Sixth through a Paradiatonic "narrow whole tone" motion Has the potential to move to the Greater Supermajor Seventh through a type of Chromatic semitone motion Is the closest approximation of 17edo's Neutral Seventh 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
141	1064.1509434	kkM7, RN7, kd8	Lesser Submajor Seventh, Diptolemaic Major Seventh, Retroptolemaic Diminished Octave	Ct>/, C#↓↓, Db↓	−4	3	This interval… Approximates the Grave Major Seventh, and thus… It frequently acts as the inverse of a diatonic semitone in Western-Classical-based harmony Approximates the Tridecimal Neutral Seventh Is found in 53edo as that system's Submajor Seventh, and can thus be used to create identical-sounding melodic and harmonic gestures in this system
142	1071.6981132	Kn7, Rkd8	Greater Submajor Seventh, Artoretromean Diminished Octave	Ct<↑, Db↓/	−3	4	This interval… Approximates the Tridecimal Submajor Seventh and a similar 11-limit interval that acts as the Submajor counterpart to the Undecimal Supraminor Seventh, and thus… It demonstrates leading-tone functionality Approximates a complex yet rooted 17-limit interval relative to the Tonic and can be used… As an unexpected option for a Diatonic Major Seventh in Western-Classical-based harmony Is the closest approximation of 19edo's Major Seventh 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
143	1079.2452830	rkM7, KN7, rd8	Narrow Major Seventh, Tendoretromean Diminished Octave	Ct>↑, C#↓\, Db\	−2	5	This interval… Approximates the Septimal Grave Major Seventh, and thus… It functions as both a type of Diminished Oetave and a type of Major Seventh in septimal harmony Is the closest approximation of 10edo's Major Seventh 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
144	1086.7924528	kM7, d8	Ptolemaic Major Seventh, Pythagorean Diminished Octave	Db, C#↓	−2	5	This interval… Approximates the Classic Major Seventh or Ptolemaic Major Seventh, 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 crowds the octave, and is thus prone to decomposition It readily serves as the traditional Major Seventh in 5-limit Western-Classical-based functional harmony and thus… Has the potential to move directly up to the Tonic as a Lead, 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 Dominant due to being located at roughly a Ptolemaic Major Third away from it Approximates the Pythagorean Diminished Octave Is the closest approximation of 31edo's own Major Seventh 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
145	1094.3396226	RkM7, Rd8	Artomean Major Seventh, Artomean Diminished Octave	Db/, C#↓/	−2	5	This interval… Approximates the Small Septendecimal Major Seventh or Octave-Reduced Seventeenth Subharmonic, and thus… Can be used as an unexpected option for a Major Seventh in Western-Classical-based harmony Approximates the Alpharabian Diminished Octave and thus… Can be used as a type of Diminished Octave in undecimal harmony Is the closest approximation of 22edo's Lesser Major Seventh 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
146	1101.8867925	rM7, rKd8	Tendomean Major Seventh, Tendomean Diminished Octave	C#\, Db↑\	−3	5	This interval… Approximates the Large Septendecimal Major Seventh, and thus… Can be used as an unexpected option for a Diminished Octave in Western-Classical-based harmony Approximates the Axirabian Major Seventh, and thus… Can be used as a type of Major Seventh in undecimal harmony Is the closest approximation of the 12edo Major Seventh 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
147	1109.4339622	M7, Kd8	Pythagorean Major Seventh, Ptolemaic Diminished Octave	C#, Db↑	−3	5	This interval… Approximates the Pythagorean Major Seventh, 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 Major Seventh in both Western-Classical-based functional harmony and Neo-Medieval harmony in general, and thus… Has the potential to move directly up to the Tonic as a Lead, though in non-meantone systems like this one, such a gesture using this interval has a slightly more tense feel Approximates the Ptolemaic Diminished Octave, and as such… It serves as a Diminished Octave in the 5-limit Diatonic settings that are common to Western-Classical-based harmony Is the closest approximation of 13edo's own Major Seventh 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
148	1116.9811321	RM7, kUd8	Wide Major Seventh, Lesser Super-Diminished Octave	C#/, Dd<↓	−3	5	This interval… Approximates the Septimal Acute Major Seventh, and thus… It serves as a type of Major Seventh when resolving septimal harmony constructions to classic harmony constructions Approximates the Large Undecimal Diminished Octave, and thus… It serves as a type of Diminished Octave in undecimal harmony constructions
149	1124.5283019	rKM7, RkUd8	Narrow Supermajor Seventh, Greater Super-Diminished Octave	C#↑\, Dd>↓	−4	5	This interval… Approximates multiple complex 17-limit intervals relative to the Tonic and can be used… As an unexpected option for a Diminished Octave in Western-Classical-based harmony Is the closest approximation of 31edo's Supermajor Seventh 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 Supermajor Seventh 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
150	1132.0754717	km2, RuA1, kkA1	Lesser Supermajor Seventh, Diptolemaic Diminished Octave	C#↑, Db↑↑	−4	5	This interval… Approximates the Classic Diminished Octave or Diptolemaic Diminished Octave, and thus… It frequently acts as a Diminished Octave in Western-Classical-based harmony Approximates the Lesser Tridecimal Diminished Octave and the Greater Tridecimal Diminished Octave Is the closest approximation of 17edo's Major Seventh 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
151	1139.6226415	SM7, kUM7, Ud8	Greater Supermajor Seventh, Narrow Infraoctave, Ultra-Diminished Octave	Dd<, C#↑/	−4	5	This interval… Approximates the Septimal Supermajor Seventh, and thus… Is the narrowest interval that has the potential to move directly up to the Tonic as a Lead in Western-Classical-based harmony and Neo-Medieval harmony Compared to other options, it has a markedly more tense feel Can be used as an unexpected option for a Diminished Octave in Western-Classical-based harmony Is the closest approximation of 19edo's Diminished Octave 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 Can be used for tonality-flux-based chord progressions
152	1147.1698113	u8, RkUM7	Infraoctave, Wide Supermajor Seventh	Dd>, Ct#>↓	−5	5	This interval… Approximates the Alpharabian Infraoctave, and as such… It functions as the inverse of 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 up to the Tonic through a Parachromatic quartertone motion Has the potential to move away from the Tonic back towards either a Lead or Subtonic harmony through a type of Diatonic or Paradiatonic semitone motion Is the closest approximation of 22edo's Greater Major Seventh 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 Infraoctave 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
153	1154.7169811	KKM7, rUM7, Ru8	Narrow Ultramajor Seventh, Wide Infraoctave	C#↑↑, Dd>/	−5	5	This interval… Approximates the Tridecimal Ultramajor Seventh It functions like an Infraoctave in that… It has the potential to move directly up to the Tonic through a parachromatic motion It has the potential to move away from the Tonic back towards either a Lead or Subtonic 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 Ultramajor Seventh 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
154	1162.2641509	UM7, rKu8	Ultramajor Seventh, Wide Superprime	Ct#<, Dd<↑	−5	5	This interval… Approximates the Undecimal Suboctave Approximates a complex 11-limit Paradiatonic quartertone that is the namesake of 24edo's own Ultramajor Seventh Is the closest approximation of 31edo's own Suboctave 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 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
155	1169.8113208	s8, Ku8	Lesser Suboctave, Wide Ultramajor Seventh	Ct#>, Dd>↑	−5	2	This interval… Approximates the Archytas suboctave, and thus… Can function as both a type of Suboctave and a type of Ultramajor Seventh in this system Approximates the telepathmic suboctave, and thus… Can be considered a type of Suboctave Is a dissonance to be avoided in Western-Classical-based harmony unless… 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
156	1177.3584906	k8	Greater Suboctave	D↓	−5	−2	This interval… Approximates the syntonic suboctave Approximates the Pythagorean Diminished Ninth 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
157	1184.9056604	Rk8	Wide Suboctave	D↓/	−5	−5	This interval… Approximates the ptolemismic suboctave and the biyatismic suboctave Is useful for slight dissonances that create noticeable tension Can only be approached in melodic lines indirectly with one or more intervening notes
158	1192.4528302	r8	Narrow Octave	D\	0	0	This interval… Approximates the rastmic narrow octave Approximates the marvelous narrow octave 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
159	1200	P8	Perfect Octave	D	5	5	This interval… Is the Perfect Octave, and thus… Is the reduplication of a chord's root in this system Is the reduplication of the Tonic in this system Is one of four perfect consonances in this system Is the most common equave due in part to the properties human hearing in relation to pitch-chroma matching

Harmonies

Harmonies in 159edo frequently have to follow a variation on the Dinner Party Rules. However, working with these rules in a system like this requires a more detailed list of "friends" and "enemies". Thus, what will be listed here are a series of basic trines, triads and tetrads.

First, the trines, of which there are already a noticeable abundance.

Table of 159edo Trines

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

Next, the basic triads, which end up inheriting the base trine's type, and as a consequence, there are even more triads than there are trines, though this list will only cover the triads that build off of the Otonal Perfect Trine for the sake of ease. Of course, it should be mentioned that suspensions occur where there's overlap between thirds and fourths, and these are excluded from this list along with augmented and diminished triads and variations thereof.

Table of 159edo Triads

Name	Notation (from D)	Steps	Approximate JI	Notes
	D, F#↓\, A	0, 50, 93
	D, F↑/, A	0, 43, 93
Ptolemaic Major	D, F#↓, A	0, 51, 93	4:5:6	This is the first of two triads that can be considered fully-resolved in Western Classical Harmony
Ptolemaic Minor	D, F↑, A	0, 42, 93	1/(4:5:6)	This is the second of two triads that can be considered fully-resolved in Western Classical Harmony
Pythagorean Major	D, F#, A	0, 54, 93	1/(54:64:81)	This dissonant triad is common in Western Classical, Medieval, and Neo-Medieval Harmony
Pythagorean Minor	D, F, A	0, 39, 93	54:64:81	This dissonant triad is common in Western Classical, Medieval, and Neo-Medieval Harmony
Neo-Gothic Major	D, F#/, A	0, 55, 93	22:28:33 1/(22:26:33)	This ambisonant triad is very useful in Neo-Medieval Harmony
Neo-Gothic Minor	D, F\, A	0, 38, 93	1/(22:28:33) 22:26:33	This ambisonant triad is very useful in Neo-Medieval Harmony
Neo-Gothic Supermajor	D, F#↑\, A	0, 56, 93	1/(34:40:51)	This triad combines an imitation of the qualities of 17edo's Major third with an accurate fifth
Neo-Gothic Subminor	D, F↓/, A	0, 37, 93	34:40:51	This triad combines an imitation of the qualities of 17edo's Minor third with an accurate fifth
Retroptolemaic Supermajor	D, F#↑, A	0, 57, 93	1(100:117:150)	This supermajor triad is inherited from 53edo, so if you're familiar enough with that system, you should know how this works
Retroptolemaic Subminor	D, F↓, A	0, 36, 93	100:117:150	This subminor triad is inherited from 53edo, so if you're familiar enough with that system, you should know how this works