159edo/Interval names and harmonies

Revision as of 19:36, 8 May 2022 by Aura (talk | contribs) (Added information)

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

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

References