159edo/Interval names and harmonies: Difference between revisions
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[[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]]. | {{breadcrumb}} | ||
[[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 10 is a full-blown friend relative to the root and −10 if a full-blown enemy relative to the root. Note that the Harmonic Compatibility and Melodic Compatibility ratings are based on octave-equivalence, and that some of the ratings are still speculative. | |||
== Interval chart == | == Interval chart == | ||
{| class="mw-collapsible mw-collapsed wikitable center-1" | {| class="mw-collapsible mw-collapsed wikitable center-1" | ||
|+ style=white-space:nowrap | Table of 159edo intervals | |+ style="font-size: 105%; white-space: nowrap;" | Table of 159edo intervals | ||
|- | |- | ||
! Step | ! rowspan="2" | Step | ||
! Cents | ! rowspan="2" | Cents | ||
! colspan="3"| Interval | ! rowspan="2" colspan="3" | Interval names | ||
! Notes | ! colspan="2" | Compatibility rating | ||
! rowspan="2" | Notes | |||
|- | |||
! Harmonic | |||
! Melodic | |||
|- | |- | ||
| 0 | | 0 | ||
Line 16: | Line 20: | ||
| Perfect Unison | | Perfect Unison | ||
| D | | D | ||
| 10 | |||
| 10 | |||
| This interval… | | This interval… | ||
* Is the [[1/1| | * Is the [[1/1|perfect unison]], and thus… | ||
:* Is the basic representation of a given chord's root | :* Is the basic representation of a given chord's root | ||
:* Is the basic representation of the Tonic | :* Is the basic representation of the Tonic | ||
Line 28: | Line 34: | ||
| Wide Prime | | Wide Prime | ||
| D/ | | D/ | ||
| 0 | |||
| 0 | |||
| This interval… | | This interval… | ||
* Approximates the [[rastma]], and thus… | * Approximates the [[rastma]], and thus… | ||
:* Is useful for defining [[11-limit]] subchromatic alterations in the Western-Classical-based functional harmony of this system | :* Is useful for defining [[11-limit]] subchromatic alterations in the Western-Classical-based functional harmony of this system | ||
* Approximates the [[marvel comma]], and thus… | * Approximates the [[marvel comma]], and thus… | ||
:* Can function as both a type of subchroma and a type of | :* 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 | * 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 only be approached in melodic lines indirectly with one or more intervening notes | ||
Line 42: | Line 50: | ||
| Narrow Superprime | | Narrow Superprime | ||
| D↑\ | | D↑\ | ||
| -10 | |||
| -10 | |||
| This interval… | | This interval… | ||
* Approximates the [[ptolemisma]] and the [[biyatisma]] | * Approximates the [[ptolemisma]] and the [[biyatisma]] | ||
Line 52: | Line 62: | ||
| Lesser Superprime | | Lesser Superprime | ||
| D↑ | | D↑ | ||
| -10 | |||
| -3 | |||
| This interval… | | This interval… | ||
* Approximates the [[syntonic comma]], and as such… | * 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 | :* 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… | * Approximates the [[Pythagorean comma]], and thus… | ||
:* Can be considered a type of | :* 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 a dissonance to be avoided in Western-Classical-based harmony unless deliberately used for expressive purposes | ||
* Is useful in melody as… | * Is useful in melody as… | ||
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| Greater Superprime, Narrow Inframinor Second | | Greater Superprime, Narrow Inframinor Second | ||
| Edb<, Dt<↓ | | Edb<, Dt<↓ | ||
| -10 | |||
| 3 | |||
| This interval… | | This interval… | ||
* Approximates the [[septimal comma|Archytas comma]], and thus… | * Approximates the [[septimal comma|Archytas comma]], and thus… | ||
Line 88: | Line 102: | ||
| Inframinor Second, Wide Superprime | | Inframinor Second, Wide Superprime | ||
| Edb>, Dt>↓ | | Edb>, Dt>↓ | ||
| -9 | |||
| 10 | |||
| This interval… | | This interval… | ||
* Approximates the [[45/44|Undecimal Fifth-Tone]] | * Approximates the [[45/44|Undecimal Fifth-Tone]] | ||
* Approximates [[8192/8019| | * Approximates the [[8192/8019|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 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 [[SpiralProgressions|spiral progressions]] | :* Is capable of being used in progressions reminiscent of that system's [[SpiralProgressions|spiral progressions]] | ||
Line 107: | Line 123: | ||
| Wide Inframinor Second, Narrow Ultraprime | | Wide Inframinor Second, Narrow Ultraprime | ||
| Eb↓↓, Dt<\ | | Eb↓↓, Dt<\ | ||
| -9 | |||
| 10 | |||
| This interval… | | This interval… | ||
* Approximates the [[40/39|Tridecimal Minor Diesis]] | * Approximates the [[40/39|Tridecimal Minor Diesis]] | ||
Line 124: | Line 142: | ||
| Ultraprime, Narrow Subminor Second | | Ultraprime, Narrow Subminor Second | ||
| Dt<, Edb<↑ | | Dt<, Edb<↑ | ||
| -9 | |||
| 10 | |||
| This interval… | | This interval… | ||
* Approximates the [[33/32|Al-Farabi Quartertone]], and as such… | * Approximates the [[33/32|Al-Farabi Quartertone]], and as such… | ||
Line 143: | Line 163: | ||
| Lesser Subminor Second, Wide Ultraprime, Infra-Augmented Prime | | Lesser Subminor Second, Wide Ultraprime, Infra-Augmented Prime | ||
| Dt>, Eb↓\ | | Dt>, Eb↓\ | ||
| -8 | |||
| 10 | |||
| This interval… | | This interval… | ||
* Approximates the [[28/27|Septimal Subminor Second]], and thus… | * Approximates the [[28/27|Septimal Subminor Second]], and thus… | ||
Line 159: | Line 181: | ||
| Greater Subminor Second, Diptolemaic Augmented Prime | | Greater Subminor Second, Diptolemaic Augmented Prime | ||
| Eb↓, D#↓↓ | | Eb↓, D#↓↓ | ||
| -8 | |||
| 9 | |||
| This interval… | | This interval… | ||
* Approximates the [[25/24|Classic Chroma]] or Diptolemaic Chroma, and thus… | * Approximates the [[25/24|Classic Chroma]] or Diptolemaic Chroma, and thus… | ||
:* It frequently acts as a chromatic semitone in Western-Classical-based harmony | :* It frequently acts as a chromatic semitone in Western-Classical-based harmony | ||
* Approximates the [[26/25|Large Tridecimal Third-Tone]] and the [[27/26|Small Tridecimal Third-Tone]], and thus… | * Approximates the [[26/25|Large Tridecimal Third-Tone]] and the [[27/26|Small Tridecimal Third-Tone]], and thus… | ||
:* It demonstrates third-tone | :* 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 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 | :* 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 | ||
Line 172: | Line 196: | ||
| Wide Subminor Second, Lesser Sub-Augmented Prime | | Wide Subminor Second, Lesser Sub-Augmented Prime | ||
| Eb↓/, Dt<↑ | | Eb↓/, Dt<↑ | ||
| -7 | |||
| 9 | |||
| This interval… | | This interval… | ||
* Approximates multiple complex [[17-limit]] intervals relative to the Tonic and can be used… | * Approximates multiple complex [[17-limit]] intervals relative to the Tonic and can be used… | ||
Line 185: | Line 211: | ||
| Narrow Minor Second, Greater Sub-Augmented Prime | | Narrow Minor Second, Greater Sub-Augmented Prime | ||
| Eb\, Dt>↑ | | Eb\, Dt>↑ | ||
| -7 | |||
| 9 | |||
| This interval… | | This interval… | ||
* Approximates the [[21/20|Septimal Minor Semitone]], and thus… | * Approximates the [[21/20|Septimal Minor Semitone]], and thus… | ||
Line 197: | Line 225: | ||
| Pythagorean Minor Second, Ptolemaic Augmented Prime | | Pythagorean Minor Second, Ptolemaic Augmented Prime | ||
| Eb, D#↓ | | Eb, D#↓ | ||
| -6 | |||
| 10 | |||
| This interval… | | This interval… | ||
* Approximates the [[256/243|Pythagorean Limma]] or Pythagorean Minor Second, and as such… | * Approximates the [[256/243|Pythagorean Limma]] or Pythagorean Minor Second, and as such… | ||
Line 216: | Line 246: | ||
| Artomean Minor Second, Artomean Augmented Prime | | Artomean Minor Second, Artomean Augmented Prime | ||
| Eb/, D#↓/ | | Eb/, D#↓/ | ||
| -6 | |||
| 10 | |||
| This interval… | | This interval… | ||
* Approximates the [[18/17|Small Septendecimal Semitone]], and thus… | * Approximates the [[18/17|Small Septendecimal Semitone]], and thus… | ||
Line 230: | Line 262: | ||
| Tendomean Minor Second, Tendomean Augmented Prime | | Tendomean Minor Second, Tendomean Augmented Prime | ||
| D#\, Eb↑\ | | D#\, Eb↑\ | ||
| -5 | |||
| 10 | |||
| This interval… | | This interval… | ||
* Approximates the [[17/16|Large Septendecimal Semitone]] or [[octave reduction|Octave-Reduced]] Seventeenth Harmonic, and thus… | * Approximates the [[17/16|Large Septendecimal Semitone]] or [[octave reduction|Octave-Reduced]] Seventeenth Harmonic, and thus… | ||
Line 244: | Line 278: | ||
| Ptolemaic Minor Second, Pythagorean Augmented Prime | | Ptolemaic Minor Second, Pythagorean Augmented Prime | ||
| D#, Eb↑ | | D#, Eb↑ | ||
| -5 | |||
| 10 | |||
| This interval… | | This interval… | ||
* Approximates the [[16/15|Classic Minor Second]] or Ptolemaic Minor Second, and as such… | * Approximates the [[16/15|Classic Minor Second]] or Ptolemaic Minor Second, and as such… | ||
Line 263: | Line 299: | ||
| Wide Minor Second, Artoretromean Augmented Prime | | Wide Minor Second, Artoretromean Augmented Prime | ||
| Ed<↓, Eb↑/, D#/ | | Ed<↓, Eb↑/, D#/ | ||
| -5 | |||
| 9 | |||
| This interval… | | This interval… | ||
* Approximates the [[15/14|Septimal Major Semitone]], and thus… | * Approximates the [[15/14|Septimal Major Semitone]], and thus… | ||
Line 275: | Line 313: | ||
| Lesser Supraminor Second, Tendoretromean Augmented Prime | | Lesser Supraminor Second, Tendoretromean Augmented Prime | ||
| Ed>↓, D#↑\ | | Ed>↓, D#↑\ | ||
| -6 | |||
| 8 | |||
| This interval… | | This interval… | ||
* Approximates the [[14/13|Tridecimal Supraminor Second]] and a similar 11-limit interval that acts as the Supraminor counterpart to the Undecimal Submajor Second, and thus… | * Approximates the [[14/13|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 can be thought of as something along the lines of a "wide semitone" in voice-leading | ||
:* It demonstrates trienthird | :* 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… | * 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 | :* As an unexpected option for a Diatonic-type semitone in Western-Classical-based harmony | ||
Line 288: | Line 328: | ||
| KKm2, rn2, KA1 | | KKm2, rn2, KA1 | ||
| Greater Supraminor Second, Diptolemaic Limma, Retroptolemaic Augmented Prime | | Greater Supraminor Second, Diptolemaic Limma, Retroptolemaic Augmented Prime | ||
| Ed<\, Eb↑↑, D#↑ | | Ed<\, Eb↑↑, D#↑ | ||
| -7 | |||
| 6 | |||
| This interval… | | This interval… | ||
* Approximates the [[27/25|Large Limma]], and thus… | * Approximates the [[27/25|Large Limma]], and thus… | ||
:* It frequently acts as a Diatonic semitone in Western-Classical-based harmony | :* It frequently acts as a Diatonic semitone in Western-Classical-based harmony | ||
* Approximates the [[13/12|Tridecimal Neutral Second]], and thus… | * Approximates the [[13/12|Tridecimal Neutral Second]], and thus… | ||
:* It demonstrates two-third-tone | :* 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 | :* 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 | * 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 | ||
|- | |- | ||
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| Artoneutral Second, Lesser Super-Augmented Prime | | Artoneutral Second, Lesser Super-Augmented Prime | ||
| Ed<, Dt#<↓ | | Ed<, Dt#<↓ | ||
| -8 | |||
| 5 | |||
| This interval… | | This interval… | ||
* Approximates the [[88/81|Alpharabian Artoneutral Second]] or 2nd Undecimal Neutral Second, and as such… | * Approximates the [[88/81|Alpharabian Artoneutral Second]] or 2nd Undecimal Neutral Second, and as such… | ||
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| Tendoneutral Second, Greater Super-Augmented Prime | | Tendoneutral Second, Greater Super-Augmented Prime | ||
| Ed>, Dt#>↓ | | Ed>, Dt#>↓ | ||
| -7 | |||
| 6 | |||
| This interval… | | This interval… | ||
* Approximates the [[12/11|Alpharabian Tendoneutral Second]], which is the traditional, [[low-complexity JI|low complexity]] Undecimal Neutral Second, and as such… | * Approximates the [[12/11|Alpharabian Tendoneutral Second]], which is the traditional, [[low-complexity JI|low complexity]] Undecimal Neutral Second, and as such… | ||
Line 334: | Line 380: | ||
| 158.4905660 | | 158.4905660 | ||
| kkM2, RN2, rUA1 | | kkM2, RN2, rUA1 | ||
| Lesser Submajor Second, | | Lesser Submajor Second, Retrodiptolemaic Augmented Prime | ||
| Ed>/, E↓↓, Dt#>↓/, D#↑↑ | | Ed>/, E↓↓, Dt#>↓/, D#↑↑ | ||
| -6 | |||
| 8 | |||
| This interval… | | This interval… | ||
* Is one half of this system's approximation of the Classic Minor Third | * Is one half of this system's approximation of the Classic Minor Third | ||
Line 347: | Line 395: | ||
| Greater Submajor Second, Ultra-Augmented Prime | | Greater Submajor Second, Ultra-Augmented Prime | ||
| Ed<↑, Dt#<, Fb↓/ | | Ed<↑, Dt#<, Fb↓/ | ||
| -5 | |||
| 9 | |||
| This interval… | | This interval… | ||
* Approximates the [[11/10|Undecimal Submajor Second]] and a similar 13-limit interval that acts as the Submajor counterpart to the Tridecimal Supraminor Second, and thus… | * Approximates the [[11/10|Undecimal Submajor Second]] and a similar 13-limit interval that acts as the Submajor counterpart to the Tridecimal Supraminor Second, and thus… | ||
Line 361: | Line 411: | ||
| Narrow Major Second | | Narrow Major Second | ||
| Ed>↑, E↓\, Dt#>, Fb\ | | Ed>↑, E↓\, Dt#>, Fb\ | ||
| -4 | |||
| 10 | |||
| This interval… | | This interval… | ||
* Is one half of the approximation of the traditional, [[low-complexity JI|low complexity]] Undecimal Neutral Third in this system | * Is one half of the approximation of the traditional, [[low-complexity JI|low complexity]] Undecimal Neutral Third in this system | ||
Line 372: | Line 424: | ||
| Ptolemaic Major Second | | Ptolemaic Major Second | ||
| E↓, Fb | | E↓, Fb | ||
| -3 | |||
| 10 | |||
| This interval… | | This interval… | ||
* Approximates the [[10/9|Classic Major Second]] or Ptolemaic Major Second, and as such… | * Approximates the [[10/9|Classic Major Second]] or Ptolemaic Major Second, and as such… | ||
Line 389: | Line 443: | ||
| Artomean Major Second | | Artomean Major Second | ||
| E↓/, Fb/ | | E↓/, Fb/ | ||
| -3 | |||
| 10 | |||
| This interval… | | This interval… | ||
* Approximates the [[143/128|Grossmic Whole Tone]], and thus… | * Approximates the [[143/128|Grossmic Whole Tone]], and thus… | ||
Line 401: | Line 457: | ||
| Tendomean Major Second | | Tendomean Major Second | ||
| E\, Fb↑\ | | E\, Fb↑\ | ||
| -2 | |||
| 10 | |||
| This interval… | | This interval… | ||
* Approximates the [[28/25|Middle Major Second]] | * Approximates the [[28/25|Middle Major Second]] | ||
Line 413: | Line 471: | ||
| Pythagorean Major Second | | Pythagorean Major Second | ||
| E, Fb↑ | | E, Fb↑ | ||
| -2 | |||
| 10 | |||
| This interval… | | This interval… | ||
* Approximates the [[9/8|Pythagorean Major Second]], and as such… | * Approximates the [[9/8|Pythagorean Major Second]], and as such… | ||
Line 419: | Line 479: | ||
:* It readily serves as a Diatonic whole tone in both Western-Classical-based functional harmony and Neo-Medieval harmony in general, since… | :* 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 | ::* 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 [[ | :* Is the whole tone that is used as a reference interval in [[diatonic, chromatic, enharmonic, subchromatic|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 | ::* 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 one fourth of this system's approximation of the Classic Minor Sixth as a consequence of the schisma being tempered out in this system | ||
Line 431: | Line 491: | ||
| Wide Major Second | | Wide Major Second | ||
| E/, Fd<↓ | | E/, Fd<↓ | ||
| -1 | |||
| 10 | |||
| This interval… | | This interval… | ||
* Approximates the [[44/39|Tridecimal Major Second]], and thus… | * Approximates the [[44/39|Tridecimal Major Second]], and thus… | ||
Line 443: | Line 505: | ||
| Narrow Supermajor Second | | Narrow Supermajor Second | ||
| E↑\, Fd>↓ | | E↑\, Fd>↓ | ||
| -1 | |||
| 10 | |||
| This interval… | | This interval… | ||
* Approximates the [[17/15|Septendecimal Whole Tone]], and thus… | * Approximates the [[17/15|Septendecimal Whole Tone]], and thus… | ||
Line 458: | Line 522: | ||
| Lesser Supermajor Second | | Lesser Supermajor Second | ||
| E↑, Fd<\, Fb↑↑, Dx | | E↑, Fd<\, Fb↑↑, Dx | ||
| -1 | |||
| 9 | |||
| This interval… | | This interval… | ||
* Approximates the [[256/225|Neapolitan Diminished Third]], and thus… | * Approximates the [[256/225|Neapolitan Diminished Third]], and thus… | ||
Line 463: | Line 529: | ||
* Approximates a complex 5-limit interval formed by stacking a syntonic comma on top of a Pythagorean Major Second, and thus… | * 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 | :* 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 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 | :* 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 | ||
Line 472: | Line 537: | ||
| Greater Supermajor Second, Narrow Inframinor Third | | Greater Supermajor Second, Narrow Inframinor Third | ||
| Fd<, Et<↓, E↑/ | | Fd<, Et<↓, E↑/ | ||
| 0 | |||
| 9 | |||
| This interval… | | This interval… | ||
* Approximates the [[8/7|Septimal Supermajor Second]] or Octave-Reduced Seventh Subharmonic, and as such… | * Approximates the [[8/7|Septimal Supermajor Second]] or Octave-Reduced Seventh Subharmonic, and as such… | ||
Line 488: | Line 555: | ||
| Inframinor Third, Wide Supermajor Second | | Inframinor Third, Wide Supermajor Second | ||
| Fd>, Et>↓ | | Fd>, Et>↓ | ||
| -1 | |||
| 8 | |||
| This interval… | | This interval… | ||
* Approximates a complex 11-limit Paradiatonic interval that functions as a syntactic third that sounds more like a second, and as such… | * Approximates a complex 11-limit Paradiatonic interval that functions as a syntactic third that sounds more like a second, and as such… | ||
Line 501: | Line 570: | ||
| Wide Inframinor Third, Narrow Ultramajor Second, Semifourth | | Wide Inframinor Third, Narrow Ultramajor Second, Semifourth | ||
| Fd>/, Et<\, F↓↓, E↑↑ | | Fd>/, Et<\, F↓↓, E↑↑ | ||
| 0 | |||
| 8 | |||
| This interval… | | This interval… | ||
* Approximates the [[15/13|Tridecimal Semifourth]], and thus… | * Approximates the [[15/13|Tridecimal Semifourth]], and thus… | ||
Line 515: | Line 586: | ||
| Ultramajor Second, Narrow Subminor Third | | Ultramajor Second, Narrow Subminor Third | ||
| Et<, Fd<↑ | | Et<, Fd<↑ | ||
| -1 | |||
| 7 | |||
| This interval… | | This interval… | ||
* Approximates a complex 11-limit Paradiatonic interval that functions as a syntactic second that sounds more like a third, and as such… | * Approximates a complex 11-limit Paradiatonic interval that functions as a syntactic second that sounds more like a third, and as such… | ||
Line 527: | Line 600: | ||
| Lesser Subminor Third, Wide Ultramajor Second | | Lesser Subminor Third, Wide Ultramajor Second | ||
| Et>, Fd>↑, F↓\ | | Et>, Fd>↑, F↓\ | ||
| 0 | |||
| 7 | |||
| This interval… | | This interval… | ||
* Approximates the [[7/6|Septimal Subminor Third]], and as such… | * Approximates the [[7/6|Septimal Subminor Third]], and as such… | ||
Line 542: | Line 617: | ||
| Greater Subminor Third | | Greater Subminor Third | ||
| F↓, Et>/, E#↓↓, Gbb | | F↓, Et>/, E#↓↓, Gbb | ||
| -1 | |||
| 7 | |||
| This interval… | | This interval… | ||
* Approximates the [[75/64|Classic Augmented Second]], and as such… | * Approximates the [[75/64|Classic Augmented Second]], and as such… | ||
Line 557: | Line 634: | ||
| Wide Subminor Third | | Wide Subminor Third | ||
| F↓/, Et<↑ | | F↓/, Et<↑ | ||
| -1 | |||
| 8 | |||
| This interval… | | This interval… | ||
* Approximates the [[20/17|Septendecimal Minor Third]] | * Approximates the [[20/17|Septendecimal Minor Third]] | ||
Line 569: | Line 648: | ||
| Narrow Minor Third | | Narrow Minor Third | ||
| F\, Et>↑ | | F\, Et>↑ | ||
| 0 | |||
| 8 | |||
| This interval… | | This interval… | ||
* Approximates the [[13/11|Neo-Gothic Minor Third]], and thus… | * Approximates the [[13/11|Neo-Gothic Minor Third]], and thus… | ||
Line 575: | Line 656: | ||
::* When it is found in what is otherwise the traditional Diatonic context of a Minor key | ::* 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 one third of this system's approximation of the Greater Tridecimal Neutral Sixth | ||
* Is very useful for essentially tempered chords such as [[gentle chords]], [[ | * Is very useful for essentially tempered chords such as [[gentle chords]], [[ainismic chords]] and [[nicolic chords]] | ||
|- | |- | ||
| 39 | | 39 | ||
Line 582: | Line 663: | ||
| Pythagorean Minor Third | | Pythagorean Minor Third | ||
| F | | F | ||
| -1 | |||
| 9 | |||
| This interval… | | This interval… | ||
* Approximates the [[32/27|Pythagorean Minor Third]], and as such… | * Approximates the [[32/27|Pythagorean Minor Third]], and as such… | ||
Line 591: | Line 674: | ||
* 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 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 | * Is reachable through stacking three of this system's approximation of the Axirabian Limma | ||
|- | |- | ||
| 40 | | 40 | ||
| 301.8867925 | | 301.8867925 | ||
Line 597: | Line 680: | ||
| Artomean Minor Third | | Artomean Minor Third | ||
| F/ | | F/ | ||
| 1 | |||
| 9 | |||
| This interval… | | This interval… | ||
* Approximates the [[25/21|Quasi-Tempered Minor Third]], and as such… | * Approximates the [[25/21|Quasi-Tempered Minor Third]], and as such… | ||
Line 609: | Line 694: | ||
| Tendomean Minor Third | | Tendomean Minor Third | ||
| F↑\ | | F↑\ | ||
| 4 | |||
| 10 | |||
| This interval… | | This interval… | ||
* Approximates the [[153/128|Septendecimal Tendomean Minor Third]] | * Approximates the [[153/128|Septendecimal Tendomean Minor Third]] | ||
Line 623: | Line 710: | ||
| Ptolemaic Minor Third | | Ptolemaic Minor Third | ||
| F↑, E# | | F↑, E# | ||
| 7 | |||
| 10 | |||
| This interval… | | This interval… | ||
* Approximates the [[6/5|Classic Minor Third]], and as such… | * Approximates the [[6/5|Classic Minor Third]], and as such… | ||
Line 639: | Line 728: | ||
| Wide Minor Third | | Wide Minor Third | ||
| Ft<↓, F↑/, Gdb< | | Ft<↓, F↑/, Gdb< | ||
| 4 | |||
| 9 | |||
| This interval… | | This interval… | ||
* Approximates the [[135/112|Marvelous Minor Third]], and as such… | * Approximates the [[135/112|Marvelous Minor Third]], and as such… | ||
Line 652: | Line 743: | ||
| Lesser Supraminor Third, Infra-Diminished Fourth | | Lesser Supraminor Third, Infra-Diminished Fourth | ||
| Ft>↓, Gdb> | | Ft>↓, Gdb> | ||
| 1 | |||
| 9 | |||
| This interval… | | This interval… | ||
* Approximates the [[40/33|Undecimal Supraminor Third]], and thus… | * Approximates the [[40/33|Undecimal Supraminor Third]], and thus… | ||
Line 661: | Line 754: | ||
| 339.6226415 | | 339.6226415 | ||
| KKm3, rn3, Rud4 | | KKm3, rn3, Rud4 | ||
| Greater Supraminor Third, | | Greater Supraminor Third, Retrodiptolemaic Diminished Fourth | ||
| Ft<\, F↑↑, Gdb<↑\, Gb↓↓ | | Ft<\, F↑↑, Gdb<↑\, Gb↓↓ | ||
| -1 | |||
| 8 | |||
| This interval… | | This interval… | ||
* Approximates the [[39/32|Lesser Tridecimal Neutral Third]], and thus… | * Approximates the [[39/32|Lesser Tridecimal Neutral Third]], and thus… | ||
Line 678: | Line 773: | ||
| Artoneutral Third, Lesser Sub-Diminished Fourth | | Artoneutral Third, Lesser Sub-Diminished Fourth | ||
| Ft<, Gdb<↑ | | Ft<, Gdb<↑ | ||
| 0 | |||
| 7 | |||
| This interval… | | This interval… | ||
* Approximates the [[11/9|Alpharabian Artoneutral Third]], which is the traditional, low complexity Undecimal Neutral Third, and as such… | * Approximates the [[11/9|Alpharabian Artoneutral Third]], which is the traditional, low complexity Undecimal Neutral Third, and as such… | ||
Line 696: | Line 793: | ||
| Tendoneutral Third, Greater Sub-Diminished Fourth | | Tendoneutral Third, Greater Sub-Diminished Fourth | ||
| Ft>, Gdb>↑ | | Ft>, Gdb>↑ | ||
| -1 | |||
| 7 | |||
| This interval… | | This interval… | ||
* Approximates the [[27/22|Alpharabian Tendoneutral Third]] or 2nd Undecimal Neutral Third, and as such… | * Approximates the [[27/22|Alpharabian Tendoneutral Third]] or 2nd Undecimal Neutral Third, and as such… | ||
Line 711: | Line 810: | ||
| Lesser Submajor Third, Retroptolemaic Diminished Fourth | | Lesser Submajor Third, Retroptolemaic Diminished Fourth | ||
| Ft>/, F#↓↓, Gb↓ | | Ft>/, F#↓↓, Gb↓ | ||
| 0 | |||
| 8 | |||
| This interval | | This interval | ||
* Approximates the [[16/13|Greater Tridecimal Neutral Third]] or Octave-Reduced Thirteenth Subharmonic, and as such… | * Approximates the [[16/13|Greater Tridecimal Neutral Third]] or Octave-Reduced Thirteenth Subharmonic, and as such… | ||
Line 724: | Line 825: | ||
| Greater Submajor Third, Artoretromean Diminished Fourth | | Greater Submajor Third, Artoretromean Diminished Fourth | ||
| Ft<↑, Gb↓/ | | Ft<↑, Gb↓/ | ||
| -1 | |||
| 9 | |||
| This interval… | | This interval… | ||
* Approximates the [[26/21|Tridecimal Submajor Third]] and a similar 11-limit interval that acts as the Submajor counterpart to the Undecimal Supraminor Third, and thus… | * Approximates the [[26/21|Tridecimal Submajor Third]] and a similar 11-limit interval that acts as the Submajor counterpart to the Undecimal Supraminor Third, and thus… | ||
Line 736: | Line 839: | ||
| Narrow Major Third, Tendoretromean Diminished Fourth | | Narrow Major Third, Tendoretromean Diminished Fourth | ||
| Ft>↑, F#↓\, Gb\ | | Ft>↑, F#↓\, Gb\ | ||
| 3 | |||
| 9 | |||
| This interval… | | This interval… | ||
* Approximates the [[56/45|Marvelous Major Third]], and as such… | * Approximates the [[56/45|Marvelous Major Third]], and as such… | ||
Line 750: | Line 855: | ||
| Ptolemaic Major Third, Pythagorean Diminished Fourth | | Ptolemaic Major Third, Pythagorean Diminished Fourth | ||
| Gb, F#↓ | | Gb, F#↓ | ||
| 8 | |||
| 10 | |||
| This interval… | | This interval… | ||
* Approximates the [[5/4|Classic Major Third]] or Octave-Reduced Fifth Harmonic, and as such… | * Approximates the [[5/4|Classic Major Third]] or Octave-Reduced Fifth Harmonic, and as such… | ||
Line 769: | Line 876: | ||
| Artomean Major Third, Artomean Diminished Fourth | | Artomean Major Third, Artomean Diminished Fourth | ||
| Gb/, F#↓/ | | Gb/, F#↓/ | ||
| 4 | |||
| 10 | |||
| This interval… | | This interval… | ||
* Approximates the [[64/51|Septendecimal Artomean Major Third]] | * Approximates the [[64/51|Septendecimal Artomean Major Third]] | ||
Line 779: | Line 888: | ||
| Tendomean Major Third, Tendomean Diminished Fourth | | Tendomean Major Third, Tendomean Diminished Fourth | ||
| F#\, Gb↑\ | | F#\, Gb↑\ | ||
| 1 | |||
| 9 | |||
| This interval… | | This interval… | ||
* Approximates the [[63/50|Quasi-Tempered Major Third]] | * Approximates the [[63/50|Quasi-Tempered Major Third]] | ||
Line 793: | Line 904: | ||
| Pythagorean Major Third, Ptolemaic Diminished Fourth | | Pythagorean Major Third, Ptolemaic Diminished Fourth | ||
| F#, Gb↑ | | F#, Gb↑ | ||
| -1 | |||
| 9 | |||
| This interval… | | This interval… | ||
* Approximates the [[81/64|Pythagorean Major Third]], and as such… | * Approximates the [[81/64|Pythagorean Major Third]], and as such… | ||
Line 810: | Line 923: | ||
| Wide Major Third, Lesser Super-Diminished Fourth | | Wide Major Third, Lesser Super-Diminished Fourth | ||
| F#/, Gd<↓, Gb↑/ | | F#/, Gd<↓, Gb↑/ | ||
| 0 | |||
| 8 | |||
| This interval… | | This interval… | ||
* Approximates the [[14/11|Neo-Gothic Major Third]], and thus… | * Approximates the [[14/11|Neo-Gothic Major Third]], and thus… | ||
Line 824: | Line 939: | ||
| Narrow Supermajor Third, Greater Super-Diminished Fourth | | Narrow Supermajor Third, Greater Super-Diminished Fourth | ||
| F#↑\, Gd>↓ | | F#↑\, Gd>↓ | ||
| -1 | |||
| 7 | |||
| This interval… | | This interval… | ||
* Approximates the [[51/40|Septendecimal Major Third]] | * Approximates the [[51/40|Septendecimal Major Third]] | ||
Line 836: | Line 953: | ||
| Lesser Supermajor Third, Diptolemaic Diminished Fourth | | Lesser Supermajor Third, Diptolemaic Diminished Fourth | ||
| F#↑, Gd<\, Gb↑↑ | | F#↑, Gd<\, Gb↑↑ | ||
| -1 | |||
| 6 | |||
| This interval… | | This interval… | ||
* Approximates the [[32/25|Classic Diminished Fourth]] or Diptolemaic Diminished Fourth, and thus… | * Approximates the [[32/25|Classic Diminished Fourth]] or Diptolemaic Diminished Fourth, and thus… | ||
:* It is easily very useful when it comes to building chords | :* 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… | * 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 | :* It can be thought of as a type of third when acting in this capacity | ||
Line 848: | Line 967: | ||
| Greater Supermajor Third, Ultra-Diminished Fourth | | Greater Supermajor Third, Ultra-Diminished Fourth | ||
| Gd<, F#↑/ | | Gd<, F#↑/ | ||
| 0 | |||
| 5 | |||
| This interval… | | This interval… | ||
* Approximates the [[9/7|Septimal Supermajor Third]], and as such… | * Approximates the [[9/7|Septimal Supermajor Third]], and as such… | ||
Line 861: | Line 982: | ||
| Paraminor Fourth, Wide Supermajor Third | | Paraminor Fourth, Wide Supermajor Third | ||
| Gd>, Ft#>↓ | | Gd>, Ft#>↓ | ||
| -1 | |||
| 3 | |||
| This interval… | | This interval… | ||
* Approximates the [[128/99|Just Paraminor Fourth]], and as such… | * Approximates the [[128/99|Just Paraminor Fourth]], and as such… | ||
Line 868: | Line 991: | ||
::* 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 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 | ::* 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 | * 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 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 | :* 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 | ||
Line 877: | Line 1,000: | ||
| Wide Paraminor Fourth, Narrow Ultramajor Third | | Wide Paraminor Fourth, Narrow Ultramajor Third | ||
| Gd>/, F#↑↑, G↓↓ | | Gd>/, F#↑↑, G↓↓ | ||
| -2 | |||
| 1 | |||
| This interval… | | This interval… | ||
* Approximates the [[13/10|Tridecimal Semisixth]] | * Approximates the [[13/10|Tridecimal Semisixth]] | ||
Line 888: | Line 1,013: | ||
| Ultramajor Third, Narrow Grave Fourth | | Ultramajor Third, Narrow Grave Fourth | ||
| Gd<↑, Ft#< | | Gd<↑, Ft#< | ||
| -4 | |||
| -2 | |||
| This interval… | | This interval… | ||
* Approximates a complex 11-limit Paradiatonic interval that functions as a syntactic third that sounds more like a fourth, and as such… | * Approximates a complex 11-limit Paradiatonic interval that functions as a syntactic third that sounds more like a fourth, and as such… | ||
Line 901: | Line 1,028: | ||
| Lesser Grave Fourth, Wide Ultramajor Third | | Lesser Grave Fourth, Wide Ultramajor Third | ||
| Gd>↑, G↓\ | | Gd>↑, G↓\ | ||
| -7 | |||
| -4 | |||
| This Interval… | | This Interval… | ||
* Approximates the [[21/16|Septimal Subfourth]], and thus… | * Approximates the [[21/16|Septimal Subfourth]], and thus… | ||
Line 913: | Line 1,042: | ||
| Greater Grave Fourth | | Greater Grave Fourth | ||
| G↓ | | G↓ | ||
| -6 | |||
| -5 | |||
| This interval… | | This interval… | ||
* Approximates a complex 5-limit interval formed by subtracting a syntonic comma from a Perfect Fourth | * Approximates a complex 5-limit interval formed by subtracting a syntonic comma from a Perfect Fourth | ||
Line 923: | Line 1,054: | ||
| Wide Grave Fourth | | Wide Grave Fourth | ||
| G↓/ | | G↓/ | ||
| -4 | |||
| 0 | |||
| This interval… | | This interval… | ||
* Is one half of this system's approximation of the Octave-Reduced Seventh Harmonic | * Is one half of this system's approximation of the Octave-Reduced Seventh Harmonic | ||
Line 934: | Line 1,067: | ||
| Narrow Fourth | | Narrow Fourth | ||
| G\ | | G\ | ||
| 1 | |||
| 5 | |||
| This interval… | | This interval… | ||
* Approximates the [[85/64|Septendecimal Fourth]], and thus… | * Approximates the [[85/64|Septendecimal Fourth]], and thus… | ||
Line 947: | Line 1,082: | ||
| Perfect Fourth | | Perfect Fourth | ||
| G | | G | ||
| 9 | |||
| 10 | |||
| This interval… | | This interval… | ||
* Approximates the [[4/3|Perfect Fourth]] or Octave-Reduced Third Subharmonic, and as such… | * Approximates the [[4/3|Perfect Fourth]] or Octave-Reduced Third Subharmonic, and as such… | ||
Line 963: | Line 1,100: | ||
:* Virtually all of its functionality in the realm of Western-Classical-based Diatonic scales and Diatonic functional harmony | :* 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 elements to its functionality include… | ||
:* New approaches enabled by this system supporting temperaments such as [[ | :* New approaches enabled by this system supporting temperaments such as [[sextilifourths]] | ||
:* A sizable chunk of its functionality in the realm of Western-Classical-based Paradiatonic functional harmony | :* A sizable chunk of its functionality in the realm of Western-Classical-based Paradiatonic functional harmony | ||
|- | |- | ||
Line 971: | Line 1,108: | ||
| Wide Fourth | | Wide Fourth | ||
| G/ | | G/ | ||
| 1 | |||
| 8 | |||
| This interval… | | This interval… | ||
* Approximates the [[75/56|Marvelous Fourth]], and thus… | * Approximates the [[75/56|Marvelous Fourth]], and thus… | ||
Line 984: | Line 1,123: | ||
| Narrow Acute Fourth | | Narrow Acute Fourth | ||
| G↑\ | | G↑\ | ||
| -3 | |||
| 6 | |||
| This interval… | | This interval… | ||
* Approximates a complex 11-limit interval, which, in this system… | * Approximates a complex 11-limit interval, which, in this system… | ||
Line 996: | Line 1,137: | ||
| Lesser Acute Fourth | | Lesser Acute Fourth | ||
| G↑ | | G↑ | ||
| -5 | |||
| 5 | |||
| This interval… | | This interval… | ||
* Approximates the [[27/20|Classic Acute Fourth]], and as such… | * Approximates the [[27/20|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 | :* 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 [[Zarlino|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 | ::* Specifically, it is found between the Major Third and Major Sixth in the Lydian Mode of the familiar [[Zarlino|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 | ::* 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 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 | ||
Line 1,010: | Line 1,153: | ||
| Greater Acute Fourth | | Greater Acute Fourth | ||
| Gt<↓, G↑/, Adb< | | Gt<↓, G↑/, Adb< | ||
| -3 | |||
| 5 | |||
| This interval… | | This interval… | ||
* Is reachable through stacking two of this system's approximation of the Septimal Subminor Third | * Is reachable through stacking two of this system's approximation of the Septimal Subminor Third | ||
Line 1,021: | Line 1,166: | ||
| Wide Acute Fourth, Infra-Diminished Fifth | | Wide Acute Fourth, Infra-Diminished Fifth | ||
| Gt>↓, Adb> | | Gt>↓, Adb> | ||
| -2 | |||
| 5 | |||
| This interval… | | This interval… | ||
* Approximates the [[15/11|Undecimal Grave Infra-Augmented Fourth]], and thus… | * Approximates the [[15/11|Undecimal Grave Infra-Augmented Fourth]], and thus… | ||
Line 1,032: | Line 1,179: | ||
| 543.3962264 | | 543.3962264 | ||
| rM4, Rud5 | | rM4, Rud5 | ||
| Narrow Paramajor Fourth, | | Narrow Paramajor Fourth, Retrodiptolemaic Diminished Fifth | ||
| Gt<\, G↑↑, Ab↓↓ | | Gt<\, G↑↑, Ab↓↓ | ||
| -1 | |||
| 6 | |||
| This interval… | | This interval… | ||
* Is reachable through stacking three of this system's approximation of the Classic Major Second……. | * Is reachable through stacking three of this system's approximation of the Classic Major Second……. | ||
Line 1,046: | Line 1,195: | ||
| Paramajor Fourth, Lesser Sub-Diminished Fifth | | Paramajor Fourth, Lesser Sub-Diminished Fifth | ||
| Gt<, Adb<↑ | | Gt<, Adb<↑ | ||
| 0 | |||
| 7 | |||
| This interval… | | This interval… | ||
* Approximates the [[11/8|Just Paramajor Fourth]], and as such… | * Approximates the [[11/8|Just Paramajor Fourth]], and as such… | ||
Line 1,066: | Line 1,217: | ||
| Infra-Augmented Fourth, Greater Sub-Diminished Fifth | | Infra-Augmented Fourth, Greater Sub-Diminished Fifth | ||
| Gt>, Adb>↑ | | Gt>, Adb>↑ | ||
| -2 | |||
| 5 | |||
| This interval… | | This interval… | ||
* Approximates the [[112/81|Septimal Subdiminished Fifth]], and thus… | * Approximates the [[112/81|Septimal Subdiminished Fifth]], and thus… | ||
Line 1,079: | Line 1,232: | ||
| Diptolemaic Augmented Fourth, Retroptolemaic Diminished Fifth | | Diptolemaic Augmented Fourth, Retroptolemaic Diminished Fifth | ||
| G#↓↓, Ab↓ | | G#↓↓, Ab↓ | ||
| -3 | |||
| 4 | |||
| This interval… | | This interval… | ||
* Approximates the [[25/18|Classic Augmented Fourth]], and thus… | * Approximates the [[25/18|Classic Augmented Fourth]], and thus… | ||
Line 1,094: | Line 1,249: | ||
| Lesser Sub-Augmented Fourth, Artoretromean Diminished Fifth | | Lesser Sub-Augmented Fourth, Artoretromean Diminished Fifth | ||
| Gt<↑, Ab↓/ | | Gt<↑, Ab↓/ | ||
| -2 | |||
| 4 | |||
| This interval… | | This interval… | ||
* Approximates a complex 11-limit interval formed by stacking a Syntonic Comma on top of a Paramajor Fourth, and thus… | * Approximates a complex 11-limit interval formed by stacking a Syntonic Comma on top of a Paramajor Fourth, and thus… | ||
Line 1,105: | Line 1,262: | ||
| Greater Sub-Augmented Fourth, Tendoretromean Diminished Fifth | | Greater Sub-Augmented Fourth, Tendoretromean Diminished Fifth | ||
| Gt>↑, Ab\ | | Gt>↑, Ab\ | ||
| 0 | |||
| 5 | |||
| This interval… | | This interval… | ||
* Approximates the [[7/5|Lesser Septimal Tritone]] and thus… | * Approximates the [[7/5|Lesser Septimal Tritone]] and thus… | ||
Line 1,117: | Line 1,276: | ||
| Ptolemaic Augmented Fourth, Pythagorean Diminished Fifth | | Ptolemaic Augmented Fourth, Pythagorean Diminished Fifth | ||
| Ab, G#↓ | | Ab, G#↓ | ||
| -5 | |||
| 6 | |||
| This interval… | | This interval… | ||
* Approximates the [[45/32|Smaller Diatonic Tritone]], and as such… | * Approximates the [[45/32|Smaller Diatonic Tritone]], and as such… | ||
Line 1,130: | Line 1,291: | ||
| Artomean Augmented Fourth, Artomean Diminished Fifth | | Artomean Augmented Fourth, Artomean Diminished Fifth | ||
| G#↓/, Ab/ | | G#↓/, Ab/ | ||
| -9 | |||
| 7 | |||
| This interval… | | This interval… | ||
* Approximates the [[24/17|Smaller Septendecimal Tritone]], and thus… | * Approximates the [[24/17|Smaller Septendecimal Tritone]], and thus… | ||
Line 1,136: | Line 1,299: | ||
* Is one of two intervals that serve as the closest approximation of the Semioctave found in this system, and thus… | * 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 | :* 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 | * Is very useful for essentially tempered chords such as ainismic chords | ||
|- | |- | ||
| 80 | | 80 | ||
Line 1,143: | Line 1,306: | ||
| Tendomean Diminished Fifth, Tendomean Augmented Fourth | | Tendomean Diminished Fifth, Tendomean Augmented Fourth | ||
| Ab↑\, G#\ | | Ab↑\, G#\ | ||
| -9 | |||
| 7 | |||
| This interval… | | This interval… | ||
* Approximates the [[17/12|Larger Septendecimal Tritone]], and thus… | * Approximates the [[17/12|Larger Septendecimal Tritone]], and thus… | ||
Line 1,149: | Line 1,314: | ||
* Is one of two intervals that serve as the closest approximation of the Semioctave found in this system, and thus… | * 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 | :* 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 | * Is very useful for essentially tempered chords such as ainismic chords | ||
|- | |- | ||
| 81 | | 81 | ||
Line 1,156: | Line 1,321: | ||
| Ptolemaic Diminished Fifth, Pythagorean Augmented Fourth | | Ptolemaic Diminished Fifth, Pythagorean Augmented Fourth | ||
| Ab↑, G# | | Ab↑, G# | ||
| -5 | |||
| 6 | |||
| This interval… | | This interval… | ||
* Approximates the [[64/45|Larger Diatonic Tritone]], and as such… | * Approximates the [[64/45|Larger Diatonic Tritone]], and as such… | ||
Line 1,168: | Line 1,335: | ||
| kUd5, RA4 | | kUd5, RA4 | ||
| Lesser Super-Diminished Fifth, Artoretromean Augmented Fourth | | Lesser Super-Diminished Fifth, Artoretromean Augmented Fourth | ||
| Ad<↓, G#/ | | Ad<↓, G#/ | ||
| 0 | |||
| 5 | |||
| This interval… | | This interval… | ||
* Approximates the [[10/7|Greater Septimal Tritone]] and thus… | * Approximates the [[10/7|Greater Septimal Tritone]] and thus… | ||
Line 1,181: | Line 1,350: | ||
| Greater Super-Diminished Fifth, Tendoretromean Augmented Fourth | | Greater Super-Diminished Fifth, Tendoretromean Augmented Fourth | ||
| Ad>↓, G#↑\ | | Ad>↓, G#↑\ | ||
| -2 | |||
| 4 | |||
| This interval… | | This interval… | ||
* Approximates a complex 11-limit interval formed by subtracting a Syntonic Comma from a Paraminor Fifth, and thus… | * Approximates a complex 11-limit interval formed by subtracting a Syntonic Comma from a Paraminor Fifth, and thus… | ||
Line 1,192: | Line 1,363: | ||
| Diptolemaic Diminished Fifth, Retroptolemaic Augmented Fourth | | Diptolemaic Diminished Fifth, Retroptolemaic Augmented Fourth | ||
| Ab↑↑, G#↑ | | Ab↑↑, G#↑ | ||
| -3 | |||
| 4 | |||
| This interval… | | This interval… | ||
* Approximates the [[36/25|Classic Diminished Fifth]], and thus… | * Approximates the [[36/25|Classic Diminished Fifth]], and thus… | ||
Line 1,207: | Line 1,380: | ||
| Ultra-Diminished Fifth, Lesser Super-Augmented Fourth | | Ultra-Diminished Fifth, Lesser Super-Augmented Fourth | ||
| Ad<, Gt#<↓ | | Ad<, Gt#<↓ | ||
| -2 | |||
| 5 | |||
| This interval… | | This interval… | ||
* Approximates the [[81/56|Septimal Superaugmented Fourth]], and thus… | * Approximates the [[81/56|Septimal Superaugmented Fourth]], and thus… | ||
Line 1,219: | Line 1,394: | ||
| Paraminor Fifth, Greater Super-Augmented Fourth | | Paraminor Fifth, Greater Super-Augmented Fourth | ||
| Ad>, Gt#>↓ | | Ad>, Gt#>↓ | ||
| 0 | |||
| 7 | |||
| This interval… | | This interval… | ||
* Approximates the [[16/11|Just Paraminor Fifth]], and as such… | * Approximates the [[16/11|Just Paraminor Fifth]], and as such… | ||
Line 1,236: | Line 1,413: | ||
| 656.6037736 | | 656.6037736 | ||
| Rm5, rUA4 | | Rm5, rUA4 | ||
| Wide Paraminor Fifth, | | Wide Paraminor Fifth, Retrodiptolemaic Augmented Fourth | ||
| Ad | | Ad>/, G#↑, Ab↑↑ | ||
| -1 | |||
| 6 | |||
| This interval… | | This interval… | ||
* Is reachable through stacking three of this system's approximation of the Septendecimal Whole Tone | * Is reachable through stacking three of this system's approximation of the Septendecimal Whole Tone | ||
Line 1,250: | Line 1,429: | ||
| Narrow Grave Fifth, Ultra-Augmented Fourth | | Narrow Grave Fifth, Ultra-Augmented Fourth | ||
| Ad<↑, Gt#< | | Ad<↑, Gt#< | ||
| -2 | |||
| 5 | |||
| This interval… | | This interval… | ||
* Approximates the [[22/15|Undecimal Acute Ultra-Diminished Fifth]], and thus… | * Approximates the [[22/15|Undecimal Acute Ultra-Diminished Fifth]], and thus… | ||
Line 1,263: | Line 1,444: | ||
| Lesser Grave Fifth | | Lesser Grave Fifth | ||
| Ad>↑, A↓\, Gt#> | | Ad>↑, A↓\, Gt#> | ||
| -3 | |||
| 5 | |||
| This interval… | | This interval… | ||
* Is reachable through stacking four of this system's approximation of the Werckismic Subfourth and octave-reducing | * Is reachable through stacking four of this system's approximation of the Werckismic Subfourth and octave-reducing | ||
Line 1,273: | Line 1,456: | ||
| Greater Grave Fifth | | Greater Grave Fifth | ||
| A↓ | | A↓ | ||
| -5 | |||
| 5 | |||
| This interval… | | This interval… | ||
* Approximates the [[40/27|Classic Grave Fifth]], and as such… | * Approximates the [[40/27|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 | :* 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 [[Zarlino|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 | ::* Specifically, it is found between the Minor Third and Minor Seventh in the Locrian Mode of the familiar [[Zarlino|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 | ::* 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 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 | ||
Line 1,287: | Line 1,472: | ||
| Wide Grave Fifth | | Wide Grave Fifth | ||
| A↓/ | | A↓/ | ||
| -3 | |||
| 6 | |||
| This interval… | | This interval… | ||
* Approximates a complex 11-limit interval, which, in this system… | * Approximates a complex 11-limit interval, which, in this system… | ||
Line 1,299: | Line 1,486: | ||
| Narrow Fifth | | Narrow Fifth | ||
| A\ | | A\ | ||
| 1 | |||
| 8 | |||
| This interval… | | This interval… | ||
* Approximates the [[112/75|Marvelous Fifth]], and thus… | * Approximates the [[112/75|Marvelous Fifth]], and thus… | ||
Line 1,313: | Line 1,502: | ||
| Perfect Fifth | | Perfect Fifth | ||
| A | | A | ||
| 9 | |||
| 10 | |||
| This interval… | | This interval… | ||
* Approximates the [[3/2|Perfect Fifth]] or Octave-Reduced Third Harmonic, and as such… | * Approximates the [[3/2|Perfect Fifth]] or Octave-Reduced Third Harmonic, and as such… | ||
Line 1,337: | Line 1,528: | ||
| Wide Fifth | | Wide Fifth | ||
| A/ | | A/ | ||
| 1 | |||
| 5 | |||
| This interval… | | This interval… | ||
* Approximates the [[128/85|Septendecimal Fifth]], and thus… | * Approximates the [[128/85|Septendecimal Fifth]], and thus… | ||
Line 1,351: | Line 1,544: | ||
| Narrow Acute Fifth | | Narrow Acute Fifth | ||
| A↑\ | | A↑\ | ||
| -4 | |||
| 0 | |||
| This interval… | | This interval… | ||
* Is reachable through stacking five of this system's approximation of the 2nd Undecimal Neutral Second | * Is reachable through stacking five of this system's approximation of the 2nd Undecimal Neutral Second | ||
Line 1,362: | Line 1,557: | ||
| Lesser Acute Fifth | | Lesser Acute Fifth | ||
| A↑ | | A↑ | ||
| -6 | |||
| -5 | |||
| This interval… | | This interval… | ||
* Approximates a complex 5-limit interval formed by stacking a syntonic comma on top of a Perfect Fifth | * Approximates a complex 5-limit interval formed by stacking a syntonic comma on top of a Perfect Fifth | ||
Line 1,372: | Line 1,569: | ||
| Greater Acute Fifth, Narrow Inframinor Sixth | | Greater Acute Fifth, Narrow Inframinor Sixth | ||
| At<↓, A↑/ | | At<↓, A↑/ | ||
| -7 | |||
| -4 | |||
| This Interval… | | This Interval… | ||
* Approximates the [[32/21|Septimal Superfifth]], and thus… | * Approximates the [[32/21|Septimal Superfifth]], and thus… | ||
Line 1,383: | Line 1,582: | ||
| Inframinor Sixth, Wide Acute Fifth | | Inframinor Sixth, Wide Acute Fifth | ||
| At>↓, Bdb> | | At>↓, Bdb> | ||
| -4 | |||
| -2 | |||
| This interval… | | This interval… | ||
* Approximates a complex 11-limit Paradiatonic interval that functions as a syntactic sixth that sounds more like a fifth, and as such… | * Approximates a complex 11-limit Paradiatonic interval that functions as a syntactic sixth that sounds more like a fifth, and as such… | ||
Line 1,392: | Line 1,593: | ||
| 99 | | 99 | ||
| 747.1698113 | | 747.1698113 | ||
| Rm4, KKM3, rUM3 | | Rm4, KKM3, rUM3 | ||
| Narrow Paramajor Fifth, Wide Inframinor Sixth | | Narrow Paramajor Fifth, Wide Inframinor Sixth | ||
| At<\, Bb↓↓, A↑↑ | | At<\, Bb↓↓, A↑↑ | ||
| -2 | |||
| 1 | |||
| This interval… | | This interval… | ||
* Approximates the [[20/13|Tridecimal Semitenth]] | * Approximates the [[20/13|Tridecimal Semitenth]] | ||
Line 1,407: | Line 1,609: | ||
| Paramajor Fifth, Narrow Subminor Sixth | | Paramajor Fifth, Narrow Subminor Sixth | ||
| At<, Bdb<↑ | | At<, Bdb<↑ | ||
| -1 | |||
| 3 | |||
| This interval… | | This interval… | ||
* Approximates the [[99/64|Just Paramajor Fifth]], and as such… | * Approximates the [[99/64|Just Paramajor Fifth]], and as such… | ||
Line 1,414: | Line 1,618: | ||
::* 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 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 | ::* 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 | * 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 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 | :* 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 | ||
Line 1,423: | Line 1,627: | ||
| Lesser Subminor Sixth, Infra-Augmented Fifth | | Lesser Subminor Sixth, Infra-Augmented Fifth | ||
| At>, Bb↓\ | | At>, Bb↓\ | ||
| 0 | |||
| 5 | |||
| This interval… | | This interval… | ||
* Approximates the [[14/9|Septimal Subminor Sixth]], and as such… | * Approximates the [[14/9|Septimal Subminor Sixth]], and as such… | ||
Line 1,435: | Line 1,641: | ||
| Greater Subminor Sixth, Diptolemaic Augmented Fifth | | Greater Subminor Sixth, Diptolemaic Augmented Fifth | ||
| Bb↓, At>/, A#↓↓ | | Bb↓, At>/, A#↓↓ | ||
| -1 | |||
| 6 | |||
| This interval… | | This interval… | ||
* Approximates the [[25/16|Classic Augmented Fifth]] or Diptolemaic Augmented Fifth, and thus… | * Approximates the [[25/16|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 | :* 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 | :* 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 | :* 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 | ::* 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… | * Approximates a complex 5-limit interval formed by subtracting a syntonic comma from a Pythagorean Minor Sixth, and thus… | ||
Line 1,450: | Line 1,658: | ||
| Wide Subminor Sixth, Lesser Sub-Augmented Fifth | | Wide Subminor Sixth, Lesser Sub-Augmented Fifth | ||
| Bb↓/, At<↑ | | Bb↓/, At<↑ | ||
| -1 | |||
| 7 | |||
| This interval… | | This interval… | ||
* Approximates the [[80/51|Septendecimal Minor Sixth]] | * Approximates the [[80/51|Septendecimal Minor Sixth]] | ||
Line 1,462: | Line 1,672: | ||
| Narrow Minor Sixth, Greater Sub-Augmented Fifth | | Narrow Minor Sixth, Greater Sub-Augmented Fifth | ||
| Bb\, At>↑, A#↓\ | | Bb\, At>↑, A#↓\ | ||
| 0 | |||
| 8 | |||
| This interval… | | This interval… | ||
* Approximates the [[ | * Approximates the [[11/7|Neo-Gothic Minor Sixth]], and thus… | ||
:* Can be used in Western-Classical-based harmony and Neo-Medieval harmony very easily | :* Can be used in Western-Classical-based harmony and Neo-Medieval harmony very easily | ||
:* Has additional applications in Paradiatonic harmony, particularly… | :* Has additional applications in Paradiatonic harmony, particularly… | ||
Line 1,475: | Line 1,687: | ||
| Pythagorean Minor Sixth, Ptolemaic Augmented Fifth | | Pythagorean Minor Sixth, Ptolemaic Augmented Fifth | ||
| Bb, A#↓ | | Bb, A#↓ | ||
| -1 | |||
| 9 | |||
| This interval… | | This interval… | ||
* Approximates the [[128/81|Pythagorean Minor Sixth]], and as such… | * Approximates the [[128/81|Pythagorean Minor Sixth]], and as such… | ||
Line 1,490: | Line 1,704: | ||
| Artomean Minor Sixth, Artomean Augmented Fifth | | Artomean Minor Sixth, Artomean Augmented Fifth | ||
| Bb/, A#↓/ | | Bb/, A#↓/ | ||
| 1 | |||
| 9 | |||
| This interval… | | This interval… | ||
* Approximates the [[100/63|Quasi-Tempered Minor Sixth]] | * Approximates the [[100/63|Quasi-Tempered Minor Sixth]] | ||
Line 1,502: | Line 1,718: | ||
| Tendomean Minor Sixth, Tendomean Augmented Fifth | | Tendomean Minor Sixth, Tendomean Augmented Fifth | ||
| A#\, Bb↑\ | | A#\, Bb↑\ | ||
| 4 | |||
| 10 | |||
| This interval… | | This interval… | ||
* Approximates the [[51/32|Septendecimal Tendomean Minor Sixth]] | * Approximates the [[51/32|Septendecimal Tendomean Minor Sixth]] | ||
Line 1,511: | Line 1,729: | ||
| Ptolemaic Minor Sixth, Pythagorean Augmented Fifth | | Ptolemaic Minor Sixth, Pythagorean Augmented Fifth | ||
| A#, Bb↑ | | A#, Bb↑ | ||
| 8 | |||
| 10 | |||
| This interval… | | This interval… | ||
* Approximates the [[8/5|Classic Minor Sixth]] or Octave-Reduced Fifth Subharmonic, and as such… | * Approximates the [[8/5|Classic Minor Sixth]] or Octave-Reduced Fifth Subharmonic, and as such… | ||
Line 1,530: | Line 1,750: | ||
|Wide Minor Sixth, Artoretromean Augmented Fifth | |Wide Minor Sixth, Artoretromean Augmented Fifth | ||
| Bd<↓, Bb↑/, A#/ | | Bd<↓, Bb↑/, A#/ | ||
| 3 | |||
| 9 | |||
| This interval… | | This interval… | ||
* Approximates the [[45/28|Marvelous Minor Sixth]], and as such… | * Approximates the [[45/28|Marvelous Minor Sixth]], and as such… | ||
Line 1,543: | Line 1,765: | ||
| Lesser Supraminor Sixth, Tendoretromean Augmented Fifth | | Lesser Supraminor Sixth, Tendoretromean Augmented Fifth | ||
| Bd>↓, A#↑\ | | Bd>↓, A#↑\ | ||
| -1 | |||
| 9 | |||
| This interval… | | This interval… | ||
* Approximates the [[21/13|Tridecimal Supraminor Sixth]] and a similar 11-limit interval that acts as the Supraminor counterpart to the Undecimal Submajor Sixth | * Approximates the [[21/13|Tridecimal Supraminor Sixth]] and a similar 11-limit interval that acts as the Supraminor counterpart to the Undecimal Submajor Sixth | ||
Line 1,554: | Line 1,778: | ||
| Greater Supraminor Sixth, Retroptolemaic Augmented Fifth | | Greater Supraminor Sixth, Retroptolemaic Augmented Fifth | ||
| Bd<\, Bb↑↑, A#↑ | | Bd<\, Bb↑↑, A#↑ | ||
| 0 | |||
| 8 | |||
| This interval | | This interval | ||
* Approximates the [[13/8|Lesser Tridecimal Neutral Sixth]] or Octave-Reduced Thirteenth Harmonic, and as such… | * Approximates the [[13/8|Lesser Tridecimal Neutral Sixth]] or Octave-Reduced Thirteenth Harmonic, and as such… | ||
Line 1,567: | Line 1,793: | ||
| Artoneutral Sixth, Lesser Super-Augmented Fifth | | Artoneutral Sixth, Lesser Super-Augmented Fifth | ||
| Bd<, At#<↓ | | Bd<, At#<↓ | ||
| -1 | |||
| 7 | |||
| This interval… | | This interval… | ||
* Approximates the [[44/27|Alpharabian Artoneutral Sixth]] or 2nd Undecimal Neutral Sixth, and as such… | * Approximates the [[44/27|Alpharabian Artoneutral Sixth]] or 2nd Undecimal Neutral Sixth, and as such… | ||
Line 1,581: | Line 1,809: | ||
| Tendoneutral Sixth, Greater Super-Augmented Fifth | | Tendoneutral Sixth, Greater Super-Augmented Fifth | ||
| Bd>, At#>↓ | | Bd>, At#>↓ | ||
| 0 | |||
| 7 | |||
| This interval… | | This interval… | ||
* Approximates the [[18/11|Alpharabian Tendoneutral Sixth]], which is the traditional, low complexity Undecimal Neutral Sixth, and as such… | * Approximates the [[18/11|Alpharabian Tendoneutral Sixth]], which is the traditional, low complexity Undecimal Neutral Sixth, and as such… | ||
Line 1,597: | Line 1,827: | ||
| 860.3773585 | | 860.3773585 | ||
| kkM6, RN6, rUA5 | | kkM6, RN6, rUA5 | ||
| Lesser Submajor Sixth, | | Lesser Submajor Sixth, Retrodiptolemaic Augmented Fifth | ||
| Bd>/, B↓↓, At#>↓/, A#↑↑ | | Bd>/, B↓↓, At#>↓/, A#↑↑ | ||
| -1 | |||
| 8 | |||
| This interval… | | This interval… | ||
* Approximates the [[64/39|Greater Tridecimal Neutral Sixth]] | * Approximates the [[64/39|Greater Tridecimal Neutral Sixth]] | ||
Line 1,613: | Line 1,845: | ||
| Greater Submajor Sixth, Ultra-Augmented Fifth | | Greater Submajor Sixth, Ultra-Augmented Fifth | ||
| Bd<↑, At#< | | Bd<↑, At#< | ||
| 1 | |||
| 9 | |||
| This interval… | | This interval… | ||
* Approximates the [[33/20|Undecimal Submajor Sixth]] | * Approximates the [[33/20|Undecimal Submajor Sixth]] | ||
Line 1,622: | Line 1,856: | ||
| Narrow Major Sixth | | Narrow Major Sixth | ||
| Bd>↑, B↓\, At#> | | Bd>↑, B↓\, At#> | ||
| 4 | |||
| 9 | |||
| This interval… | | This interval… | ||
* Approximates the [[224/135|Marvelous Major Sixth]], and as such… | * Approximates the [[224/135|Marvelous Major Sixth]], and as such… | ||
Line 1,633: | Line 1,869: | ||
| Ptolemaic Major Sixth | | Ptolemaic Major Sixth | ||
| B↓, Cb | | B↓, Cb | ||
| 7 | |||
| 10 | |||
| This interval… | | This interval… | ||
* Approximates the [[5/3|Classic Major Sixth]], and as such… | * Approximates the [[5/3|Classic Major Sixth]], and as such… | ||
Line 1,650: | Line 1,888: | ||
| Artomean Major Sixth | | Artomean Major Sixth | ||
| B↓/ | | B↓/ | ||
| 4 | |||
| 10 | |||
| This interval… | | This interval… | ||
* Approximates the [[256/153|Septendecimal Artomean Major Sixth]] | * Approximates the [[256/153|Septendecimal Artomean Major Sixth]] | ||
Line 1,661: | Line 1,901: | ||
| Tendomean Major Sixth | | Tendomean Major Sixth | ||
| B\ | | B\ | ||
| 1 | |||
| 9 | |||
| This interval… | | This interval… | ||
* Approximates the [[42/25|Quasi-Tempered Major Sixth]], and as such… | * Approximates the [[42/25|Quasi-Tempered Major Sixth]], and as such… | ||
Line 1,672: | Line 1,914: | ||
| Pythagorean Major Sixth | | Pythagorean Major Sixth | ||
| B | | B | ||
| -1 | |||
| 9 | |||
| This interval… | | This interval… | ||
* Approximates the [[27/16|Pythagorean Major Sixth]], and as such… | * Approximates the [[27/16|Pythagorean Major Sixth]], and as such… | ||
Line 1,688: | Line 1,932: | ||
| Wide Major Sixth | | Wide Major Sixth | ||
| B/, Cd<↓ | | B/, Cd<↓ | ||
| 0 | |||
| 8 | |||
| This interval… | | This interval… | ||
* Approximates the [[22/13|Neo-Gothic Major Sixth]], and thus… | * Approximates the [[22/13|Neo-Gothic Major Sixth]], and thus… | ||
Line 1,693: | Line 1,939: | ||
:* Has additional applications in Paradiatonic harmony, particularly… | :* Has additional applications in Paradiatonic harmony, particularly… | ||
::* When it is found in what is otherwise the traditional Diatonic context of a Major key | ::* 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, | * Is very useful for essentially tempered chords such as gentle chords, ainismic chords and nicolic chords | ||
|- | |- | ||
| 122 | | 122 | ||
Line 1,700: | Line 1,946: | ||
| Narrow Supermajor Sixth | | Narrow Supermajor Sixth | ||
| B↑\, Cd>↓ | | B↑\, Cd>↓ | ||
| -1 | |||
| 8 | |||
| This interval… | | This interval… | ||
* Approximates the [[17/10|Septendecimal Major Sixth]] | * Approximates the [[17/10|Septendecimal Major Sixth]] | ||
Line 1,712: | Line 1,960: | ||
| Lesser Supermajor Sixth | | Lesser Supermajor Sixth | ||
| B↑, Cd<\, Cb↑↑, A## | | B↑, Cd<\, Cb↑↑, A## | ||
| -1 | |||
| 7 | |||
| This interval… | | This interval… | ||
* Approximates the [[128/75|Classic Diminished Seventh]], and as such… | * Approximates the [[128/75|Classic Diminished Seventh]], and as such… | ||
Line 1,726: | Line 1,976: | ||
| SM6, kUM6 | | SM6, kUM6 | ||
| Greater Supermajor Second, Narrow Inframinor Seventh | | Greater Supermajor Second, Narrow Inframinor Seventh | ||
| | | Cd<, Bt<↓, B↑/ | ||
| 0 | |||
| 7 | |||
| This interval… | | This interval… | ||
* Approximates the [[12/7|Septimal Supermajor Sixth]], and as such… | * Approximates the [[12/7|Septimal Supermajor Sixth]], and as such… | ||
Line 1,741: | Line 1,993: | ||
| Inframinor Seventh, Wide Supermajor Sixth | | Inframinor Seventh, Wide Supermajor Sixth | ||
| Cd>, Bt>↓ | | Cd>, Bt>↓ | ||
| -1 | |||
| 7 | |||
| This interval… | | This interval… | ||
* Approximates a complex 11-limit Paradiatonic interval that functions as a syntactic seventh that sounds more like a sixth, and as such… | * Approximates a complex 11-limit Paradiatonic interval that functions as a syntactic seventh that sounds more like a sixth, and as such… | ||
Line 1,752: | Line 2,006: | ||
| Narrow Ultramajor Sixth, Wide Inframinor Seventh, Semitwelfth | | Narrow Ultramajor Sixth, Wide Inframinor Seventh, Semitwelfth | ||
| Bt<\, Cd>/, B↑↑, C↓↓ | | Bt<\, Cd>/, B↑↑, C↓↓ | ||
| 0 | |||
| 8 | |||
| This interval… | | This interval… | ||
* Approximates the [[26/15|Tridecimal Semitwelfth]], and thus… | * Approximates the [[26/15|Tridecimal Semitwelfth]], and thus… | ||
Line 1,766: | Line 2,022: | ||
| Ultramajor Sixth, Narrow Subminor Seventh | | Ultramajor Sixth, Narrow Subminor Seventh | ||
| Bt<, Cd<↑ | | Bt<, Cd<↑ | ||
| -1 | |||
| 8 | |||
| This interval… | | This interval… | ||
* Approximates a complex 11-limit Paradiatonic interval that functions as a syntactic sixth that sounds more like a seventh, and as such… | * Approximates a complex 11-limit Paradiatonic interval that functions as a syntactic sixth that sounds more like a seventh, and as such… | ||
Line 1,778: | Line 2,036: | ||
| Lesser Subminor Seventh, Wide Ultramajor Sixth | | Lesser Subminor Seventh, Wide Ultramajor Sixth | ||
| Bt>, Cd>↑, C↓\ | | Bt>, Cd>↑, C↓\ | ||
| 0 | |||
| 9 | |||
| This interval… | | This interval… | ||
* Approximates the [[7/4|Septimal Subminor Seventh]] or Octave-Reduced Seventh Harmonic, and as such… | * Approximates the [[7/4|Septimal Subminor Seventh]] or Octave-Reduced Seventh Harmonic, and as such… | ||
Line 1,793: | Line 2,053: | ||
| Greater Subminor Seventh | | Greater Subminor Seventh | ||
| C↓, Bt>/, B#↓↓, Dbb | | C↓, Bt>/, B#↓↓, Dbb | ||
| -1 | |||
| 9 | |||
| This interval… | | This interval… | ||
* Approximates the [[225/128|Neapolitan Augmented Sixth]], and thus… | * Approximates the [[225/128|Neapolitan Augmented Sixth]], and thus… | ||
Line 1,798: | Line 2,060: | ||
* Approximates a complex 5-limit interval formed by subtracting a syntonic comma from a Pythagorean Minor Seventh, and thus… | * 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 | :* 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 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 | :* 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 | ||
Line 1,807: | Line 2,068: | ||
| Wide Subminor Seventh | | Wide Subminor Seventh | ||
| C↓/, Bt<↑ | | C↓/, Bt<↑ | ||
| -1 | |||
| 10 | |||
| This interval… | | This interval… | ||
* Approximates the [[30/17|Septendecimal Minor Seventh]], and thus… | * Approximates the [[30/17|Septendecimal Minor Seventh]], and thus… | ||
Line 1,815: | Line 2,078: | ||
* Is the closest approximation of 22edo's Lesser Minor Seventh in this system, and thus… | * 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 | :* 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 | | 131 | ||
| 988.6792458 | | 988.6792458 | ||
Line 1,821: | Line 2,084: | ||
| Narrow Minor Seventh | | Narrow Minor Seventh | ||
| C\, Bt>↑ | | C\, Bt>↑ | ||
| -1 | |||
| 10 | |||
| This interval… | | This interval… | ||
* Approximates the [[39/22|Tridecimal Minor Seventh]], and thus… | * Approximates the [[39/22|Tridecimal Minor Seventh]], and thus… | ||
Line 1,832: | Line 2,097: | ||
| Pythagorean Minor Seventh | | Pythagorean Minor Seventh | ||
| C, B#↓ | | C, B#↓ | ||
| -2 | |||
| 10 | |||
| This interval… | | This interval… | ||
* Approximates the [[16/9|Pythagorean Minor Seventh]], and as such… | * Approximates the [[16/9|Pythagorean Minor Seventh]], and as such… | ||
Line 1,847: | Line 2,114: | ||
| Artomean Minor Seventh | | Artomean Minor Seventh | ||
| C/, B#↓/ | | C/, B#↓/ | ||
| -2 | |||
| 10 | |||
| This interval… | | This interval… | ||
* Approximates the [[25/14|Middle Minor Seventh]] | * Approximates the [[25/14|Middle Minor Seventh]] | ||
Line 1,859: | Line 2,128: | ||
| Tendomean Minor Seventh | | Tendomean Minor Seventh | ||
| C↑\, B#\ | | C↑\, B#\ | ||
| -3 | |||
| 10 | |||
| This interval… | | This interval… | ||
* Approximates the [[256/143|Grossmic Minor Seventh]], and thus… | * Approximates the [[256/143|Grossmic Minor Seventh]], and thus… | ||
Line 1,870: | Line 2,141: | ||
| Ptolemaic Minor Seventh | | Ptolemaic Minor Seventh | ||
| C↑, B# | | C↑, B# | ||
| -3 | |||
| 10 | |||
| This interval… | | This interval… | ||
* Approximates the [[9/5|Classic Minor Seventh]] or Ptolemaic Minor Seventh, and as such… | * Approximates the [[9/5|Classic Minor Seventh]] or Ptolemaic Minor Seventh, and as such… | ||
Line 1,885: | Line 2,158: | ||
| Wide Minor Seventh | | Wide Minor Seventh | ||
| Ct<↓, C↑/, Ddb<, B#/ | | Ct<↓, C↑/, Ddb<, B#/ | ||
| -4 | |||
| 10 | |||
| This interval… | | This interval… | ||
* Is reachable through stacking eight of this system's approximation of the Tridecimal Supraminor Second | * Is reachable through stacking eight of this system's approximation of the Tridecimal Supraminor Second | ||
Line 1,895: | Line 2,170: | ||
| Lesser Supraminor Seventh, Infra-Diminished Octave | | Lesser Supraminor Seventh, Infra-Diminished Octave | ||
| Ct>↓, Ddb>, B#↑\ | | Ct>↓, Ddb>, B#↑\ | ||
| -5 | |||
| 9 | |||
| This interval… | | This interval… | ||
* Approximates the [[20/11|Undecimal Supraminor Seventh]] and a similar 13-limit interval that acts as the Supraminor counterpart to the Tridecimal Submajor Seventh | * Approximates the [[20/11|Undecimal Supraminor Seventh]] and a similar 13-limit interval that acts as the Supraminor counterpart to the Tridecimal Submajor Seventh | ||
Line 1,906: | Line 2,183: | ||
| 1041.5094340 | | 1041.5094340 | ||
| KKm7, rn7, Rud8 | | KKm7, rn7, Rud8 | ||
| Greater Supraminor Seventh, | | Greater Supraminor Seventh, Retrodiptolemaic Diminished Octave | ||
| Ct<\, C↑↑, Ddb<↑\, Db↓↓ | | Ct<\, C↑↑, Ddb<↑\, Db↓↓ | ||
| -6 | |||
| 8 | |||
| This interval… | | This interval… | ||
* Is the closest approximation of 31edo's own Middle Seventh found in this system, and thus… | * Is the closest approximation of 31edo's own Middle Seventh found in this system, and thus… | ||
Line 1,918: | Line 2,197: | ||
| Artoneutral Seventh, Lesser Sub-Diminished Octave | | Artoneutral Seventh, Lesser Sub-Diminished Octave | ||
| Ct<, Ddb<↑ | | Ct<, Ddb<↑ | ||
| -7 | |||
| 6 | |||
| This interval… | | This interval… | ||
* Approximates the [[11/6|Alpharabian Artoneutral Seventh]], which is the traditional, [[low-complexity JI|low complexity]] Undecimal Neutral Seventh, and as such… | * Approximates the [[11/6|Alpharabian Artoneutral Seventh]], which is the traditional, [[low-complexity JI|low complexity]] Undecimal Neutral Seventh, and as such… | ||
Line 1,933: | Line 2,214: | ||
| Tendoneutral Seventh, Greater Sub-Diminished Octave | | Tendoneutral Seventh, Greater Sub-Diminished Octave | ||
| Ct>, Ddb>↑ | | Ct>, Ddb>↑ | ||
| -8 | |||
| 5 | |||
| This interval… | | This interval… | ||
* Approximates the [[81/44|Alpharabian Tendoneutral Seventh]] or 2nd Undecimal Neutral Seventh, and as such… | * Approximates the [[81/44|Alpharabian Tendoneutral Seventh]] or 2nd Undecimal Neutral Seventh, and as such… | ||
Line 1,946: | Line 2,229: | ||
| kkM7, RN7, kd8 | | kkM7, RN7, kd8 | ||
| Lesser Submajor Seventh, Diptolemaic Major Seventh, Retroptolemaic Diminished Octave | | Lesser Submajor Seventh, Diptolemaic Major Seventh, Retroptolemaic Diminished Octave | ||
| Ct>/, C#↓↓, Db↓ | | Ct>/, C#↓↓, Db↓ | ||
| -7 | |||
| 6 | |||
| This interval… | | This interval… | ||
* Approximates the [[50/27|Grave Major Seventh]], and thus… | * Approximates the [[50/27|Grave Major Seventh]], and thus… | ||
Line 1,958: | Line 2,243: | ||
| Greater Submajor Seventh, Artoretromean Diminished Octave | | Greater Submajor Seventh, Artoretromean Diminished Octave | ||
| Ct<↑, Db↓/ | | Ct<↑, Db↓/ | ||
| -6 | |||
| 8 | |||
| This interval… | | This interval… | ||
* Approximates the [[13/7|Tridecimal Submajor Seventh]] and a similar 11-limit interval that acts as the Submajor counterpart to the Undecimal Supraminor Seventh, and thus… | * Approximates the [[13/7|Tridecimal Submajor Seventh]] and a similar 11-limit interval that acts as the Submajor counterpart to the Undecimal Supraminor Seventh, and thus… | ||
Line 1,971: | Line 2,258: | ||
| Narrow Major Seventh, Tendoretromean Diminished Octave | | Narrow Major Seventh, Tendoretromean Diminished Octave | ||
| Ct>↑, C#↓\, Db\ | | Ct>↑, C#↓\, Db\ | ||
| -5 | |||
| 9 | |||
| This interval… | | This interval… | ||
* Approximates the [[28/15|Septimal Grave Major Seventh]], and thus… | * Approximates the [[28/15|Septimal Grave Major Seventh]], and thus… | ||
Line 1,982: | Line 2,271: | ||
| Ptolemaic Major Seventh, Pythagorean Diminished Octave | | Ptolemaic Major Seventh, Pythagorean Diminished Octave | ||
| Db, C#↓ | | Db, C#↓ | ||
| -5 | |||
| 10 | |||
| This interval… | | This interval… | ||
* Approximates the [[15/8|Classic Major Seventh]] or Ptolemaic Major Seventh, and as such… | * Approximates the [[15/8|Classic Major Seventh]] or Ptolemaic Major Seventh, and as such… | ||
Line 1,998: | Line 2,289: | ||
| Artomean Major Seventh, Artomean Diminished Octave | | Artomean Major Seventh, Artomean Diminished Octave | ||
| Db/, C#↓/ | | Db/, C#↓/ | ||
| -5 | |||
| 10 | |||
| This interval… | | This interval… | ||
* Approximates the [[32/17|Small Septendecimal Major Seventh]] or Octave-Reduced Seventeenth Subharmonic, and thus… | * Approximates the [[32/17|Small Septendecimal Major Seventh]] or Octave-Reduced Seventeenth Subharmonic, and thus… | ||
Line 2,012: | Line 2,305: | ||
| Tendomean Major Seventh, Tendomean Diminished Octave | | Tendomean Major Seventh, Tendomean Diminished Octave | ||
| C#\, Db↑\ | | C#\, Db↑\ | ||
| -6 | |||
| 10 | |||
| This interval… | | This interval… | ||
* Approximates the [[17/9|Large Septendecimal Major Seventh]], and thus… | * Approximates the [[17/9|Large Septendecimal Major Seventh]], and thus… | ||
Line 2,025: | Line 2,320: | ||
| Pythagorean Major Seventh, Ptolemaic Diminished Octave | | Pythagorean Major Seventh, Ptolemaic Diminished Octave | ||
| C#, Db↑ | | C#, Db↑ | ||
| -6 | |||
| 10 | |||
| This interval… | | This interval… | ||
* Approximates the [[243/128|Pythagorean Major Seventh]], and as such… | * Approximates the [[243/128|Pythagorean Major Seventh]], and as such… | ||
Line 2,041: | Line 2,338: | ||
| Wide Major Seventh, Lesser Super-Diminished Octave | | Wide Major Seventh, Lesser Super-Diminished Octave | ||
| C#/, Dd<↓ | | C#/, Dd<↓ | ||
| -7 | |||
| 9 | |||
| This interval… | | This interval… | ||
* Approximates the [[40/21|Septimal Acute Major Seventh]], and thus… | * Approximates the [[40/21|Septimal Acute Major Seventh]], and thus… | ||
Line 2,052: | Line 2,351: | ||
| Narrow Supermajor Seventh, Greater Super-Diminished Octave | | Narrow Supermajor Seventh, Greater Super-Diminished Octave | ||
| C#↑\, Dd>↓ | | C#↑\, Dd>↓ | ||
| -7 | |||
| 9 | |||
| This interval… | | This interval… | ||
* Approximates multiple complex [[17-limit]] intervals relative to the Tonic and can be used… | * Approximates multiple complex [[17-limit]] intervals relative to the Tonic and can be used… | ||
Line 2,065: | Line 2,366: | ||
| Lesser Supermajor Seventh, Diptolemaic Diminished Octave | | Lesser Supermajor Seventh, Diptolemaic Diminished Octave | ||
| C#↑, Db↑↑ | | C#↑, Db↑↑ | ||
| -8 | |||
| 9 | |||
| This interval… | | This interval… | ||
* Approximates the [[48/25|Classic Diminished Octave]] or Diptolemaic Diminished Octave, and thus… | * Approximates the [[48/25|Classic Diminished Octave]] or Diptolemaic Diminished Octave, and thus… | ||
Line 2,077: | Line 2,380: | ||
| Greater Supermajor Seventh, Narrow Infraoctave, Ultra-Diminished Octave | | Greater Supermajor Seventh, Narrow Infraoctave, Ultra-Diminished Octave | ||
| Dd<, C#↑/ | | Dd<, C#↑/ | ||
| -8 | |||
| 10 | |||
| This interval… | | This interval… | ||
* Approximates the [[27/14|Septimal Supermajor Seventh]], and thus… | * Approximates the [[27/14|Septimal Supermajor Seventh]], and thus… | ||
Line 2,091: | Line 2,396: | ||
| Infraoctave, Wide Supermajor Seventh | | Infraoctave, Wide Supermajor Seventh | ||
| Dd>, Ct#>↓ | | Dd>, Ct#>↓ | ||
| -9 | |||
| 10 | |||
| This interval… | | This interval… | ||
* Approximates the [[64/33|Alpharabian Infraoctave]], and as such… | * Approximates the [[64/33|Alpharabian Infraoctave]], and as such… | ||
Line 2,109: | Line 2,416: | ||
| Narrow Ultramajor Seventh, Wide Infraoctave | | Narrow Ultramajor Seventh, Wide Infraoctave | ||
| C#↑↑, Dd>/ | | C#↑↑, Dd>/ | ||
| -9 | |||
| 10 | |||
| This interval… | | This interval… | ||
* Approximates the [[39/20|Tridecimal Ultramajor Seventh]] | * Approximates the [[39/20|Tridecimal Ultramajor Seventh]] | ||
Line 2,125: | Line 2,434: | ||
| Ultramajor Seventh, Wide Superprime | | Ultramajor Seventh, Wide Superprime | ||
| Ct#<, Dd<↑ | | Ct#<, Dd<↑ | ||
| -9 | |||
| 10 | |||
| This interval… | | This interval… | ||
* Approximates the [[88/45|Undecimal Suboctave]] | * Approximates the [[88/45|Undecimal Suboctave]] | ||
Line 2,143: | Line 2,454: | ||
| Lesser Suboctave, Wide Ultramajor Seventh | | Lesser Suboctave, Wide Ultramajor Seventh | ||
| Ct#>, Dd>↑ | | Ct#>, Dd>↑ | ||
| -10 | |||
| 3 | |||
| This interval… | | This interval… | ||
* Approximates the [[septimal suboctave|Archytas suboctave]], and thus… | * Approximates the [[septimal suboctave|Archytas suboctave]], and thus… | ||
Line 2,162: | Line 2,475: | ||
| Greater Suboctave | | Greater Suboctave | ||
| D↓ | | D↓ | ||
| -10 | |||
| -3 | |||
| This interval… | | This interval… | ||
* Approximates the [[syntonic suboctave]] | * Approximates the [[syntonic suboctave]] | ||
Line 2,176: | Line 2,491: | ||
| Wide Suboctave | | Wide Suboctave | ||
| D↓/ | | D↓/ | ||
| -10 | |||
| -10 | |||
| This interval… | | This interval… | ||
* Approximates the [[ptolemismic suboctave]] and the [[biyatismic suboctave]] | * Approximates the [[ptolemismic suboctave]] and the [[biyatismic suboctave]] | ||
Line 2,186: | Line 2,503: | ||
| Narrow Octave | | Narrow Octave | ||
| D\ | | D\ | ||
| 0 | |||
| 0 | |||
| This interval… | | This interval… | ||
* Approximates the [[rastmic narrow octave]] | * Approximates the [[rastmic narrow octave]] | ||
Line 2,198: | Line 2,517: | ||
| Perfect Octave | | Perfect Octave | ||
| D | | D | ||
| 10 | |||
| 10 | |||
| This interval… | | This interval… | ||
* Is the [[2/1|Perfect Octave]], and thus… | * Is the [[2/1|Perfect Octave]], and thus… | ||
Line 2,204: | Line 2,525: | ||
:* Is one of four perfect consonances 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 | * Is the most common [[equave]] due in part to the properties human hearing in relation to pitch-chroma matching | ||
|} | |} | ||
Line 2,213: | Line 2,533: | ||
{| class="mw-collapsible mw-collapsed wikitable center-1" | {| class="mw-collapsible mw-collapsed wikitable center-1" | ||
|+ style=white-space:nowrap | Table of 159edo Trines | |+ style="font-size: 105%; white-space: nowrap;" | Table of 159edo Trines | ||
|- | |- | ||
! Name | ! Name | ||
Line 2,472: | Line 2,792: | ||
| 1/(12:17:24) | | 1/(12:17:24) | ||
| This trine is very likely to be used as a partial basis for suspended chords | | This trine is very likely to be used as a partial basis for suspended chords | ||
|} | |} | ||
Next, the triads, which end up inheriting the base trine's type, and as a consequence, there are even more triads than there are trines. Of course, it should be mentioned that suspensions occur where there's overlap between thirds and fourths, and these are excluded from this list. | 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. | ||
{| class="mw-collapsible mw-collapsed wikitable center-1" | {| class="mw-collapsible mw-collapsed wikitable center-1" | ||
|+ style=white-space:nowrap | Table of 159edo Triads | |+ style="font-size: 105%; white-space: nowrap;" | Table of 159edo Triads | ||
|- | |- | ||
! Name | ! Name | ||
Line 2,485: | Line 2,804: | ||
! Approximate JI | ! Approximate JI | ||
! Notes | ! Notes | ||
|- | |||
| | |||
| D, F#↓\, A | |||
| 0, 50, 93 | |||
| | |||
| | |||
|- | |||
| | |||
| D, F↑/, A | |||
| 0, 43, 93 | |||
| | |||
| | |||
|- | |- | ||
| Ptolemaic Major | | Ptolemaic Major | ||
Line 2,495: | Line 2,826: | ||
| D, F↑, A | | D, F↑, A | ||
| 0, 42, 93 | | 0, 42, 93 | ||
| | | 1/(4:5:6) | ||
| This is the second of two triads that can be considered fully-resolved in Western Classical Harmony | | This is the second of two triads that can be considered fully-resolved in Western Classical Harmony | ||
|- | |- | ||
Line 2,501: | Line 2,832: | ||
| D, F#, A | | D, F#, A | ||
| 0, 54, 93 | | 0, 54, 93 | ||
| 64:81 | | 1/(54:64:81) | ||
| This dissonant triad is common in Western Classical, Medieval, and Neo-Medieval Harmony | | This dissonant triad is common in Western Classical, Medieval, and Neo-Medieval Harmony | ||
|- | |- | ||
Line 2,522: | Line 2,853: | ||
| This ambisonant triad is very useful in Neo-Medieval Harmony | | 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 | |||
|} | |} | ||
[[Category:159edo]] | [[Category:159edo]] | ||
[[Category:Interval naming]] | [[Category:Interval naming]] |
Latest revision as of 15:48, 16 May 2025
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 10 is a full-blown friend relative to the root and −10 if a full-blown enemy relative to the root. Note that the Harmonic Compatibility and Melodic Compatibility ratings are based on octave-equivalence, and that some of the ratings are still speculative.
Interval chart
Step | Cents | Interval names | Compatibility rating | Notes | |||
---|---|---|---|---|---|---|---|
Harmonic | Melodic | ||||||
0 | 0 | P1 | Perfect Unison | D | 10 | 10 | This interval…
|
1 | 7.5471698 | R1 | Wide Prime | D/ | 0 | 0 | This interval…
|
2 | 15.0943396 | rK1 | Narrow Superprime | D↑\ | -10 | -10 | This interval…
|
3 | 22.6415094 | K1 | Lesser Superprime | D↑ | -10 | -3 | This interval…
|
4 | 30.1886792 | S1, kU1 | Greater Superprime, Narrow Inframinor Second | Edb<, Dt<↓ | -10 | 3 | This interval…
|
5 | 37.7358491 | um2, RkU1 | Inframinor Second, Wide Superprime | Edb>, Dt>↓ | -9 | 10 | This interval…
|
6 | 45.2830189 | kkm2, Rum2, rU1 | Wide Inframinor Second, Narrow Ultraprime | Eb↓↓, Dt<\ | -9 | 10 | This interval…
|
7 | 52.8301887 | U1, rKum2 | Ultraprime, Narrow Subminor Second | Dt<, Edb<↑ | -9 | 10 | This interval…
|
8 | 60.3773585 | sm2, Kum2, uA1 | Lesser Subminor Second, Wide Ultraprime, Infra-Augmented Prime | Dt>, Eb↓\ | -8 | 10 | This interval…
|
9 | 67.9245283 | km2, RuA1, kkA1 | Greater Subminor Second, Diptolemaic Augmented Prime | Eb↓, D#↓↓ | -8 | 9 | This interval…
|
10 | 75.4716981 | Rkm2, rKuA1 | Wide Subminor Second, Lesser Sub-Augmented Prime | Eb↓/, Dt<↑ | -7 | 9 | This interval…
|
11 | 83.0188679 | rm2, KuA1 | Narrow Minor Second, Greater Sub-Augmented Prime | Eb\, Dt>↑ | -7 | 9 | This interval…
|
12 | 90.5660377 | m2, kA1 | Pythagorean Minor Second, Ptolemaic Augmented Prime | Eb, D#↓ | -6 | 10 | This interval…
|
13 | 98.1132075 | Rm2, RkA1 | Artomean Minor Second, Artomean Augmented Prime | Eb/, D#↓/ | -6 | 10 | This interval…
|
14 | 105.6603774 | rKm2, rA1 | Tendomean Minor Second, Tendomean Augmented Prime | D#\, Eb↑\ | -5 | 10 | This interval…
|
15 | 113.2075472 | Km2, A1 | Ptolemaic Minor Second, Pythagorean Augmented Prime | D#, Eb↑ | -5 | 10 | This interval…
|
16 | 120.7547170 | RKm2, kn2, RA1 | Wide Minor Second, Artoretromean Augmented Prime | Ed<↓, Eb↑/, D#/ | -5 | 9 | This interval…
|
17 | 128.3018868 | kN2, rKA1 | Lesser Supraminor Second, Tendoretromean Augmented Prime | Ed>↓, D#↑\ | -6 | 8 | This interval…
|
18 | 135.8490566 | KKm2, rn2, KA1 | Greater Supraminor Second, Diptolemaic Limma, Retroptolemaic Augmented Prime | Ed<\, Eb↑↑, D#↑ | -7 | 6 | This interval…
|
19 | 143.3962264 | n2, SA1 | Artoneutral Second, Lesser Super-Augmented Prime | Ed<, Dt#<↓ | -8 | 5 | This interval…
|
20 | 150.9433962 | N2, RkUA1 | Tendoneutral Second, Greater Super-Augmented Prime | Ed>, Dt#>↓ | -7 | 6 | This interval…
|
21 | 158.4905660 | kkM2, RN2, rUA1 | Lesser Submajor Second, Retrodiptolemaic Augmented Prime | Ed>/, E↓↓, Dt#>↓/, D#↑↑ | -6 | 8 | This interval…
|
22 | 166.0377358 | Kn2, UA1 | Greater Submajor Second, Ultra-Augmented Prime | Ed<↑, Dt#<, Fb↓/ | -5 | 9 | This interval…
|
23 | 173.5849057 | rkM2, KN2 | Narrow Major Second | Ed>↑, E↓\, Dt#>, Fb\ | -4 | 10 | This interval…
|
24 | 181.1320755 | kM2 | Ptolemaic Major Second | E↓, Fb | -3 | 10 | This interval…
|
25 | 188.6792458 | RkM2 | Artomean Major Second | E↓/, Fb/ | -3 | 10 | This interval…
|
26 | 196.2264151 | rM2 | Tendomean Major Second | E\, Fb↑\ | -2 | 10 | This interval…
|
27 | 203.7735849 | M2 | Pythagorean Major Second | E, Fb↑ | -2 | 10 | This interval…
|
28 | 211.3207547 | RM2 | Wide Major Second | E/, Fd<↓ | -1 | 10 | This interval…
|
29 | 218.8679245 | rKM2 | Narrow Supermajor Second | E↑\, Fd>↓ | -1 | 10 | This interval…
|
30 | 226.4150943 | KM2 | Lesser Supermajor Second | E↑, Fd<\, Fb↑↑, Dx | -1 | 9 | This interval…
|
31 | 233.9622642 | SM2, kUM2 | Greater Supermajor Second, Narrow Inframinor Third | Fd<, Et<↓, E↑/ | 0 | 9 | This interval…
|
32 | 241.5094340 | um3, RkUM2 | Inframinor Third, Wide Supermajor Second | Fd>, Et>↓ | -1 | 8 | This interval…
|
33 | 249.0566038 | kkm3, KKM2, Rum3, rUM2 | Wide Inframinor Third, Narrow Ultramajor Second, Semifourth | Fd>/, Et<\, F↓↓, E↑↑ | 0 | 8 | This interval…
|
34 | 256.6037736 | UM2, rKum3 | Ultramajor Second, Narrow Subminor Third | Et<, Fd<↑ | -1 | 7 | This interval…
|
35 | 264.1509434 | sm3, Kum3 | Lesser Subminor Third, Wide Ultramajor Second | Et>, Fd>↑, F↓\ | 0 | 7 | This interval…
|
36 | 271.6981132 | km3 | Greater Subminor Third | F↓, Et>/, E#↓↓, Gbb | -1 | 7 | This interval…
|
37 | 279.2452830 | Rkm3 | Wide Subminor Third | F↓/, Et<↑ | -1 | 8 | This interval…
|
38 | 286.7924528 | rm3 | Narrow Minor Third | F\, Et>↑ | 0 | 8 | This interval…
|
39 | 294.3396226 | m3 | Pythagorean Minor Third | F | -1 | 9 | This interval…
|
40 | 301.8867925 | Rm3 | Artomean Minor Third | F/ | 1 | 9 | This interval…
|
41 | 309.4339622 | rKm3 | Tendomean Minor Third | F↑\ | 4 | 10 | This interval…
|
42 | 316.9811321 | Km3 | Ptolemaic Minor Third | F↑, E# | 7 | 10 | This interval…
|
43 | 324.5283019 | RKm3, kn3 | Wide Minor Third | Ft<↓, F↑/, Gdb< | 4 | 9 | This interval…
|
44 | 332.0754717 | kN3, ud4 | Lesser Supraminor Third, Infra-Diminished Fourth | Ft>↓, Gdb> | 1 | 9 | This interval…
|
45 | 339.6226415 | KKm3, rn3, Rud4 | Greater Supraminor Third, Retrodiptolemaic Diminished Fourth | Ft<\, F↑↑, Gdb<↑\, Gb↓↓ | -1 | 8 | This interval…
|
46 | 347.1698113 | n3, rKud4 | Artoneutral Third, Lesser Sub-Diminished Fourth | Ft<, Gdb<↑ | 0 | 7 | This interval…
|
47 | 354.7169811 | N3, sd4, Kud4 | Tendoneutral Third, Greater Sub-Diminished Fourth | Ft>, Gdb>↑ | -1 | 7 | This interval…
|
48 | 362.2641509 | kkM3, RN3, kd4 | Lesser Submajor Third, Retroptolemaic Diminished Fourth | Ft>/, F#↓↓, Gb↓ | 0 | 8 | This interval
|
49 | 369.8113208 | Kn3, Rkd4 | Greater Submajor Third, Artoretromean Diminished Fourth | Ft<↑, Gb↓/ | -1 | 9 | This interval…
|
50 | 377.3584906 | rkM3, KN3, rd4 | Narrow Major Third, Tendoretromean Diminished Fourth | Ft>↑, F#↓\, Gb\ | 3 | 9 | This interval…
|
51 | 384.9056604 | kM3, d4 | Ptolemaic Major Third, Pythagorean Diminished Fourth | Gb, F#↓ | 8 | 10 | This interval…
|
52 | 392.4528302 | RkM3, Rd4 | Artomean Major Third, Artomean Diminished Fourth | Gb/, F#↓/ | 4 | 10 | This interval…
|
53 | 400 | rM3, rKd4 | Tendomean Major Third, Tendomean Diminished Fourth | F#\, Gb↑\ | 1 | 9 | This interval…
|
54 | 407.5471698 | M3, Kd4 | Pythagorean Major Third, Ptolemaic Diminished Fourth | F#, Gb↑ | -1 | 9 | This interval…
|
55 | 415.0943396 | RM3, kUd4 | Wide Major Third, Lesser Super-Diminished Fourth | F#/, Gd<↓, Gb↑/ | 0 | 8 | This interval…
|
56 | 422.6415094 | rKM3, RkUd4 | Narrow Supermajor Third, Greater Super-Diminished Fourth | F#↑\, Gd>↓ | -1 | 7 | This interval…
|
57 | 430.1886792 | KM3, rUd4, KKd4 | Lesser Supermajor Third, Diptolemaic Diminished Fourth | F#↑, Gd<\, Gb↑↑ | -1 | 6 | This interval…
|
58 | 437.7358491 | SM3, kUM3, rm4, Ud4 | Greater Supermajor Third, Ultra-Diminished Fourth | Gd<, F#↑/ | 0 | 5 | This interval…
|
59 | 445.2830189 | m4, RkUM3 | Paraminor Fourth, Wide Supermajor Third | Gd>, Ft#>↓ | -1 | 3 | This interval…
|
60 | 452.8301887 | Rm4, KKM3, rUM3 | Wide Paraminor Fourth, Narrow Ultramajor Third | Gd>/, F#↑↑, G↓↓ | -2 | 1 | This interval…
|
61 | 460.3773585 | UM3, rKm4 | Ultramajor Third, Narrow Grave Fourth | Gd<↑, Ft#< | -4 | -2 | This interval…
|
62 | 467.9245283 | s4, Km4 | Lesser Grave Fourth, Wide Ultramajor Third | Gd>↑, G↓\ | -7 | -4 | This Interval…
|
63 | 475.4716981 | k4 | Greater Grave Fourth | G↓ | -6 | -5 | This interval…
|
64 | 483.0188679 | Rk4 | Wide Grave Fourth | G↓/ | -4 | 0 | This interval…
|
65 | 490.5660377 | r4 | Narrow Fourth | G\ | 1 | 5 | This interval…
|
66 | 498.1132075 | P4 | Perfect Fourth | G | 9 | 10 | This interval…
|
67 | 505.6603774 | R4 | Wide Fourth | G/ | 1 | 8 | This interval…
|
68 | 513.2075472 | rK4 | Narrow Acute Fourth | G↑\ | -3 | 6 | This interval…
|
69 | 520.7547170 | K4 | Lesser Acute Fourth | G↑ | -5 | 5 | This interval…
|
70 | 528.3018868 | S4, kM4 | Greater Acute Fourth | Gt<↓, G↑/, Adb< | -3 | 5 | This interval…
|
71 | 535.8490566 | RkM4, ud5 | Wide Acute Fourth, Infra-Diminished Fifth | Gt>↓, Adb> | -2 | 5 | This interval…
|
72 | 543.3962264 | rM4, Rud5 | Narrow Paramajor Fourth, Retrodiptolemaic Diminished Fifth | Gt<\, G↑↑, Ab↓↓ | -1 | 6 | This interval…
|
73 | 550.9433962 | M4, rKud5 | Paramajor Fourth, Lesser Sub-Diminished Fifth | Gt<, Adb<↑ | 0 | 7 | This interval…
|
74 | 558.4905660 | RM4, uA4, Kud5 | Infra-Augmented Fourth, Greater Sub-Diminished Fifth | Gt>, Adb>↑ | -2 | 5 | This interval…
|
75 | 566.0377358 | kkA4, RuA4, kd5 | Diptolemaic Augmented Fourth, Retroptolemaic Diminished Fifth | G#↓↓, Ab↓ | -3 | 4 | This interval…
|
76 | 573.5849057 | rKuA4, Rkd5 | Lesser Sub-Augmented Fourth, Artoretromean Diminished Fifth | Gt<↑, Ab↓/ | -2 | 4 | This interval…
|
77 | 581.1320755 | KuA4, rd5 | Greater Sub-Augmented Fourth, Tendoretromean Diminished Fifth | Gt>↑, Ab\ | 0 | 5 | This interval…
|
78 | 588.6792458 | kA4, d5 | Ptolemaic Augmented Fourth, Pythagorean Diminished Fifth | Ab, G#↓ | -5 | 6 | This interval…
|
79 | 596.2264151 | RkA4, Rd5 | Artomean Augmented Fourth, Artomean Diminished Fifth | G#↓/, Ab/ | -9 | 7 | This interval…
|
80 | 603.7735849 | rKd5, rA4 | Tendomean Diminished Fifth, Tendomean Augmented Fourth | Ab↑\, G#\ | -9 | 7 | This interval…
|
81 | 611.3207547 | Kd5, A4 | Ptolemaic Diminished Fifth, Pythagorean Augmented Fourth | Ab↑, G# | -5 | 6 | This interval…
|
82 | 618.8679245 | kUd5, RA4 | Lesser Super-Diminished Fifth, Artoretromean Augmented Fourth | Ad<↓, G#/ | 0 | 5 | This interval…
|
83 | 626.4150943 | RkUd5, rKA4 | Greater Super-Diminished Fifth, Tendoretromean Augmented Fourth | Ad>↓, G#↑\ | -2 | 4 | This interval…
|
84 | 633.9622642 | KKd5, rUDd5, KA4 | Diptolemaic Diminished Fifth, Retroptolemaic Augmented Fourth | Ab↑↑, G#↑ | -3 | 4 | This interval…
|
85 | 641.5094340 | rm5, Ud5, kUA4 | Ultra-Diminished Fifth, Lesser Super-Augmented Fourth | Ad<, Gt#<↓ | -2 | 5 | This interval…
|
86 | 649.0566038 | m5, RkUA4 | Paraminor Fifth, Greater Super-Augmented Fourth | Ad>, Gt#>↓ | 0 | 7 | This interval…
|
87 | 656.6037736 | Rm5, rUA4 | Wide Paraminor Fifth, Retrodiptolemaic Augmented Fourth | Ad>/, G#↑, Ab↑↑ | -1 | 6 | This interval…
|
88 | 664.1509434 | rKm5, UA4 | Narrow Grave Fifth, Ultra-Augmented Fourth | Ad<↑, Gt#< | -2 | 5 | This interval…
|
89 | 671.6981132 | s5, Km5 | Lesser Grave Fifth | Ad>↑, A↓\, Gt#> | -3 | 5 | This interval…
|
90 | 679.2452830 | k5 | Greater Grave Fifth | A↓ | -5 | 5 | This interval…
|
91 | 686.7924528 | Rk5 | Wide Grave Fifth | A↓/ | -3 | 6 | This interval…
|
92 | 694.3396226 | r5 | Narrow Fifth | A\ | 1 | 8 | This interval…
|
93 | 701.8867925 | P5 | Perfect Fifth | A | 9 | 10 | This interval…
|
94 | 709.4339622 | R5 | Wide Fifth | A/ | 1 | 5 | This interval…
|
95 | 716.9811321 | rK5 | Narrow Acute Fifth | A↑\ | -4 | 0 | This interval…
|
96 | 724.5283019 | K5 | Lesser Acute Fifth | A↑ | -6 | -5 | This interval…
|
97 | 732.0754717 | S5, kM5 | Greater Acute Fifth, Narrow Inframinor Sixth | At<↓, A↑/ | -7 | -4 | This Interval…
|
98 | 739.6226415 | um6, RkM5 | Inframinor Sixth, Wide Acute Fifth | At>↓, Bdb> | -4 | -2 | This interval…
|
99 | 747.1698113 | Rm4, KKM3, rUM3 | Narrow Paramajor Fifth, Wide Inframinor Sixth | At<\, Bb↓↓, A↑↑ | -2 | 1 | This interval…
|
100 | 754.7169811 | M5, rKum6 | Paramajor Fifth, Narrow Subminor Sixth | At<, Bdb<↑ | -1 | 3 | This interval…
|
101 | 762.2641509 | sm6, Kum6, RM5, uA5 | Lesser Subminor Sixth, Infra-Augmented Fifth | At>, Bb↓\ | 0 | 5 | This interval…
|
102 | 769.8113208 | km6, RuA5, kkA5 | Greater Subminor Sixth, Diptolemaic Augmented Fifth | Bb↓, At>/, A#↓↓ | -1 | 6 | This interval…
|
103 | 777.3584906 | Rkm6, rKuA5 | Wide Subminor Sixth, Lesser Sub-Augmented Fifth | Bb↓/, At<↑ | -1 | 7 | This interval…
|
104 | 784.9056604 | rm6, KuA5 | Narrow Minor Sixth, Greater Sub-Augmented Fifth | Bb\, At>↑, A#↓\ | 0 | 8 | This interval…
|
105 | 792.4528302 | m6, kA5 | Pythagorean Minor Sixth, Ptolemaic Augmented Fifth | Bb, A#↓ | -1 | 9 | This interval…
|
106 | 800 | Rm6, RkA5 | Artomean Minor Sixth, Artomean Augmented Fifth | Bb/, A#↓/ | 1 | 9 | This interval…
|
107 | 807.5471698 | rKm6, rA5 | Tendomean Minor Sixth, Tendomean Augmented Fifth | A#\, Bb↑\ | 4 | 10 | This interval…
|
108 | 815.0943396 | Km6, A5 | Ptolemaic Minor Sixth, Pythagorean Augmented Fifth | A#, Bb↑ | 8 | 10 | This interval…
|
109 | 822.6415094 | RKm6, kn6, RA5 | Wide Minor Sixth, Artoretromean Augmented Fifth | Bd<↓, Bb↑/, A#/ | 3 | 9 | This interval…
|
110 | 830.1886792 | kN6, rKA5 | Lesser Supraminor Sixth, Tendoretromean Augmented Fifth | Bd>↓, A#↑\ | -1 | 9 | This interval…
|
111 | 837.7358491 | KKm6, rn6, KA5 | Greater Supraminor Sixth, Retroptolemaic Augmented Fifth | Bd<\, Bb↑↑, A#↑ | 0 | 8 | This interval
|
112 | 845.2830189 | n6, SA5, kUA5 | Artoneutral Sixth, Lesser Super-Augmented Fifth | Bd<, At#<↓ | -1 | 7 | This interval…
|
113 | 852.8301887 | N6, RkUA5 | Tendoneutral Sixth, Greater Super-Augmented Fifth | Bd>, At#>↓ | 0 | 7 | This interval…
|
114 | 860.3773585 | kkM6, RN6, rUA5 | Lesser Submajor Sixth, Retrodiptolemaic Augmented Fifth | Bd>/, B↓↓, At#>↓/, A#↑↑ | -1 | 8 | This interval…
|
115 | 867.9245283 | Kn6, UA5 | Greater Submajor Sixth, Ultra-Augmented Fifth | Bd<↑, At#< | 1 | 9 | This interval…
|
116 | 875.4716981 | rkM6, KN6 | Narrow Major Sixth | Bd>↑, B↓\, At#> | 4 | 9 | This interval…
|
117 | 883.0188679 | kM6 | Ptolemaic Major Sixth | B↓, Cb | 7 | 10 | This interval…
|
118 | 890.5660377 | RkM6 | Artomean Major Sixth | B↓/ | 4 | 10 | This interval…
|
119 | 898.1132075 | rM6 | Tendomean Major Sixth | B\ | 1 | 9 | This interval…
|
120 | 905.6603774 | M6 | Pythagorean Major Sixth | B | -1 | 9 | This interval…
|
121 | 913.2075472 | RM6 | Wide Major Sixth | B/, Cd<↓ | 0 | 8 | This interval…
|
122 | 920.7547170 | rKM6 | Narrow Supermajor Sixth | B↑\, Cd>↓ | -1 | 8 | This interval…
|
123 | 928.3018868 | KM6 | Lesser Supermajor Sixth | B↑, Cd<\, Cb↑↑, A## | -1 | 7 | This interval…
|
124 | 935.8490566 | SM6, kUM6 | Greater Supermajor Second, Narrow Inframinor Seventh | Cd<, Bt<↓, B↑/ | 0 | 7 | This interval…
|
125 | 943.3962264 | um7, RkUM6 | Inframinor Seventh, Wide Supermajor Sixth | Cd>, Bt>↓ | -1 | 7 | This interval…
|
126 | 950.9433962 | KKM6, kkm7, rUM6, Rum7 | Narrow Ultramajor Sixth, Wide Inframinor Seventh, Semitwelfth | Bt<\, Cd>/, B↑↑, C↓↓ | 0 | 8 | This interval…
|
127 | 958.4905660 | UM6, rKum7 | Ultramajor Sixth, Narrow Subminor Seventh | Bt<, Cd<↑ | -1 | 8 | This interval…
|
128 | 966.0377358 | sm7, Kum7 | Lesser Subminor Seventh, Wide Ultramajor Sixth | Bt>, Cd>↑, C↓\ | 0 | 9 | This interval…
|
129 | 973.5849057 | km7 | Greater Subminor Seventh | C↓, Bt>/, B#↓↓, Dbb | -1 | 9 | This interval…
|
130 | 981.1320755 | Rkm7 | Wide Subminor Seventh | C↓/, Bt<↑ | -1 | 10 | This interval…
|
131 | 988.6792458 | rm7 | Narrow Minor Seventh | C\, Bt>↑ | -1 | 10 | This interval…
|
132 | 996.2264151 | m7 | Pythagorean Minor Seventh | C, B#↓ | -2 | 10 | This interval…
|
133 | 1003.7735849 | Rm7 | Artomean Minor Seventh | C/, B#↓/ | -2 | 10 | This interval…
|
134 | 1011.3207547 | rKm7 | Tendomean Minor Seventh | C↑\, B#\ | -3 | 10 | This interval…
|
135 | 1018.8679245 | kM2 | Ptolemaic Minor Seventh | C↑, B# | -3 | 10 | This interval…
|
136 | 1026.4150943 | RKm7, kn7 | Wide Minor Seventh | Ct<↓, C↑/, Ddb<, B#/ | -4 | 10 | This interval…
|
137 | 1033.9622642 | kN7, ud8 | Lesser Supraminor Seventh, Infra-Diminished Octave | Ct>↓, Ddb>, B#↑\ | -5 | 9 | This interval…
|
138 | 1041.5094340 | KKm7, rn7, Rud8 | Greater Supraminor Seventh, Retrodiptolemaic Diminished Octave | Ct<\, C↑↑, Ddb<↑\, Db↓↓ | -6 | 8 | This interval…
|
139 | 1049.0566038 | n7, rKud8 | Artoneutral Seventh, Lesser Sub-Diminished Octave | Ct<, Ddb<↑ | -7 | 6 | This interval…
|
140 | 1056.6037736 | N7, sd8 | Tendoneutral Seventh, Greater Sub-Diminished Octave | Ct>, Ddb>↑ | -8 | 5 | This interval…
|
141 | 1064.1509434 | kkM7, RN7, kd8 | Lesser Submajor Seventh, Diptolemaic Major Seventh, Retroptolemaic Diminished Octave | Ct>/, C#↓↓, Db↓ | -7 | 6 | This interval…
|
142 | 1071.6981132 | Kn7, Rkd8 | Greater Submajor Seventh, Artoretromean Diminished Octave | Ct<↑, Db↓/ | -6 | 8 | This interval…
|
143 | 1079.2452830 | rkM7, KN7, rd8 | Narrow Major Seventh, Tendoretromean Diminished Octave | Ct>↑, C#↓\, Db\ | -5 | 9 | This interval…
|
144 | 1086.7924528 | kM7, d8 | Ptolemaic Major Seventh, Pythagorean Diminished Octave | Db, C#↓ | -5 | 10 | This interval…
|
145 | 1094.3396226 | RkM7, Rd8 | Artomean Major Seventh, Artomean Diminished Octave | Db/, C#↓/ | -5 | 10 | This interval…
|
146 | 1101.8867925 | rM7, rKd8 | Tendomean Major Seventh, Tendomean Diminished Octave | C#\, Db↑\ | -6 | 10 | This interval…
|
147 | 1109.4339622 | M7, Kd8 | Pythagorean Major Seventh, Ptolemaic Diminished Octave | C#, Db↑ | -6 | 10 | This interval…
|
148 | 1116.9811321 | RM7, kUd8 | Wide Major Seventh, Lesser Super-Diminished Octave | C#/, Dd<↓ | -7 | 9 | This interval…
|
149 | 1124.5283019 | rKM7, RkUd8 | Narrow Supermajor Seventh, Greater Super-Diminished Octave | C#↑\, Dd>↓ | -7 | 9 | This interval…
|
150 | 1132.0754717 | km2, RuA1, kkA1 | Lesser Supermajor Seventh, Diptolemaic Diminished Octave | C#↑, Db↑↑ | -8 | 9 | This interval…
|
151 | 1139.6226415 | SM7, kUM7, Ud8 | Greater Supermajor Seventh, Narrow Infraoctave, Ultra-Diminished Octave | Dd<, C#↑/ | -8 | 10 | This interval…
|
152 | 1147.1698113 | u8, RkUM7 | Infraoctave, Wide Supermajor Seventh | Dd>, Ct#>↓ | -9 | 10 | This interval…
|
153 | 1154.7169811 | KKM7, rUM7, Ru8 | Narrow Ultramajor Seventh, Wide Infraoctave | C#↑↑, Dd>/ | -9 | 10 | This interval…
|
154 | 1162.2641509 | UM7, rKu8 | Ultramajor Seventh, Wide Superprime | Ct#<, Dd<↑ | -9 | 10 | This interval…
|
155 | 1169.8113208 | s8, Ku8 | Lesser Suboctave, Wide Ultramajor Seventh | Ct#>, Dd>↑ | -10 | 3 | This interval…
|
156 | 1177.3584906 | k8 | Greater Suboctave | D↓ | -10 | -3 | This interval…
|
157 | 1184.9056604 | Rk8 | Wide Suboctave | D↓/ | -10 | -10 | This interval…
|
158 | 1192.4528302 | r8 | Narrow Octave | D\ | 0 | 0 | This interval…
|
159 | 1200 | P8 | Perfect Octave | D | 10 | 10 | This interval…
|
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.
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.
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 |