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SHEFKHED interval names: Difference between revisions

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*A tuning's best approximation to 3/2 is labelled P5, for perfect fifth, and it's octave-complement labelled P4, for perfect fourth.
*A tuning's best approximation to 3/2 is labelled P5, for perfect fifth, and it's octave-complement labelled P4, for perfect fourth.
*From the Pythagorean diatonic scale using a tuning's best 3/2 fifth, the two sizes of second, third, sixth and seventh are labelled major, or 'M', for the larger, and minor, or 'm' for the smaller.
*From the Pythagorean diatonic scale using a tuning's best 3/2 fifth, the two sizes of second, third, sixth and seventh are labelled major, or 'M', for the larger, and minor, or 'm' for the smaller.
*Any perfect or major interval raised by the apotome, the interval between the major and minor intervals of a single interval-class is labelled 'A' for augmented, and any perfect or minor interval lowered by the same is labelled 'd' for diminished.
*Any perfect or major interval raised by the apotome, the interval between the major and minor intervals of a single interval-class is labelled 'Aug' for augmented, and any perfect or minor interval lowered by the same is labelled 'dim' for diminished.
*Any augmented interval may be made doubly augmented, with short-hand 'AA' by the further raising of an apotome and any diminished interval made doubly diminished, with short-hand 'dd' by the further lowering of an apotome. This process may be iterated ad nauseum. At this stage we have simply rigorously defined diatonic interval names. Thankfully what remains of the definition leads to more desirable alternatives for most occasions in which one might find these iteratively diminished and augmented intervals.
*Any augmented interval may be made doubly augmented, with short-hand 'AA' by the further raising of an apotome and any diminished interval made doubly diminished, with short-hand 'dd' by the further lowering of an apotome. This process may be iterated ad nauseum. At this stage we have simply rigorously defined diatonic interval names. Thankfully what remains of the definition leads to more desirable alternatives for most occasions in which one might find these iteratively diminished and augmented intervals.
*Perfect, major and augmented intervals may be given the prefix 'super', with shorthand 'S' which infers an augmentation by the septimal comma, 64/63, whereas perfect, minor and diminished intervals are lowered by the same interval when given the prefix 'sub', with short-form 's'.
*Perfect, major and augmented intervals may be given the prefix 'super', with shorthand 'S' which infers an augmentation by the septimal comma, 64/63, whereas perfect, minor and diminished intervals are lowered by the same interval when given the prefix 'sub', with short-form 's'.
*Major and augmented intervals may be given the prefix 'classic', with short-form 'c', inferring a diminution by the syntonic comma, 81/80, whereas minor and diminished may also be given the prefix 'classic' but with short-hand 'C', inferring an augmentation by 81/80. This results in the labeling of 10/9, 6/5, 5/4, 8/5, 5/3 and 9/5 as classic major second, classic major third, classic minor third, classic minor sixth, classic major sixth and classic minor seventh, as per Keenan's suggestion when a comparison to Pythagorean is needed.
*Major and augmented intervals may be given the prefix 'classic', with short-form 'cla', inferring a diminution by the syntonic comma, 81/80, whereas minor and diminished may also be given the prefix 'classic' but with short-hand 'Cla', inferring an augmentation by 81/80. This results in the labeling of 10/9, 6/5, 5/4, 8/5, 5/3 and 9/5 as classic major second, classic major third, classic minor third, classic minor sixth, classic major sixth and classic minor seventh, as per Keenan's suggestion when a comparison to Pythagorean is needed.
*Perfect intervals may also be given the prefixes 'C' and 'c' to imply augmentation and diminution by the syntonic comma, however, where this interval is given the name 'comma' by Smith and Bosanquet, and where Smith's interval-naming scheme involves prefixes of ''m''/''n''-comma sharp and ''m''/''n''-comma flat, the long-form for these prefixes when applied to perfect intervals is 'comma-sharp' and 'comma-flat' respectively.
*81/80 is referred to by Smith and Bosanquet simple as 'comma'. Smith's interval-naming scheme involves prefixes of ''m''/''n''-comma sharp and ''m''/''n''-comma flat. Using 'wide' and 'narrow' instead of 'sharp' and 'flat', we add long-form prefixes that may be used before perfect intervals, 'comma-wide' and 'comma-narrow' to infer movement up or down a syntonic comma, with short form 'Co-wde' and 'co-nrw'. These names are chosen because they start with the same letter as the other prefixes that infer alteration by 81/80.
*For seconds, thirds, sixths and sevenths, any interval exactly half-way major and minor is labelled 'neutral', with short-form 'N'.
*For seconds, thirds, sixths and sevenths, any interval exactly half-way major and minor is labelled 'neutral', with short-form 'N'.
*This is extended such that for the intervals exactly half-way between the perfect unison and the augmented unison, the perfect fourth and the augmented fourth, the perfect fifth and the diminished fifth; and the perfect octave and the diminished octave are also given the label 'neutral' with short-form 'N'. In all cases 'N' marks a splitting-in-half or the apotome, and it's presence implies neutral temperament and the tempering out of 243/242. Accordingly it implies a diminution from perfect, major, augmented of 33/32, as well as an augmentation from perfect, minor or diminished of 33/32, but may not be used to imply those alterations in any other cases.
*Where neutral intervals split the apotome, to pair with neutral when acting on perfect intervals are 'hemi augmented' and 'hemi diminished', with short form 'h-Aug' and 'h-dim'. 'Hemi' is used instead of 'semi' of 'half' because 'half diminished' is a type of chord, and 'semi' begins with the letter 's', which has been associated with alterations of 64/63. In all cases it's presence implies neutral temperament and the tempering out of 243/242. Accordingly it implies a diminution from perfect, major, augmented of 33/32, as well as an augmentation from perfect, minor or diminished of 33/32, but may not be used to imply those alterations in any other cases.  


*To extend to the 13-limit, we add that to cP, cM and cA intervals may be added the 'sub' or 's' prefix, in this instance indicating a diminution of [[65/64]], and that to CP, Cm and Cd intervals may be added the 'super' or 'S' prefix, indication an augmentation of the same interval. Accordingly the difference between 65/64 and 64/63, 4096/4095, the ''tridecimal schisma'', is tempered out. Accordingly 16/13 is labelled a 'sub classic major third', or scM3. In tunings where the syntonic comma is tempered out, such that (cP, cM, cA, CP, Cm, Cd) = (P, M, A, P, m, d), the 'c' or 'C' prefixes are dropped in the short-form.
*To extend to the 13-limit, we add that to co-n P, co-n M and co-n A intervals may be added the 'sub' or 's' prefix, in this instance indicating a diminution of [[65/64]], and that to Co-w P, Co-w m and Co-w dim intervals may be added the 'super' or 'S' prefix, indication an augmentation of the same interval. Accordingly the difference between 65/64 and 64/63, 4096/4095, the ''tridecimal schisma'', is tempered out. Accordingly 16/13 is labelled a 'sub classic major third', or scla M3. In tunings where the syntonic comma is tempered out, such that (co-n P, cla M, cla Aug, Co-w P, Cla m, Cla d) = (P, M, Aug, P, m, dim), the 'cla', 'Cla', 'co-n' and 'Co-w' prefixes are dropped in the short-form.
*Where N indicates a splitting of the apotome and of the perfect fifth, interval names indicating the splitting of the limma and of the perfect fourth are included for remaining unnamed intervals, reflecting limited, but existing practice. The interval half-way between P1 and m2 is given the short-form '1-2', and long-form 'unison-second' and mid-form 'unicond'. Similarly the interval half-way between M7 and P8 is given the short-form '7-8', long-form 'seventh-octave' and mid-form 'sevtave'. The interval splitting the fourth, lying half-way between M2 and m3 is given the short-form '2-3', long-form 'second-third', and mid-form 'serd', and it's octave complement, lying half-way between M6 and m7 is given the short-form '6-7', long-form 'sixth-seventh', and mid-form 'sinth'. The interval half-way between M3 and P4 is given the short-form '3-4', long-form 'third-fourth' and mid-form 'thourth', and it's octave-complement, the interval half-way between P5 and m6 is given the short-form '5-6', long-form 'fifth-sixth' and mid-form 'fixth'. These interval names can be associated with [[The Archipelago|Barbados]] temperament, indicating the tempering out of 676/675, generated by 2-3, half of the fourth, associated with the ratio 15/13. These ''intermediates'' lie 40/39 above major intervals and the perfect unison and fifth, and below minor intervals and the perfect fourth and octave. 3-4, for example, is associated with the ratio 13/10.
*Where N indicates a splitting of the apotome and of the perfect fifth, interval names indicating the splitting of the limma and of the perfect fourth are included for remaining unnamed intervals, reflecting limited, but existing practice. The interval half-way between P1 and m2 is given the short-form '1-2', and long-form 'unison-second' and mid-form 'unicond'. Similarly the interval half-way between M7 and P8 is given the long-form 'seventh-octave' and short-form '7-8', that may be said 'sevtave'. The interval splitting the fourth, lying half-way between M2 and m3 is given the long-form 'second-third' and short-form '2-3' that may be said 'serd', and it's octave complement, lying half-way between M6 and m7 is given the long-form 'sixth-seventh', with short-form '6-7' that may be said 'sinth'. The interval half-way between M3 and P4 is given the long-form 'third-fourth', with short-form '3-4', that may be said 'thourth', and it's octave-complement, the interval half-way between P5 and m6 is given the long-form 'fifth-sixth', with short-form '5-6', that may be saud 'fixth'. These interval names can be associated with [[The Archipelago|Barbados]] temperament, indicating the tempering out of 676/675, generated by 2-3, half of the fourth, associated with the ratio 15/13. These ''intermediates'' lie 40/39 above major intervals and the perfect unison and fifth, and below minor intervals and the perfect fourth and octave. 3-4, for example, is associated with the ratio 13/10.
*For completeness, '4-5', with long-form 'fourth-fifth' and short-form 'firth' is added, though it is separate to the other intermediates, splitting not the limma, but the dieses (between A4 and d5), or the octave. It does not map to any particular ratios and is not needed as a primary interval name, apart from in 16edo, and is included mostly to be used as an optional secondary interval name when there are no others.
*For completeness, the  'fourth-fifth', with '4-5' that may be said 'firth' is added, though it is separate to the other intermediates, splitting not the limma, but the dieses (between Aug4 and dim5), or the octave. It does not map to any particular ratios and is not needed as a primary interval name, apart from in 16edo, and is included mostly to be used as an optional secondary interval name when there are no others.
*In any prefix is used before 'P' then 'P' is removed in both the short-form and long-form names.
*In any prefix is used before 'P' then 'P' is removed in both the short-form and long-form names.
*The prefixes so far take us as far as 53edo, which is considered a 'commatic' scale by many, and as far as extended-diatonic function, which I hope to reflect with this scheme, could be considered to apply. Keenan's functional names take us to 31edo, after which 'narrow' and 'wide' prefixes are added to differentiate different intervals in medium to large sized edos of the same function. Ups and Downs takes function as far as regular diatonic and mids (equivalent to neutrals), which will give us most of a well-ordered interval name set for 17edo (if mids were extended as I have extended neutrals, all the notes would be obtainable) without up or down prefixes, and only functional names, or all of 19edo or 26edo, since these are meantone edos with the apotome subtended by a single degree and may be given a well-ordered interval names set using only regular diatonic interval names. The up and down prefixes are not functional, and specify movement instead by a single step of an edo. If the naming of systems with more than one interval per function is desired, then 'up' and 'down' prefixes, with short form '^' and 'v' respectively are to be employed. This also allows the notation of intervals for which intermediates are the only available functional interval name. Note: For regular diatonic intervals, I consider function only to go as far as singly diminished or augmented intervals, and never use multiply diminished or augmented intervals for my interval names.
*The prefixes so far take us as far as 53edo, which is considered a 'commatic' scale by many, and as far as extended-diatonic function, which I hope to reflect with this scheme, could be considered to apply. Keenan's functional names take us to 31edo, after which 'narrow' and 'wide' prefixes are added to differentiate different intervals in medium to large sized edos of the same function. Ups and Downs takes function as far as regular diatonic and mids (equivalent to neutrals), which will give us most of a well-ordered interval name set for 17edo (if mids were extended as I have extended neutrals, all the notes would be obtainable) without up or down prefixes, and only functional names, or all of 19edo or 26edo, since these are meantone edos with the apotome subtended by a single degree and may be given a well-ordered interval names set using only regular diatonic interval names. The up and down prefixes are not functional, and specify movement instead by a single step of an edo. If the naming of systems with more than one interval per function is desired, then 'up' and 'down' prefixes, with short form '^' and 'v' respectively are to be employed. This also allows the notation of intervals for which intermediates are the only available functional interval name. Note: For regular diatonic intervals, I consider function only to go as far as singly diminished or augmented intervals, and never use multiply diminished or augmented intervals for my interval names.
*Mid-form for prefixes is added for chord-naming, and wherever else desired, considering A, d and C/c are notes. The long, mid, and short-form for all prefixes is detailed below.
{| class="wikitable"
{| class="wikitable"
|+Prefixes summary
|+Prefixes summary
!Long-form
!Long-form
!Mid-form
!Short-form
!Short-form
!Alteration
!Alteration
|-
|-
|perfect
|perfect
|Prf
|P
|P
|none
|none
|-
|-
|major
|major
|Maj
|M
|M
|none
|none
|-
|-
|minor
|minor
|min
|m
|m
|none
|none
|-
|augmented
|Aug
|up an apotome
|-
|diminished
|dim
|down an apotome
|-
|-
|super
|super
|Spr
|S
|S
|up 64/63 (or 65/64)
|up 64/63 (or 65/64)
|-
|-
|sub
|sub
|sub
|s
|s
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|classic
|classic
|Cla
|Cla
|C
|up 81/80
|up 81/80
|-
|-
|classic
|classic
|cla
|cla
|c
|down 81/80
|down 81/80
|-
|-
|comma-sharp
|comma-wide
|Co-shp
|Co-w
|C
|up 81/80
|up 81/80
|-
|-
|comma-flat
|comma-narrow
|co-flt
|co-n
|c
|down 81/80
|down 81/80
|-
|-
|up
|wide
|up
|W
|^
|up a single degree
|up a single degree
|-
|-
|down
|narrow
|dwn
|n
|v
|down a single degree
|down a single degree
|}
|}
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|+Intermediates and neutrals summary
|+Intermediates and neutrals summary
!Long-form
!Long-form
!Mid-form
!Short-form
!Short-form
!Description
!Description
|-
|-
|unison-second
|unison-second
|unicond
|1-2 (unicond)
|1-2
|half of the limma, half way between P1 and m2
|half of the limma, half way between P1 and m2
|-
|-
|second-third
|second-third
|serd
|2-3 (serd)
|2-3
|half of P4, halfway between M2 and m3  
|half of P4, halfway between M2 and m3  
|-
|-
|third-fourth
|third-fourth
|thourth
|3-4 (thourth)
|3-4
|half of M6, halfway between M3 and P4
|half of M6, halfway between M3 and P4
|-
|-
|fourth-fifth
|fourth-fifth
|firth
|4-5 (firth)
|4-5
|half of the octave, halfway between Aug 4 and dim 5
|half of the octave, halfway between A4 and d5
|-
|-
|fifth-sixth
|fifth-sixth
|fixth
|5-6 (fixth)
|5-6
|half of M10, halfway between P5 and m6  
|half of M10, halfway between P5 and m6  
|-
|-
|sixth-seventh
|sixth-seventh
|sinth
|6-7 (sinth)
|6-7
|halfway between P5 and P8, halfway between M6 and m7
|halfway between P5 and P8, halfway between M6 and m7
|-
|-
|seventh-octave
|seventh-octave
|sevtave
|7-8 (sevtave)
|7-8
|halfway between M7 and P8
|halfway between M7 and P8
|-
|-
|neutral
|neutral
|ntl
|N
|N
|halfway between M and m (seconds, thirds, sixths, seventh)
|halfway between M and m
halfway between P and A (unison, fourth)
|-
 
|hemi augmented
halfway between P and d (fifth, octave)
|h-Aug
|halfway between P and Aug
|-
|hemi diminished
|h-dim
|halfway between P and dim
|}
|}


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As in Keenan/Fokker and Ups and Downs, intervals may be given multiple names. The following details the order to which certain names are privileged above others.
As in Keenan/Fokker and Ups and Downs, intervals may be given multiple names. The following details the order to which certain names are privileged above others.


Interval names are ranked in eight tiers.
Interval names are ranked in nine tiers.
#Perfect
#Perfect and neutral
#Neutral
#Major, minor, Aug 4 and dim 5.
#Major, minor, A4 and d5.
#h-Aug 4 and h-dim 5
#'S', 's', 'C', 'c' 'SC' and 'sc' prefixes to major, minor, perfect intervals and to A4 and d5
#'S', 's', 'Cla', 'cla', 'Co-w', 'co-n', 'SCla', 'scla', 'SCo-w' and 'sco-n' prefixes to major, minor, perfect intervals and to Aug 4 and dim 5
#h-Aug 1 and h-dim 8 (plus any other h-Augs and h-dims if needed)
#Intermediates
#Intermediates
#Remaining augmented and diminished intervals (for when the chroma is subtended by more than a single (positive) step of the edo)
#Remaining augmented and diminished intervals (for when the chroma is subtended by more than a single (positive) step of the edo)
#'S', 's', 'C', 'c' 'SC' and 'sc' prefixes to augmented and diminished intervals
#'S', 's', 'Cla', 'cla', 'Co-w', 'co-n', 'SCla', 'scla', 'SCo-w' and 'sco-n' prefixes to augmented and diminished intervals
#Intervals augmented and diminished more than singularly
#Intervals augmented and diminished more than singularly
When more than one interval name corresponds to a specific interval, the names are privileged in order of the tiers. By this ordering, the first available name is the ‘primary’ for that interval, the second available ‘secondary’ and third 'tertiary'.
When more than one interval name corresponds to a specific interval, the names are privileged in order of the tiers. By this ordering, the first available name is the ‘primary’ for that interval, the second available ‘secondary’ and third 'tertiary'.


Where the same interval may be named c4 or s4, s4 is preferred and where the same interval may be named C4 or S4, C4 is preferred. Similarly, where the same interval may be named C5 or S5, S5 is preferred and where the same interval may be named c5 or s5, c5 is preferred. This is to ensure the interval is named after the simpler ratio. On top of this, well-ordered interval-name sets are desired, leading to interval names in lower tires being used in preference to higher-tier names in some cases.
Where the same interval may be named co-n 4 or s4, s4 is preferred and where the same interval may be named Co-w 4 or S4, Co-w 4 is preferred. Similarly, where the same interval may be named Co-w 5 or S5, S5 is preferred and where the same interval may be named co-n 5 or s5, co-n 5 is preferred. This is to ensure the interval is named after the simpler ratio. On top of this, well-ordered interval-name sets are desired, leading to interval names in lower tires being used in preference to higher-tier names in some cases.


=== Regular diatonic edos ===
=== Regular diatonic edos ===
All ''regular diatonic'' edos (edos whose best fifth is greater than 4 degrees of 7edo and less than 3 degrees of 5edo, such that the diatonic scale has 5 large and 2 small steps) up to 46 can be simply given primary well-ordered interval names. All of those that I've seen used have their primary well-ordered interval-names below, with the addition of 53edo, which is as far as I want to go and can go with this system without extending it further.
All ''regular diatonic'' edos (edos whose best fifth is greater than 4 degrees of 7edo and less than 3 degrees of 5edo, such that the diatonic scale has 5 large and 2 small steps) up to 46 can be simply given primary well-ordered interval names. All of those that I've seen used have their primary well-ordered interval-names below, with the addition of 53edo, which is as far as I want to go and can go with this system without extending it further. In the interest of parsimony, these lists will use single letter abbreviations for all prefixes, except h-Aug, and h-dim, which will be written as 'hA' and 'hd'.


12edo: P1 m2 M2 m3 M3 P4 A4/d5 P5 m6 M6 m7 M7 P8
12edo: P1 m2 M2 m3 M3 P4 A4/d5 P5 m6 M6 m7 M7 P8


17edo: P1 N1 N2 M2 m3 N3 M3 P4 N4 N5 P5 m6 N6 M6 m7 N7 N8 P8
17edo: P1 m2 N2 M2 m3 N3 M3 P4 hA4 hd5 P5 m6 N6 M6 m7 N7 M7 P8


19edo: P1 S1/sm2 m2 M2 SM2/sm3 m3 M3 SM3/s4 P4 A4 d5 P5 S5/sm6 m6 M6 SM6/sm7 m7 M7/s8 P8
19edo: P1 S1/sm2 m2 M2 SM2/sm3 m3 M3 SM3/s4 P4 A4 d5 P5 S5/sm6 m6 M6 SM6/sm7 m7 M7/s8 P8
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22edo: P1 m2 Cm2 cM2 M2 m3 Cm3 cM3 M3 P4 C4 cA4/Cd5 c5 P5 m6 Cm6 cM6 M6 m7 Cm7 cM7 M7 P8
22edo: P1 m2 Cm2 cM2 M2 m3 Cm3 cM3 M3 P4 C4 cA4/Cd5 c5 P5 m6 Cm6 cM6 M6 m7 Cm7 cM7 M7 P8


24edo: P1 S1/sm2 m2 N2 M2 SM2/sm3 m3 N3 M3 SM3/s4 P4 N4 A4/d5 N5 P5 S5/sm6 m6 N6 M6 SM6/sm7 m7 N7 M7 SM7/s8 P8
24edo: P1 S1/sm2 m2 N2 M2 SM2/sm3 m3 N3 M3 SM3/s4 P4 hA4 A4/d5 N5 P5 S5/sm6 m6 N6 M6 SM6/sm7 m7 N7 M7 SM7/s8 P8


26edo: P1 S1 sm2 m2 M2 SM2 sm3 m3 M3 SM3 s4 P4 A4 SA4/sd5 d5 P5 S5 sm6 m6 M6 SM6 sm7 m7 M7 SM7 s8 P8
26edo: P1 S1 sm2 m2 M2 SM2 sm3 m3 M3 SM3 s4 P4 A4 SA4/sd5 d5 P5 S5 sm6 m6 M6 SM6 sm7 m7 M7 SM7 s8 P8
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29edo: P1 C1/S1/sm2 m2 Cm2 cM2 M2 SM2/sm3 m3 Cm3 cM3 M3 SM3/s4 P4 C4 cA4/d5 A4/Cd5 c5 P5 S5/sm6 m6 Cm6 cM6 M6 SM6/sm7 m7 Cm7 cM7 M7 SM7/S8/c8 P8
29edo: P1 C1/S1/sm2 m2 Cm2 cM2 M2 SM2/sm3 m3 Cm3 cM3 M3 SM3/s4 P4 C4 cA4/d5 A4/Cd5 c5 P5 S5/sm6 m6 Cm6 cM6 M6 SM6/sm7 m7 Cm7 cM7 M7 SM7/S8/c8 P8


31edo: P1 N1 sm2 m2 N2 M2 SM2 sm3 m3 N3 M3 SM3 s4 P4 N4 A4 d5 N4 P5 S5 sm6 m6 N6 M6 SM6 sm7 m7 N7 M7 SM7 N8 P8
31edo: P1 S1 sm2 m2 N2 M2 SM2 sm3 m3 N3 M3 SM3 s4 P4 hA4 A4 d5 hd5 P5 S5 sm6 m6 N6 M6 SM6 sm7 m7 N7 M7 SM7 s8 P8


34edo: P1 C1 m2 Cm2 N2 cM2 M2 2-3 m3 Cm3 N3 cM3 M3 3-4 P4 C4 N4/d5 cA4/Cd5 A4/N5 c5 P5 5-6 m6 Cm6 N6 cM6 M6 6-7 m7 Cm7 N7 cM7 M7 c8 P8
34edo: P1 C1 m2 Cm2 N2 cM2 M2 2-3 m3 Cm3 N3 cM3 M3 3-4 P4 C4 N4/d5 cA4/Cd5 A4/N5 c5 P5 5-6 m6 Cm6 N6 cM6 M6 6-7 m7 Cm7 N7 cM7 M7 c8 P8
Line 623: Line 615:
36edo: P1 S1 sm2 m2 Sm2 sM2 M2 SM2 sm3 m3 Sm3 sM3 M3 SM3 s4 P4 S4 sA4 A4/d5 SA4 s5 P5 S5 sm6 m6 Sm6 sM6 M6 SM6 sm7 m7 Sm7 sM7 M7 SM7 s8 P8
36edo: P1 S1 sm2 m2 Sm2 sM2 M2 SM2 sm3 m3 Sm3 sM3 M3 SM3 s4 P4 S4 sA4 A4/d5 SA4 s5 P5 S5 sm6 m6 Sm6 sM6 M6 SM6 sm7 m7 Sm7 sM7 M7 SM7 s8 P8


38edo: P1 S1 A1 sm2 m2 N2 M2 SM2 A2/d3 sm3 m3 N3 M3 SM3 d4 s4 P4 N4 A4 SA4/sd5 d5 N5 P5 S5 A5 sm6 m6 N6 M6 SM6 A6/d7 sm7 m7 N7 M7 SM7 d8 s8 P8
38edo: P1 S1 A1 sm2 m2 N2 M2 SM2 A2/d3 sm3 m3 N3 M3 SM3 d4 s4 P4 hA4 A4 SA4/sd5 d5 hd5 P5 S5 A5 sm6 m6 N6 M6 SM6 A6/d7 sm7 m7 N7 M7 SM7 d8 s8 P8


41edo: P1 C1/S1 N1 m2 Cm2 N2 cM2 M2 SM2 sm3 m3 Cm3 N3 cM3 M3 SM3 s4 P4 C4 N4 cA4 Cd5 N5 c5 P5 S5 sm6 m6 Cm6 N6 cM6 M6 SM6 sm7 m7 Cm7 N7 cM7 M7 N8 c8/s8 P8
41edo: P1 C1/S1 hA1 m2 Cm2 N2 cM2 M2 SM2 sm3 m3 Cm3 N3 cM3 M3 SM3 s4 P4 C4 hA4 cA4 Cd5 hd5 c5 P5 S5 sm6 m6 Cm6 N6 cM6 M6 SM6 sm7 m7 Cm7 N7 cM7 M7 hd8 c8/s8 P8


43edo: P1 S1 1-2 sm2 m2 Sm2 sM2 M2 SM2 2-3 sm3 m3 Sm3 sM3 M3 SM3 3-4 s4 P4 S4 sA4 A4 d5 Sd5 s5 P5 S5 5-6 sm6 m6 Sm6 sM6 M6 SM6 6-7 sm7 m7 Sm7 sM7 M7 SM7 7-8 s8 P8
43edo: P1 S1 1-2 sm2 m2 Sm2 sM2 M2 SM2 2-3 sm3 m3 Sm3 sM3 M3 SM3 3-4 s4 P4 S4 sA4 A4 d5 Sd5 s5 P5 S5 5-6 sm6 m6 Sm6 sM6 M6 SM6 6-7 sm7 m7 Sm7 sM7 M7 SM7 7-8 s8 P8
Line 634: Line 626:


We can see that
We can see that
*17edo, 24edo, 27edo, 31edo, 34edo (through 17edo), and 38edo are neutral tunings from the use of 'N'. We can find the MOS scale Neutral[17] 8|8: P1 N1 N2 M2 m3 N3 M3 P4 N4 N5 P5 m6 N6 M6 m7 N7 N8 P8 in all of these edos.
*17edo, 24edo, 27edo, 31edo, 34edo (through 17edo), and 38edo are neutral tunings from the use of 'N'. We can find the MOS scale Neutral[10] 5|4: P1 N2 M2 N3 P4 hd5 P5 N6 m7 N7 P8 in all of these edos.
*19edo, 24edo (through 12edo), 26edo, 31edo, 36edo (through 12edo), 38edo (through 19edo) and 43edo are meantone tunings through the use of 'S' and 's'.
*19edo, 24edo (through 12edo), 26edo, 31edo, 36edo (through 12edo), 38edo (through 19edo) and 43edo are meantone tunings through the use of 'S' and 's'.
*22edo, 27edo and 34edo (through 17edo) are superpythagorean tunings from the use of 'C' and 'c'.
*22edo, 27edo and 34edo (through 17edo) are superpythagorean tunings from the use of 'C' and 'c'.
Line 666: Line 658:
|m2
|m2
|comma-sharp unison/super minor second
|comma-sharp unison/super minor second
|C1/sm2
|Co-w 1 / sm2
|54.55
|54.55
|33/32, 34/33, 25/24, 81/80
|33/32, 34/33, 25/24, 81/80
Line 672: Line 664:
|2
|2
|classic minor second
|classic minor second
|Cm2
|Cla m2
|diminished third
|diminished third
|d3
|dim 3
|109.09
|109.09
|18/17, 17/16, 16/15, 15/14
|18/17, 17/16, 16/15, 15/14
Line 680: Line 672:
|3
|3
|classic major second
|classic major second
|cM2
|cla M2
|augmented unison
|augmented unison
|A1
|Aug 1
|163.64
|163.64
|11/10, 10/9
|11/10, 10/9
Line 704: Line 696:
|6
|6
|classic minor third
|classic minor third
|Cm3
|Cla m3
|diminished fourth
|diminished fourth
|d4
|dim 4
|327.27
|327.27
|6/5, 17/14, 11/9
|6/5, 17/14, 11/9
Line 712: Line 704:
|7
|7
|classic major third
|classic major third
|cM3
|cla M3
|augmented second
|augmented second
|A2
|Aug 2
|381.82
|381.82
|5/4
|5/4
Line 738: Line 730:
|C4
|C4
|diminished fifth
|diminished fifth
|d5
|dim 5
|545.45
|545.45
|11/8, 15/11, 27/20
|11/8, 15/11, 27/20
Line 745: Line 737:
|classic augmented fourth
|classic augmented fourth
classic diminished fifth
classic diminished fifth
|cA4
|cla Aug 4
Cd5
Cla dim 5
|fourth-fifth
|fourth-fifth
|4-5
|4-5 (firth)
|600
|600
|7/5, 17/12, 45/32
|7/5, 17/12, 45/32
Line 757: Line 749:
|c5
|c5
|augmented fourth
|augmented fourth
|A4
|Aug 4
|654.55
|654.55
|16/11, 22/15, 40/27
|16/11, 22/15, 40/27
Line 779: Line 771:
|15
|15
|classic minor sixth
|classic minor sixth
|Cm6
|Cla m6
|diminished seventh
|diminished seventh
|d7
|dim 7
|818.18
|818.18
|8/5
|8/5
Line 787: Line 779:
|16
|16
|classic major sixth
|classic major sixth
|cM6
|cla M6
|augmented fifth
|augmented fifth
|A5
|Aug 5
|872.73
|872.73
|5/3, 18/11, 28/17
|5/3, 18/11, 28/17
Line 811: Line 803:
|19
|19
|classic minor seventh
|classic minor seventh
|Cm7
|Cla m7
|diminished octave
|diminished octave
|d8
|dim 8
|1036.36
|1036.36
|20/11, 9/5
|20/11, 9/5
Line 819: Line 811:
|20
|20
|classic major seventh
|classic major seventh
|cM7
|cla M7
|augmented sixth
|augmented sixth
|A6
|Aug 6
|1090.91
|1090.91
|15/8, 32/17, 17/9, 28/15
|15/8, 32/17, 17/9, 28/15
Line 829: Line 821:
|M7
|M7
|super major seventh / comma-flat octave
|super major seventh / comma-flat octave
|SM7/d8
|SM7 / co-n 8
|1145.45
|1145.45
|33/17, 64/33, 48/25, 160/81
|33/17, 64/33, 48/25, 160/81
Line 856: Line 848:
|-
|-
|1
|1
|comma-sharp unison/super unison
|comma-wide unison/super unison
|C1/S1
|Co-w 1 / S1
|29.27
|29.27
|[[81/80]], 64/63
|[[81/80]], 64/63
|-
|-
|2
|2
|neutral unison, subminor second
|hemi-augmented unison, subminor second
|N1, sm2
|h-Aug 1, sm2
|58.54
|58.54
|[[25/24]], [[28/27]], [[33/32]]
|[[25/24]], [[28/27]], [[33/32]]
Line 875: Line 867:
|4
|4
|classic minor second, augmented unison
|classic minor second, augmented unison
|Cm2, A1
|Cla m2, Aug 1
|117.07
|117.07
|[[16/15]], [[15/14]]
|[[16/15]], [[15/14]]
Line 887: Line 879:
|6
|6
|classic major second, diminished third
|classic major second, diminished third
|cM2, d3
|cla M2, dim 3
|175.61
|175.61
|[[10/9]], [[11/10]]
|[[10/9]], [[11/10]]
Line 917: Line 909:
|11
|11
|classic minor third, augmented second
|classic minor third, augmented second
|Cm3, A2
|Cla m3, Aug 2
|321.95
|321.95
|[[6/5]]
|[[6/5]]
Line 929: Line 921:
|13
|13
|classic major third, diminished fourth
|classic major third, diminished fourth
|cM3, d4
|cla M3, dim 4
|380.49
|380.49
|[[5/4]]
|[[5/4]]
Line 958: Line 950:
|-
|-
|18
|18
|comma-sharp fourth, augmented third
|comma-wide fourth, augmented third
|C4, A3
|Co-w 4, Aug 3
|526.83
|526.83
|[[15/11]], [[27/20]]
|[[15/11]], [[27/20]]
|-
|-
|19
|19
|neutral fourth
|hemi-augmented fourth
|N4
|h-Aug 4
|556.10
|556.10
|[[11/8]]
|[[11/8]]
Line 971: Line 963:
|20
|20
|classic augmented fourth, diminished fifth
|classic augmented fourth, diminished fifth
|cA4, d5
|cla Aug 4, dim 5
|585.37
|585.37
|[[7/5]], 45/32
|[[7/5]], 45/32
Line 977: Line 969:
|21
|21
|classic diminished fifth, augmented fourth
|classic diminished fifth, augmented fourth
|Cd5, A4
|Cla dim 5, Aug 4
|614.63
|614.63
|[[10/7]], 64/45
|[[10/7]], 64/45
|-
|-
|22
|22
|neutral fifth
|hemi-diminished fifth
|N5
|h-dim 5
|643.90
|643.90
|[[16/11]]
|[[16/11]]
|-
|-
|23
|23
|comma-flat fifth, diminished sixth
|comma-narrow fifth, diminished sixth
|c5, d6
|co-n 5, dim 6
|673.17
|673.17
|[[22/15]], [[40/27]]
|[[22/15]], [[40/27]]
Line 1,019: Line 1,011:
|28
|28
|classic minor sixth, augmented fifth
|classic minor sixth, augmented fifth
|Cm6, A5
|Cla m6, Aug 5
|819.51
|819.51
|[[8/5]]
|[[8/5]]
Line 1,031: Line 1,023:
|30
|30
|classic major sixth, diminished seventh
|classic major sixth, diminished seventh
|cM6, d7
|cla M6, dim 7
|878.05
|878.05
|[[5/3]]
|[[5/3]]
Line 1,061: Line 1,053:
|35
|35
|classic minor seventh, augmented sixth
|classic minor seventh, augmented sixth
|Cm7, A6
|Cla m7, Aug 6
|1024.39
|1024.39
|[[9/5]], [[20/11]]
|[[9/5]], [[20/11]]
Line 1,073: Line 1,065:
|37
|37
|classic major seventh, diminished octave
|classic major seventh, diminished octave
|cM7, d8
|cla M7, dim 8
|1082.93
|1082.93
|[[15/8]]
|[[15/8]]
Line 1,084: Line 1,076:
|-
|-
|39
|39
|neutral octave, super major seventh
|hemi-diminished octave, super major seventh
|N8, SM7
|h-dim 8, SM7
|1141.46
|1141.46
|[[48/25]], [[27/14]], 64/33
|[[48/25]], [[27/14]], 64/33
|-
|-
|40
|40
|comma-flat octave/sub octave
|comma-narrow octave/sub octave
|c8/s8
|co-n 8 / s8
|1170.73
|1170.73
|[[160/81]], 63/32
|[[160/81]], 63/32
Line 1,119: Line 1,111:
|-
|-
|1
|1
|comma-sharp unison/super unison
|comma-narrow unison/super unison
|C1/S1
|Co-w 1 / S1
|22.64
|22.64
|81/80, 64/63, 65/64, 50/49
|81/80, 64/63, 65/64, 50/49
Line 1,144: Line 1,136:
|5
|5
|classic minor second, augmented unison
|classic minor second, augmented unison
|Cm2, A1
|Cla m2, Aug 1
|113.21
|113.21
|16/15, 15/14
|16/15, 15/14
Line 1,150: Line 1,142:
|6
|6
|super classic minor second
|super classic minor second
|SCm2
|SCla m2
|135.85
|135.85
|14/13, 13/12, 27/25
|14/13, 13/12, 27/25
Line 1,156: Line 1,148:
|7
|7
|sub classic minor second
|sub classic minor second
|scM2
|scla M2
|158.49
|158.49
|12/11, 11/10, 800/729
|12/11, 11/10, 800/729
Line 1,162: Line 1,154:
|8
|8
|classic minor second, diminished third
|classic minor second, diminished third
|cM2, d3
|cla M2, dim 3
|181.13
|181.13
|10/9
|10/9
Line 1,198: Line 1,190:
|14
|14
|classic minor third, augmented second
|classic minor third, augmented second
|Cm3, A2
|Cla m3, Aug 2
|316.98
|316.98
|6/5
|6/5
Line 1,204: Line 1,196:
|15
|15
|super classic minor third
|super classic minor third
|SCm3
|SCla m3
|339.62
|339.62
|11/9, 243/200
|11/9, 243/200
Line 1,210: Line 1,202:
|16
|16
|sub classic major third
|sub classic major third
|scM3
|scla M3
|362.26
|362.26
|16/13, 100/81
|16/13, 100/81
Line 1,216: Line 1,208:
|17
|17
|classic major third, diminished fourth
|classic major third, diminished fourth
|cM3, d4
|cla M3, dim 4
|384.91
|384.91
|5/4
|5/4
Line 1,251: Line 1,243:
|-
|-
|23
|23
|comma-sharp fourth, augmented third
|comma-wide fourth, augmented third
|C4, A3
|Co-w 4, Aug 3
|520.75
|520.75
|27/20
|27/20
|-
|-
|24
|24
|super comma-sharp fourth
|super comma-wide fourth
|SC4
|SCo-w 4
|543.40
|543.40
|11/8, 15/11
|11/8, 15/11
|-
|-
|25
|25
|sub comma-flat augmented fourth
|sub classic augmented fourth
|scA4
|scla Aug 4
|566.04
|566.04
|18/13
|18/13
Line 1,270: Line 1,262:
|26
|26
|classic augmented fourth, diminished fifth
|classic augmented fourth, diminished fifth
|cA4, d5
|cla Aug 4, dim 5
|588.68
|588.68
|7/5, 45/32
|7/5, 45/32
Line 1,276: Line 1,268:
|27
|27
|classic diminished fifth, augmented fourth
|classic diminished fifth, augmented fourth
|Cd5, A4
|Cla dim 5, Aug 4
|611.32
|611.32
|10/7, 64/45
|10/7, 64/45
Line 1,282: Line 1,274:
|28
|28
|super classic diminished fifth
|super classic diminished fifth
|SCd5
|SCla dim 5
|633.96
|633.96
|13/9
|13/9
|-
|-
|29
|29
|sub comma-flat fifth
|sub comma-narrow fifth
|sc5
|sco-n 5
|656.60
|656.60
|16/11, 22/15
|16/11, 22/15
|-
|-
|30
|30
|comma-flat fifth
|comma-narrow fifth
|c5
|co-n 5
|679.25
|679.25
|40/27
|40/27
Line 1,330: Line 1,322:
|36
|36
|classic minor sixth, augmented fifth
|classic minor sixth, augmented fifth
|Cm6, A5
|Cla m6, Aug 5
|815.09
|815.09
|8/5
|8/5
Line 1,336: Line 1,328:
|37
|37
|super classic minor sixth
|super classic minor sixth
|SCm6
|SCla m6
|837.74
|837.74
|13/8, 81/50
|13/8, 81/50
Line 1,342: Line 1,334:
|38
|38
|sub classic major sixth
|sub classic major sixth
|scM6
|scla M6
|860.38
|860.38
|18/11, 400/243
|18/11, 400/243
Line 1,348: Line 1,340:
|39
|39
|classic major sixth, diminished seventh
|classic major sixth, diminished seventh
|cM6, d7
|cla M6, dim 7
|883.02
|883.02
|5/3
|5/3
Line 1,390: Line 1,382:
|46
|46
|super classic minor seventh
|super classic minor seventh
|SCm7
|SCla m7
|1041.51
|1041.51
|11/6, 20/11, 729/400
|11/6, 20/11, 729/400
Line 1,396: Line 1,388:
|47
|47
|sub classic major seventh
|sub classic major seventh
|scM7
|scla M7
|1064.15
|1064.15
|13/7, 24/13, 50/27
|13/7, 24/13, 50/27
Line 1,402: Line 1,394:
|48
|48
|classic major seventh, diminished octave
|classic major seventh, diminished octave
|cM7, d8
|cla M7, dim 8
|1086.79
|1086.79
|15/8
|15/8
Line 1,425: Line 1,417:
|-
|-
|52
|52
|comma-flat octave/sub octave
|comma-narrow octave/sub octave
|c8/S8
|co-n 8 / S8
|1177.36
|1177.36
|160/81, 63/32, 128/65
|160/81, 63/32, 128/65
Line 1,532: Line 1,524:
|0.82
|0.82
|}
|}
The patent val, 25p performs best here. We may still use either 25b or 25d if we desire, however if we want to use 25p, we may employ ups and downs to name the intervals that do not carry a separate function under this mapping:
The patent val, 25p performs best here. We may still use either 25b or 25d if we desire, however if we want to use 25p, we may narrow and wide prefixes to name the intervals that do not carry a separate function under this mapping:


25edo: P1 ^P1/^m2 Cm2 cM2 vM2 M2/m3 ^m3 Cm3 cM3 vM3/v4 P4 ^4 C4 c5 v5 P5 ^5/^m6 Cm6 cM6 vM6 M6/m7 ^m7 Cm7 cM7 vM7/vP8 P8
25edo: P1 WP1/Wm2 Cm2 cM2 nM2 M2/m3 Wm3 Cm3 cM3 nM3/n4 P4 W4 C4 c5 n5 P5 W5/nm6 Cm6 cM6 nM6 M6/m7 Wm7 Cm7 cM7 nM7/nP8 P8


In 25edo 81/80 is represented by 2 degrees rather than by a single degree, so our scheme doesn't completely work for 25edo, but our scheme is based on the diatonic scale, which in 25edo has pretty much completely broken down.
In 25edo 81/80 is represented by 2 degrees rather than by a single degree, so our scheme doesn't completely work for 25edo, but our scheme is based on the diatonic scale, which in 25edo has pretty much completely broken down.


=== 7''n''-edos ===
=== 7''n''-edos ===
At the other limit, in 7edo the large and small steps of the diatonic scale are the same size, and the apotome is tempered out, hence every degree may be written as a neutral, however the interval-name ordering rules lead us to the following primary interval name set:
At the other limit, in 7edo the large and small steps of the diatonic scale are the same size, and the apotome is tempered out and therefore major and minor are equated with each other, and therefor with neutral:


P1 N2 N3 P4 P5 N6 N7 P8, equivalent to Neutral[7] 3|3.
7edo: P1 N2 N3 P4 P5 N6 N7 P8, equivalent to Neutral[7] 3|3.


It is easy to apply our scheme to 14edo:
It is easy to apply our scheme to 14edo:
Line 1,555: Line 1,547:
21edo: P1 S1 sm2 N2 SM2 sm3 N3 SM3 s4 P4 SA4 sd5 P5 S5 sm6 N6 SM6 sm7 N7 SM7 d8 P8
21edo: P1 S1 sm2 N2 SM2 sm3 N3 SM3 s4 P4 SA4 sd5 P5 S5 sm6 N6 SM6 sm7 N7 SM7 d8 P8


In 28edo, 81/80 is also subtended by -1 degrees, but since 64/63 is subtended by 2 degrees we cannot label all of our intervals using 'S' and 's'. Considering the diatonic scale is all neutral, we can still build a well-ordered interval names set using alterations of 81/80:
In 28edo, 81/80 is also subtended by -1 degrees, but since 64/63 is subtended by 2 degrees we cannot label all of our intervals using 'S' and 's'. If we relabel P4 'hA4', and P5 'hd5', we can still build a well-ordered interval names set using alterations of 81/80:


28edo: P1 cA1 SA1/sm2 Cm2 N2 cM2 SM2/sm3 Cm3 N3 cM3 SM3/s4 C4 N4 cA4 SA4/sd5 Cd5 N5 c5 S5/sm6 Cm6 N6 cM6 SM6/sm7 Cm7 N7 cM7 SM7/sd8 Cd8 P8
28edo: P1 cA1 SA1/sm2 Cm2 N2 cM2 SM2/sm3 Cm3 N3 cM3 SM3/s4 C4 hA4 cA4 SA4/sd5 Cd5 hd5 c5 S5/sm6 Cm6 N6 cM6 SM6/sm7 Cm7 N7 cM7 SM7/sd8 Cd8 P8


=== Super-flat edos ===
=== Super-flat edos ===
Retrieved from "https://en.xen.wiki/w/SHEFKHED_interval_names"