SHEFKHED interval names: Difference between revisions

Line 455:

*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.

*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 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 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'.

*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.

*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.

*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-W' and 'co-n'. These names are chosen because they start with the same letter as the other prefixes that infer alteration by 81/80~~.

*Perfect intervals may also be given the prefixes 'C' and 'c' to imply augmentation and diminution by the syntonic comma. Where 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. Following this example but using 'wide' and 'narrow' instead of 'sharp' and 'flat', we associate the long-form 'comma-wide' and 'comma-narrow' to infer movement up or down a syntonic comma from Perfect intervals.

*For seconds, thirds, sixths and sevenths, any interval exactly half-way major and minor is labelled 'neutral', with short-form 'N'.

*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.

*Where neutral intervals split the apotome, to pair with neutral when acting on perfect intervals are 'hemi augmented' and 'hemi diminished', with short form 'hA' and 'hd'. '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 ~~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.

*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', 'C', 'c' and 'C' 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 ~~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 said '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' that may be said 'unind'. Similarly the interval half-way between M7 and P8 is given the short-form '1-2' and long-form 'seventh-octave' that may be said 'sevtave'. The interval splitting the fourth, lying half-way between M2 and m3 is given the short-form '2-3', with long-form 'second-third' that may be said 'serd', and it's octave complement, lying half-way between M6 and m7 is given the short-form '6-7', with long-form 'sixth-seventh', that may be said 'sinth'. The interval half-way between M3 and P4 is given the short-form '3-4', with long-form 'third-fourth', that may be said 'thourth', and it's octave-complement, the interval half-way between P5 and m6 has short-form '5-6', with long-form 'fifth-sixth', that may be said '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, 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.

*For completeness, the interval '4-5', long form 'fourth-fifth' that may be said '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.

*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 'wide' and 'narrow' prefixes, with short form 'W' and 'n' 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.

*Where 'c', 'a' and 'd' are also note names, in some contexts short-form interval names may be confused as short-form chord names, such as Cm7, which is a minor seventh chord rooted on C. Normally context differentiates between, or it can simple be added 'the interval' or 'the chord', but if alternative abbreviations for interval-names that may not be confused with chord names are desired, such 'mid-form' abbreviations are provided in the following tables, which summarise the prefixes listed in this section.

{| class="wikitable"

|+Prefixes summary

!Long-form

!Short-form

!Mid-form

!Alteration

|-

|perfect

|P

|Prf

|none

|-

|major

|M

|Maj

|none

|-

|minor

|m

|min

|none

|-

|augmented

|A

|Aug

|up an apotome

|-

|diminished

|d

|dim

|down an apotome

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|super

|S

|Sub

|up 64/63 (or 65/64)

|-

|sub

|s

|spr

|down 64/63 (or 65/64)

|-

|classic

|C

|Cla

|up 81/80

|-

|classic

|c

|cla

|down 81/80

|-

|comma-wide

|C

|Co-W

|up 81/80

|-

|comma-narrow

|c

|co-n

|down 81/80

Line 520:

Line 533:

|wide

|W

|Wde

|up a single degree

|-

|narrow

|n

|nrw

|down a single degree

|}

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Line 545:

!Long-form

!Short-form

!Mid-form

!Description

|-

|unison-second

|1-2 (~~unicond~~)

|1-2 (unind)

|half of the limma, half way between P1 and m2

|-

|second-third

|2-3 (serd)

|half of P4, halfway between M2 and m3

|-

|third-fourth

|3-4 (thourth)

|half of M6, halfway between M3 and P4

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Line 565:

|fourth-fifth

|4-5 (firth)

|half of the octave, halfway between ~~Aug 4~~ and ~~dim 5~~

|4-5 (firth)

|half of the octave, halfway between A4 and d5

|-

|fifth-sixth

|5-6 (fixth)

|half of M10, halfway between P5 and m6

|-

|sixth-seventh

|6-7 (sinth)

|halfway between P5 and P8, halfway between M6 and m7

|-

|seventh-octave

|7-8 (sevtave)

|halfway between M7 and P8

|-

|neutral

|N

|halfway between M and m

|-

|hemi augmented

|hA

|h-Aug

|halfway between P and ~~Aug~~

|halfway between P and A

|-

|hemi diminished

|hd

|h-dim

|halfway between P and ~~dim~~

|halfway between P and d

|}

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Interval names are ranked in nine tiers.

#Perfect and neutral

#Major, minor, ~~Aug 4~~ and ~~dim 5~~.

#Major, minor, A4 and d5.

#~~h-Aug 4~~ and ~~h-dim 5~~

#hA4 and hd5

#'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~~

#'S', 's', 'C', 'c', 'SC' and 'sc' prefixes to major, minor, perfect intervals and to A4 and d5

#~~h-Aug 1~~ and ~~h-dim 8~~ (plus any other ~~h-Augs~~ and ~~h-dims~~ if needed)

#hA1 and hd8 (plus any other hAs and hds if needed)

#Intermediates

#Remaining augmented and diminished intervals (for when the chroma is subtended by more than a single (positive) step of the edo)

#'S', 's', '~~Cla~~', '~~cla~~', '~~Co-w', 'co-n', 'SCla', 'scla', 'SCo-w~~' and '~~sco-n~~' prefixes to augmented and diminished intervals

#'S', 's', 'C', 'c', 'SC' and 'sc' prefixes to augmented and diminished intervals

#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'.

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.

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.

=== 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~~. 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'~~.

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.

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

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|minor second

|m2

|comma-~~sharp~~ unison/super minor second

|comma-wide unison / super minor second

|~~Co-W 1~~ / sm2

|C1/sm2

|54.55

|33/32, 34/33, 25/24, 81/80

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|2

|classic minor second

|~~Cla m2~~

|Cm2

|diminished third

|~~dim 3~~

|d3

|109.09

|18/17, 17/16, 16/15, 15/14

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|3

|classic major second

|~~cla M2~~

|cM2

|augmented unison

|~~Aug 1~~

|A1

|163.64

|11/10, 10/9

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|6

|classic minor third

|~~Cla m3~~

|Cm3

|diminished fourth

|~~dim 4~~

|d4

|327.27

|6/5, 17/14, 11/9

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|7

|classic major third

|~~cla M3~~

|cM3

|augmented second

|~~Aug 2~~

|A2

|381.82

|5/4

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|10

|comma-wide fourth

|~~Co-W 4~~

|C4

|diminished fifth

|~~dim 5~~

|d5

|545.45

|11/8, 15/11, 27/20

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|classic augmented fourth

classic diminished fifth

|~~cla Aug 4~~

|cA4

~~Cla dim 5~~

Cd5

|fourth-fifth

|4-5 (firth)

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|12

|comma-narrow fifth

|~~co-n 5~~

|c5

|augmented fourth

|~~Aug 4~~

|A4

|654.55

|16/11, 22/15, 40/27

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|15

|classic minor sixth

|~~Cla m6~~

|Cm6

|diminished seventh

|~~dim 7~~

|d7

|818.18

|8/5

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|16

|classic major sixth

|~~cla M6~~

|cM6

|augmented fifth

|~~Aug 5~~

|A5

|872.73

|5/3, 18/11, 28/17

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|19

|classic minor seventh

|~~Cla m7~~

|Cm7

|diminished octave

|~~dim 8~~

|d8

|1036.36

|20/11, 9/5

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|20

|classic major seventh

|~~cla M7~~

|cM7

|augmented sixth

|~~Aug 6~~

|A6

|1090.91

|15/8, 32/17, 17/9, 28/15

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|major seventh

|M7

|super major seventh / comma-~~flat~~ octave

|super major seventh / comma-narrow octave

|SM7 / ~~co-n 8~~

|SM7/c8

|1145.45

|33/17, 64/33, 48/25, 160/81

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|1

|comma-wide unison/super unison

|~~Co-W 1~~ / S1

|C1/S1

|29.27

|[[81/80]], 64/63

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|2

|subminor second, hemi-augmented unison

|sm2, ~~h-Aug 1~~

|sm2, hA1

|58.54

|[[25/24]], [[28/27]], [[33/32]]

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|4

|classic minor second, augmented unison

|~~Cla m2~~, ~~Aug 1~~

|Cm2, A1

|117.07

|[[16/15]], [[15/14]]

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|6

|classic major second, diminished third

|~~cla M2~~, ~~dim 3~~

|cM2, d3

|175.61

|[[10/9]], [[11/10]]

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|11

|classic minor third, augmented second

|~~Cla m3~~, ~~Aug 2~~

|Cm3, A2

|321.95

|[[6/5]]

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|13

|classic major third, diminished fourth

|~~cla M3~~, ~~dim 4~~

|cM3, d4

|380.49

|[[5/4]]

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|18

|comma-wide fourth, augmented third

|~~Co-W 4~~, ~~Aug 3~~

|C4, A3

|526.83

|[[15/11]], [[27/20]]

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|19

|hemi-augmented fourth

|~~h-Aug 4~~

|hA4

|556.10

|[[11/8]]

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|20

|classic augmented fourth, diminished fifth

|~~cla Aug 4~~, ~~dim 5~~

|cA4, d5

|585.37

|[[7/5]], 45/32

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|21

|classic diminished fifth, augmented fourth

|~~Cla dim 5~~, ~~Aug 4~~

|Cd5, A4

|614.63

|[[10/7]], 64/45

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Line 1,001:

|22

|hemi-diminished fifth

|~~h-dim 5~~

|h5

|643.90

|[[16/11]]

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|23

|comma-narrow fifth, diminished sixth

|~~co-n 5~~, ~~dim 6~~

|c5, d6

|673.17

|[[22/15]], [[40/27]]

Line 1,011:

Line 1,037:

|28

|classic minor sixth, augmented fifth

|~~Cla m6~~, ~~Aug 5~~

|Cm6, A5

|819.51

|[[8/5]]

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Line 1,049:

|30

|classic major sixth, diminished seventh

|~~cla M6~~, ~~dim 7~~

|cM6, d7

|878.05

|[[5/3]]

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Line 1,079:

|35

|classic minor seventh, augmented sixth

|~~Cla m7~~, ~~Aug 6~~

|Cm7, A6

|1024.39

|[[9/5]], [[20/11]]

Line 1,065:

Line 1,091:

|37

|classic major seventh, diminished octave

|~~cla M7~~, ~~dim 8~~

|cM7, d8

|1082.93

|[[15/8]]

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Line 1,103:

|39

|super major seventh, hemi-diminished octave

|SM7, ~~h-dim 8~~

|SM7, hd8

|1141.46

|[[48/25]], [[27/14]], 64/33

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Line 1,109:

|40

|comma-narrow octave/sub octave

|~~co-n 8~~ / s8

|c8/s8

|1170.73

|[[160/81]], 63/32

Line 1,112:

Line 1,138:

|1

|comma-narrow unison/super unison

|~~Co-W 1~~ / S1

|C1/S1

|22.64

|81/80, 64/63, 65/64, 50/49

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Line 1,162:

|5

|classic minor second, augmented unison

|~~Cla m2~~, ~~Aug 1~~

|Cm2, A1

|113.21

|16/15, 15/14

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Line 1,168:

|6

|super classic minor second

|~~SCla m2~~

|SCm2

|135.85

|14/13, 13/12, 27/25

Line 1,148:

Line 1,174:

|7

|sub classic minor second

|~~scla M2~~

|scM2

|158.49

|12/11, 11/10, 800/729

Line 1,154:

Line 1,180:

|8

|classic minor second, diminished third

|~~cla M2~~, ~~dim 3~~

|cM2, dim3

|181.13

|10/9

Line 1,190:

Line 1,216:

|14

|classic minor third, augmented second

|~~Cla m3~~, ~~Aug 2~~

|Cm3, A2

|316.98

|6/5

Line 1,196:

Line 1,222:

|15

|super classic minor third

|~~SCla m3~~

|SCm3

|339.62

|11/9, 243/200

Line 1,202:

Line 1,228:

|16

|sub classic major third

|~~scla M3~~

|scM3

|362.26

|16/13, 100/81

Line 1,208:

Line 1,234:

|17

|classic major third, diminished fourth

|~~cla M3~~, ~~dim 4~~

|cM3, d4

|384.91

|5/4

Line 1,244:

Line 1,270:

|23

|comma-wide fourth, augmented third

|~~Co-W 4~~, ~~Aug 3~~

|C4, A3

|520.75

|27/20

Line 1,250:

Line 1,276:

|24

|super comma-wide fourth

|~~SCo-W 4~~

|SC4

|543.40

|11/8, 15/11

Line 1,256:

Line 1,282:

|25

|sub classic augmented fourth

|~~scla Aug 4~~

|scA4

|566.04

|18/13

Line 1,262:

Line 1,288:

|26

|classic augmented fourth, diminished fifth

|~~cla Aug 4~~, ~~dim 5~~

|cA4, d5

|588.68

|7/5, 45/32

Line 1,268:

Line 1,294:

|27

|classic diminished fifth, augmented fourth

|~~Cla dim 5~~, ~~Aug 4~~

|Cd5, A4

|611.32

|10/7, 64/45

Line 1,274:

Line 1,300:

|28

|super classic diminished fifth

|~~SCla dim 5~~

|SCd5

|633.96

|13/9

Line 1,280:

Line 1,306:

|29

|sub comma-narrow fifth

|~~sco-n 5~~

|sc5

|656.60

|16/11, 22/15

Line 1,286:

Line 1,312:

|30

|comma-narrow fifth

|~~co-n 5~~

|c5

|679.25

|40/27

Line 1,322:

Line 1,348:

|36

|classic minor sixth, augmented fifth

|~~Cla m6~~, ~~Aug 5~~

|Cm6, A5

|815.09

|8/5

Line 1,328:

Line 1,354:

|37

|super classic minor sixth

|~~SCla m6~~

|SCm6

|837.74

|13/8, 81/50

Line 1,334:

Line 1,360:

|38

|sub classic major sixth

|~~scla M6~~

|scM6

|860.38

|18/11, 400/243

Line 1,340:

Line 1,366:

|39

|classic major sixth, diminished seventh

|~~cla M6~~, ~~dim 7~~

|cM6, d7

|883.02

|5/3

Line 1,382:

Line 1,408:

|46

|super classic minor seventh

|~~SCla m7~~

|SCm7

|1041.51

|11/6, 20/11, 729/400

Line 1,388:

Line 1,414:

|47

|sub classic major seventh

|~~scla M7~~

|scM7

|1064.15

|13/7, 24/13, 50/27

Line 1,394:

Line 1,420:

|48

|classic major seventh, diminished octave

|~~cla M7~~, ~~dim 8~~

|cM7, d8

|1086.79

|15/8

Line 1,418:

Line 1,444:

|52

|comma-narrow octave/sub octave

|~~co-n 8~~ / S8

|c8/S8

|1177.36

|160/81, 63/32, 128/65

Line 1,554:

Line 1,580:

There are edos whose best fifth is flatter even than 4\7. In such edos major intervals are smaller than minor intervals, augmented smaller than major and diminished larger than minor. We expand our definition of well-ordered intervals to include that within each degree... ≤ dd ≤ d ≤ m ≤ M ≤ A ≤ AA ≤ ... or ... ≤ dd ≤ d ≤ P ≤ A ≤ AA ≤ ..., and where sc _ ≤ s/c_ ≤ _ ≤ S/C_ ≤ SC _ (where '_' represents any of ... dd, d, m, (P), M, A, AA ...). In order to obtain well-ordered interval-name sets, we use enharmonic equivalences, replacing diatonic intervals with altered intervals.

In super flat edos, the fifths are so flat that the major third, from four fifths approximates the classic minor third, 6/5 and the minor third approximates the classic major third, 5/4, tempering out 135/128, resulting in [[Mavila temperament]]. Mavila temperament can be defined in the 5-limit using the enharmonic equivalence cla M = m, where meantone can be defined by ~~cla M~~ = M, and schismatic by ~~cla M~~ ''n'' = ~~dim~~ ''n+1'' (superpyth in 2.3.7 can be defined by SM = M). Mavila[7] 3|3 reads the same as Meantone[7] 3|3: P1 M2 m3 P4 P5 M6 m7 P8, however Mavila[9] 4|4 has diatonic interval names:

In super flat edos, the fifths are so flat that the major third, from four fifths approximates the classic minor third, 6/5 and the minor third approximates the classic major third, 5/4, tempering out 135/128, resulting in [[Mavila temperament]]. Mavila temperament can be defined in the 5-limit using the enharmonic equivalence cla M = m, where meantone can be defined by cM = M, and schismatic by cM''n'' = d''n+1'' (superpyth in 2.3.7 can be defined by SM = M). Mavila[7] 3|3 reads the same as Meantone[7] 3|3: P1 M2 m3 P4 P5 M6 m7 P8, however Mavila[9] 4|4 has diatonic interval names:

P1 M2 M3 m3 P4 P5 M6 m6 m7 P8.

@@ Line 455: / Line 455: @@
 *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.
-*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 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 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'.
-*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.
+*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.
-*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-W' and 'co-n'. These names are chosen because they start with the same letter as the other prefixes that infer alteration by 81/80.
+*Perfect intervals may also be given the prefixes 'C' and 'c' to imply augmentation and diminution by the syntonic comma. Where 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. Following this example but using 'wide' and 'narrow' instead of 'sharp' and 'flat', we associate the long-form 'comma-wide' and 'comma-narrow' to infer movement up or down a syntonic comma from Perfect intervals.
 *For seconds, thirds, sixths and sevenths, any interval exactly half-way major and minor is labelled 'neutral', with short-form 'N'.
-*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.
+*Where neutral intervals split the apotome, to pair with neutral when acting on perfect intervals are 'hemi augmented' and 'hemi diminished', with short form 'hA' and 'hd'. '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 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.
+*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', 'C', 'c' and 'C' 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 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 said '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' that may be said 'unind'. Similarly the interval half-way between M7 and P8 is given the short-form '1-2' and long-form 'seventh-octave' that may be said 'sevtave'. The interval splitting the fourth, lying half-way between M2 and m3 is given the short-form '2-3', with long-form 'second-third' that may be said 'serd', and it's octave complement, lying half-way between M6 and m7 is given the short-form '6-7', with long-form 'sixth-seventh', that may be said 'sinth'. The interval half-way between M3 and P4 is given the short-form '3-4', with long-form 'third-fourth', that may be said 'thourth', and it's octave-complement, the interval half-way between P5 and m6 has  short-form '5-6', with long-form 'fifth-sixth', that may be said '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, 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.
+*For completeness, the interval '4-5', long form 'fourth-fifth' that may be said '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.
 *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 'wide' and 'narrow' prefixes, with short form 'W' and 'n' 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.
+*Where 'c', 'a' and 'd' are also note names, in some contexts short-form interval names may be confused as short-form chord names, such as Cm7, which is a minor seventh chord rooted on C. Normally context differentiates between, or it can simple be added 'the interval' or 'the chord', but if alternative abbreviations for interval-names that may not be confused with chord names are desired, such 'mid-form' abbreviations are provided in the following tables, which summarise the prefixes listed in this section.
 {| class="wikitable"
 |+Prefixes summary
 !Long-form
 !Short-form
+!Mid-form
 !Alteration
 |-
 |perfect
 |P
+|Prf
 |none
 |-
 |major
 |M
+|Maj
 |none
 |-
 |minor
 |m
+|min
 |none
 |-
 |augmented
+|A
 |Aug
 |up an apotome
 |-
 |diminished
+|d
 |dim
 |down an apotome
@@ Line 496: / Line 503: @@
 |super
 |S
+|Sub
 |up 64/63 (or 65/64)
 |-
 |sub
 |s
+|spr
 |down 64/63 (or 65/64)
 |-
 |classic
+|C
 |Cla
 |up 81/80
 |-
 |classic
+|c
 |cla
 |down 81/80
 |-
 |comma-wide
+|C
 |Co-W
 |up 81/80
 |-
 |comma-narrow
+|c
 |co-n
 |down 81/80
@@ Line 520: / Line 533: @@
 |wide
 |W
+|Wde
 |up a single degree
 |-
 |narrow
 |n
+|nrw
 |down a single degree
 |}
@@ Line 530: / Line 545: @@
 !Long-form
 !Short-form
+!Mid-form
 !Description
 |-
 |unison-second
-|1-2 (unicond)
+|1-2 (unind)
+|1-2 (unind)
 |half of the limma, half way between P1 and m2
 |-
 |second-third
+|2-3 (serd)
 |2-3 (serd)
 |half of P4, halfway between M2 and m3
 |-
 |third-fourth
+|3-4 (thourth)
 |3-4 (thourth)
 |half of M6, halfway between M3 and P4
@@ Line 546: / Line 565: @@
 |fourth-fifth
 |4-5 (firth)
-|half of the octave, halfway between Aug 4 and dim 5
+|4-5 (firth)
+|half of the octave, halfway between A4 and d5
 |-
 |fifth-sixth
+|5-6 (fixth)
 |5-6 (fixth)
 |half of M10, halfway between P5 and m6
 |-
 |sixth-seventh
+|6-7 (sinth)
 |6-7 (sinth)
 |halfway between P5 and P8, halfway between M6 and m7
 |-
 |seventh-octave
+|7-8 (sevtave)
 |7-8 (sevtave)
 |halfway between M7 and P8
 |-
 |neutral
+|N
 |N
 |halfway between M and m
 |-
 |hemi augmented
+|hA
 |h-Aug
-|halfway between P and Aug
+|halfway between P and A
 |-
 |hemi diminished
+|hd
 |h-dim
-|halfway between P and dim
+|halfway between P and d
 |}
@@ Line 578: / Line 604: @@
 Interval names are ranked in nine tiers.
 #Perfect and neutral
-#Major, minor, Aug 4 and dim 5.
+#Major, minor, A4 and d5.
-#h-Aug 4 and h-dim 5
+#hA4 and hd5
-#'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
+#'S', 's', 'C', 'c', 'SC' and 'sc' prefixes to major, minor, perfect intervals and to A4 and d5
-#h-Aug 1 and h-dim 8 (plus any other h-Augs and h-dims if needed)
+#hA1 and hd8 (plus any other hAs and hds if needed)
 #Intermediates
 #Remaining augmented and diminished intervals (for when the chroma is subtended by more than a single (positive) step of the edo)
-#'S', 's', 'Cla', 'cla', 'Co-w', 'co-n', 'SCla', 'scla', 'SCo-w' and 'sco-n' prefixes to augmented and diminished intervals
+#'S', 's', 'C', 'c', 'SC' and 'sc' prefixes to augmented and diminished intervals
 #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'.
-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.
+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.
 === 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. 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'.
+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.
 edo: P1 m2 M2 m3 M3 P4 A4/d5 P5 m6 M6 m7 M7 P8
@@ Line 657: / Line 683: @@
 |minor second
 |m2
-|comma-sharp unison/super minor second
+|comma-wide unison / super minor second
-|Co-W 1 / sm2
+|C1/sm2
 |54.55
 |33/32, 34/33, 25/24, 81/80
@@ Line 664: / Line 690: @@
 |2
 |classic minor second
-|Cla m2
+|Cm2
 |diminished third
-|dim 3
+|d3
 |109.09
 |18/17, 17/16, 16/15, 15/14
@@ Line 672: / Line 698: @@
 |3
 |classic major second
-|cla M2
+|cM2
 |augmented unison
-|Aug 1
+|A1
 |163.64
 |11/10, 10/9
@@ Line 696: / Line 722: @@
 |6
 |classic minor third
-|Cla m3
+|Cm3
 |diminished fourth
-|dim 4
+|d4
 |327.27
 |6/5, 17/14, 11/9
@@ Line 704: / Line 730: @@
 |7
 |classic major third
-|cla M3
+|cM3
 |augmented second
-|Aug 2
+|A2
 |381.82
 |5/4
@@ Line 728: / Line 754: @@
 |10
 |comma-wide fourth
-|Co-W 4
+|C4
 |diminished fifth
-|dim 5
+|d5
 |545.45
 |11/8, 15/11, 27/20
@@ Line 737: / Line 763: @@
 |classic augmented fourth
 classic diminished fifth
-|cla Aug 4
+|cA4
-Cla dim 5
+Cd5
 |fourth-fifth
 |4-5 (firth)
@@ Line 747: / Line 773: @@
 |12
 |comma-narrow fifth
-|co-n 5
+|c5
 |augmented fourth
-|Aug 4
+|A4
 |654.55
 |16/11, 22/15, 40/27
@@ Line 771: / Line 797: @@
 |15
 |classic minor sixth
-|Cla m6
+|Cm6
 |diminished seventh
-|dim 7
+|d7
 |818.18
 |8/5
@@ Line 779: / Line 805: @@
 |16
 |classic major sixth
-|cla M6
+|cM6
 |augmented fifth
-|Aug 5
+|A5
 |872.73
 |5/3, 18/11, 28/17
@@ Line 803: / Line 829: @@
 |19
 |classic minor seventh
-|Cla m7
+|Cm7
 |diminished octave
-|dim 8
+|d8
 |1036.36
 |20/11, 9/5
@@ Line 811: / Line 837: @@
 |20
 |classic major seventh
-|cla M7
+|cM7
 |augmented sixth
-|Aug 6
+|A6
 |1090.91
 |15/8, 32/17, 17/9, 28/15
@@ Line 820: / Line 846: @@
 |major seventh
 |M7
-|super major seventh / comma-flat octave
+|super major seventh / comma-narrow octave
-|SM7 / co-n 8
+|SM7/c8
 |1145.45
 |33/17, 64/33, 48/25, 160/81
@@ Line 849: / Line 875: @@
 |1
 |comma-wide unison/super unison
-|Co-W 1 / S1
+|C1/S1
 |29.27
 |[[81/80]], 64/63
@@ Line 855: / Line 881: @@
 |2
 |subminor second, hemi-augmented unison
-|sm2, h-Aug 1
+|sm2, hA1
 |58.54
 |[[25/24]], [[28/27]], [[33/32]]
@@ Line 867: / Line 893: @@
 |4
 |classic minor second, augmented unison
-|Cla m2, Aug 1
+|Cm2, A1
 |117.07
 |[[16/15]], [[15/14]]
@@ Line 879: / Line 905: @@
 |6
 |classic major second, diminished third
-|cla M2, dim 3
+|cM2, d3
 |175.61
 |[[10/9]], [[11/10]]
@@ Line 909: / Line 935: @@
 |11
 |classic minor third, augmented second
-|Cla m3, Aug 2
+|Cm3, A2
 |321.95
 |[[6/5]]
@@ Line 921: / Line 947: @@
 |13
 |classic major third, diminished fourth
-|cla M3, dim 4
+|cM3, d4
 |380.49
 |[[5/4]]
@@ Line 951: / Line 977: @@
 |18
 |comma-wide fourth, augmented third
-|Co-W 4, Aug 3
+|C4, A3
 |526.83
 |[[15/11]], [[27/20]]
@@ Line 957: / Line 983: @@
 |19
 |hemi-augmented fourth
-|h-Aug 4
+|hA4
 |556.10
 |[[11/8]]
@@ Line 963: / Line 989: @@
 |20
 |classic augmented fourth, diminished fifth
-|cla Aug 4, dim 5
+|cA4, d5
 |585.37
 |[[7/5]], 45/32
@@ Line 969: / Line 995: @@
 |21
 |classic diminished fifth, augmented fourth
-|Cla dim 5, Aug 4
+|Cd5, A4
 |614.63
 |[[10/7]], 64/45
@@ Line 975: / Line 1,001: @@
 |22
 |hemi-diminished fifth
-|h-dim 5
+|h5
 |643.90
 |[[16/11]]
@@ Line 981: / Line 1,007: @@
 |23
 |comma-narrow fifth, diminished sixth
-|co-n 5, dim 6
+|c5, d6
 |673.17
 |[[22/15]], [[40/27]]
@@ Line 1,011: / Line 1,037: @@
 |28
 |classic minor sixth, augmented fifth
-|Cla m6, Aug 5
+|Cm6, A5
 |819.51
 |[[8/5]]
@@ Line 1,023: / Line 1,049: @@
 |30
 |classic major sixth, diminished seventh
-|cla M6, dim 7
+|cM6, d7
 |878.05
 |[[5/3]]
@@ Line 1,053: / Line 1,079: @@
 |35
 |classic minor seventh, augmented sixth
-|Cla m7, Aug 6
+|Cm7, A6
 |1024.39
 |[[9/5]], [[20/11]]
@@ Line 1,065: / Line 1,091: @@
 |37
 |classic major seventh, diminished octave
-|cla M7, dim 8
+|cM7, d8
 |1082.93
 |[[15/8]]
@@ Line 1,077: / Line 1,103: @@
 |39
 |super major seventh, hemi-diminished octave
-|SM7, h-dim 8
+|SM7, hd8
 |1141.46
 |[[48/25]], [[27/14]], 64/33
@@ Line 1,083: / Line 1,109: @@
 |40
 |comma-narrow octave/sub octave
-|co-n 8 / s8
+|c8/s8
 |1170.73
 |[[160/81]], 63/32
@@ Line 1,112: / Line 1,138: @@
 |1
 |comma-narrow unison/super unison
-|Co-W 1 / S1
+|C1/S1
 |22.64
 |81/80, 64/63, 65/64, 50/49
@@ Line 1,136: / Line 1,162: @@
 |5
 |classic minor second, augmented unison
-|Cla m2, Aug 1
+|Cm2, A1
 |113.21
 |16/15, 15/14
@@ Line 1,142: / Line 1,168: @@
 |6
 |super classic minor second
-|SCla m2
+|SCm2
 |135.85
 |14/13, 13/12, 27/25
@@ Line 1,148: / Line 1,174: @@
 |7
 |sub classic minor second
-|scla M2
+|scM2
 |158.49
 |12/11, 11/10, 800/729
@@ Line 1,154: / Line 1,180: @@
 |8
 |classic minor second, diminished third
-|cla M2, dim 3
+|cM2, dim3
 |181.13
 |10/9
@@ Line 1,190: / Line 1,216: @@
 |14
 |classic minor third, augmented second
-|Cla m3, Aug 2
+|Cm3, A2
 |316.98
 |6/5
@@ Line 1,196: / Line 1,222: @@
 |15
 |super classic minor third
-|SCla m3
+|SCm3
 |339.62
 |11/9, 243/200
@@ Line 1,202: / Line 1,228: @@
 |16
 |sub classic major third
-|scla M3
+|scM3
 |362.26
 |16/13, 100/81
@@ Line 1,208: / Line 1,234: @@
 |17
 |classic major third, diminished fourth
-|cla M3, dim 4
+|cM3, d4
 |384.91
 |5/4
@@ Line 1,244: / Line 1,270: @@
 |23
 |comma-wide fourth, augmented third
-|Co-W 4, Aug 3
+|C4, A3
 |520.75
 |27/20
@@ Line 1,250: / Line 1,276: @@
 |24
 |super comma-wide fourth
-|SCo-W 4
+|SC4
 |543.40
 |11/8, 15/11
@@ Line 1,256: / Line 1,282: @@
 |25
 |sub classic augmented fourth
-|scla Aug 4
+|scA4
 |566.04
 |18/13
@@ Line 1,262: / Line 1,288: @@
 |26
 |classic augmented fourth, diminished fifth
-|cla Aug 4, dim 5
+|cA4, d5
 |588.68
 |7/5, 45/32
@@ Line 1,268: / Line 1,294: @@
 |27
 |classic diminished fifth, augmented fourth
-|Cla dim 5, Aug 4
+|Cd5, A4
 |611.32
 |10/7, 64/45
@@ Line 1,274: / Line 1,300: @@
 |28
 |super classic diminished fifth
-|SCla dim 5
+|SCd5
 |633.96
 |13/9
@@ Line 1,280: / Line 1,306: @@
 |29
 |sub comma-narrow fifth
-|sco-n 5
+|sc5
 |656.60
 |16/11, 22/15
@@ Line 1,286: / Line 1,312: @@
 |30
 |comma-narrow fifth
-|co-n 5
+|c5
 |679.25
 |40/27
@@ Line 1,322: / Line 1,348: @@
 |36
 |classic minor sixth, augmented fifth
-|Cla m6, Aug 5
+|Cm6, A5
 |815.09
 |8/5
@@ Line 1,328: / Line 1,354: @@
 |37
 |super classic minor sixth
-|SCla m6
+|SCm6
 |837.74
 |13/8, 81/50
@@ Line 1,334: / Line 1,360: @@
 |38
 |sub classic major sixth
-|scla M6
+|scM6
 |860.38
 |18/11, 400/243
@@ Line 1,340: / Line 1,366: @@
 |39
 |classic major sixth, diminished seventh
-|cla M6, dim 7
+|cM6, d7
 |883.02
 |5/3
@@ Line 1,382: / Line 1,408: @@
 |46
 |super classic minor seventh
-|SCla m7
+|SCm7
 |1041.51
 |11/6, 20/11, 729/400
@@ Line 1,388: / Line 1,414: @@
 |47
 |sub classic major seventh
-|scla M7
+|scM7
 |1064.15
 |13/7, 24/13, 50/27
@@ Line 1,394: / Line 1,420: @@
 |48
 |classic major seventh, diminished octave
-|cla M7, dim 8
+|cM7, d8
 |1086.79
 |15/8
@@ Line 1,418: / Line 1,444: @@
 |52
 |comma-narrow octave/sub octave
-|co-n 8 / S8
+|c8/S8
 |1177.36
 |160/81, 63/32, 128/65
@@ Line 1,554: / Line 1,580: @@
 There are edos whose best fifth is flatter even than 4\7. In such edos major intervals are smaller than minor intervals, augmented smaller than major and diminished larger than minor. We expand our definition of well-ordered intervals to include that within each degree... ≤ dd ≤ d ≤ m ≤ M ≤ A ≤ AA ≤ ... or ... ≤ dd ≤ d ≤ P ≤ A ≤ AA ≤ ..., and where sc _ ≤ s/c_ ≤ _ ≤ S/C_ ≤ SC _ (where '_' represents any of ... dd, d, m, (P), M, A, AA ...). In order to obtain well-ordered interval-name sets, we use enharmonic equivalences, replacing diatonic intervals with altered intervals.
-In super flat edos, the fifths are so flat that the major third, from four fifths approximates the classic minor third, 6/5 and the minor third approximates the classic major third, 5/4, tempering out 135/128, resulting in [[Mavila temperament]]. Mavila temperament can be defined in the 5-limit using the enharmonic equivalence cla M = m, where meantone can be defined by cla M = M, and schismatic by cla M ''n'' = dim ''n+1'' (superpyth in 2.3.7 can be defined by SM = M). Mavila[7] 3|3 reads the same as Meantone[7] 3|3: P1 M2 m3 P4 P5 M6 m7 P8, however Mavila[9] 4|4 has diatonic interval names:
+In super flat edos, the fifths are so flat that the major third, from four fifths approximates the classic minor third, 6/5 and the minor third approximates the classic major third, 5/4, tempering out 135/128, resulting in [[Mavila temperament]]. Mavila temperament can be defined in the 5-limit using the enharmonic equivalence cla M = m, where meantone can be defined by cM = M, and schismatic by cM''n'' = d''n+1'' (superpyth in 2.3.7 can be defined by SM = M). Mavila[7] 3|3 reads the same as Meantone[7] 3|3: P1 M2 m3 P4 P5 M6 m7 P8, however Mavila[9] 4|4 has diatonic interval names:
 P1 M2 M3 m3 P4 P5 M6 m6 m7 P8.