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SHEFKHED interval names: Difference between revisions

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*Prefixes that imply augmentation and diminution by a single step of an ET (Fokker/Keenan, Ups and Downs)
*Prefixes that imply augmentation and diminution by a single step of an ET (Fokker/Keenan, Ups and Downs)
*Possibility for well-ordered interval name sets (all proposals)
*Possibility for well-ordered interval name sets (all proposals)
*More than one possible name for intervals (Fokker/Keenan, Miracle interval naming, Sagittal, Keenan's most recent, Ups and Downs)
*More than one possible name for intervals (Fokker/Keenan, Miracle interval naming, Sagispeak, Keenan's most recent, Ups and Downs)
*Consistent mapping to Sagittal and HEWM notation (Sagispeak)
*Consistent mapping to Ups and Downs notation (Ups and Downs)
Such a system is developed through the extension of Fokker/Keenan Extended-diatonic Interval-names to define prefixes by alterations by specific commas as in Sagispeak, with the addition of prefix names of Smith and Keenan to enable application to Pythagorean and Superpythagorean systems. Where the prefixes of the Fokker/Keenan system were introduced by Helmholtz/Ellis, attribution to them is added, leading to Smith/Helmholtz/Ellis/Fokker/Keenan/Hearne Extended-diatonic interval names, or SHEFKHED interval names (Hearne is me).
Such a system is developed through the extension of Fokker/Keenan Extended-diatonic Interval-names to define prefixes by alterations by specific commas as in Sagispeak, with the addition of prefix names of Smith and Keenan to enable application to Pythagorean and Superpythagorean systems. Where the prefixes of the Fokker/Keenan system were introduced by Helmholtz/Ellis, attribution to them is added, leading to Smith/Helmholtz/Ellis/Fokker/Keenan/Hearne Extended-diatonic interval names, or SHEFKHED interval names (Hearne is me).
===Prefixes===
===Prefixes===
Line 460: Line 462:
*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.
*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.
*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 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.
*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' or 'unicond' for short. Similarly the interval half-way between M7 and P8 is given the short-form '7-8', and long-form 'seventh-octave' or 'sevtave' for short. The interval splitting the fourth, lying half-way between M2 and m3 is given the short-form '2-3', and long-form 'second-third', or 'serd', and it's octave complement, lying half-way between M6 and m7 is given the short-form '6-7', and long-form 'sixth-seventh', or 'sinth'. The interval half-way between M3 and P4 is given the short-form '3-4' and long-form 'third-fourth' or 'thourth', and it's octave-complement, the interval half-way between P5 and m6 is given the short-form '5-6', and long-form 'fifth-sixth', or '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 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.
*For completeness, '4-5', with long-form 'fourth-fifth' or '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, but is included to be used as an optional secondary interval name when there are no others.
*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, but is included 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.
===Application===
*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"
|+Prefixes summary
!Long-form
!Mid-form
!Short-form
!Alteration
|-
|perfect
|Prf
|P
|none
|-
|major
|Maj
|M
|none
|-
|minor
|min
|m
|none
|-
|super
|Spr
|S
|up 64/63 (or 65/64)
|-
|sub
|sub
|s
|down 64/63 (or 65/64)
|-
|classic
|Cla
|C
|up 81/80
|-
|classic
|cla
|c
|down 81/80
|-
|comma-sharp
|Co-shp
|C
|up 81/80
|-
|comma-flat
|co-flt
|c
|down 81/80
|-
|up
|up
|^
|up a single degree
|-
|down
|dwn
|v
|down a single degree
|}
{| class="wikitable"
|+Intermediates and neutrals summary
!Long-form
!Mid-form
!Short-form
!Description
|-
|unison-second
|unicond
|1-2
|half of the limma, half way between P1 and m2
|-
|second-third
|serd
|2-3
|half of P4, halfway between M2 and m3
|-
|third-fourth
|thourth
|3-4
|half of M6, halfway between M3 and P4
|-
|fourth-fifth
|firth
|4-5
|half of the octave, halfway between A4 and d5
|-
|fifth-sixth
|fixth
|5-6
|half of M10, halfway between P5 and m6
|-
|sixth-seventh
|sinth
|6-7
|halfway between P5 and P8, halfway between M6 and m7
|-
|seventh-octave
|sevtave
|7-8
|halfway between M7 and P8
|-
|neutral
|ntl
|N
|halfway between M and m (seconds, thirds, sixths, seventh)
halfway between P and A (unison, fourth)
 
halfway between P and d (fifth, octave)
|}
 
==Application==
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.


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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.
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.
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.


Line 510: Line 629:
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


46edo: P1 C1 sm2 m2 Cm2 SCm2 scM2 sM2 M2 SM2 sm3 m3 Cm3 SCm3 scM3 cM3 M3 SM3 s4 P4 C4 SC4 scA4/d5 cA4/Cd5 A4/SCd5 SA4/sc5 c5 P5 S5 sm6 m6 Cm6 SCm6 scM6 sM6 M6 SM6 sm7 m7 Cm7 SCm7 scM7 cM7 M7 SM7 c8 P8
46edo: P1 C1/S1 sm2 m2 Cm2 SCm2 scM2 sM2 M2 SM2 sm3 m3 Cm3 SCm3 scM3 cM3 M3 SM3 s4 P4 C4 SC4 scA4/d5 cA4/Cd5 A4/SCd5 SA4/sc5 c5 P5 S5 sm6 m6 Cm6 SCm6 scM6 sM6 M6 SM6 sm7 m7 Cm7 SCm7 scM7 cM7 M7 SM7 c8/s8 P8


53edo: P1 C1/S1 1-2 sm2 m2 Cm2 SCm2 scM2 sM2 M2 SM2 2-3 sm3 m3 Cm3 SCm3 scM3 cM3 M3 SM3 3-4 s4 P4 C4 SC4 scA4 cA4/d5 A4/Cd5 SCd5 SA4/sc5 c5 P5 S5 5-6 sm6 m6 Cm6 SCm6 scM6 sM6 M6 SM6 6-7 sm7 m7 Cm7 SCm7 scM7 cM7 M7 SM7 7-8 c8/s8 P8
53edo: P1 C1/S1 1-2 sm2 m2 Cm2 SCm2 scM2 sM2 M2 SM2 2-3 sm3 m3 Cm3 SCm3 scM3 cM3 M3 SM3 3-4 s4 P4 C4 SC4 scA4 cA4 Cd5 SCd5 SA4/sc5 c5 P5 S5 5-6 sm6 m6 Cm6 SCm6 scM6 sM6 M6 SM6 6-7 sm7 m7 Cm7 SCm7 scM7 cM7 M7 SM7 7-8 c8/s8 P8


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 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[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.
*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 526: Line 645:
Well-ordered primary and their secondary interval names for 22edo, 41edo and 53edo are shown below in more detail.
Well-ordered primary and their secondary interval names for 22edo, 41edo and 53edo are shown below in more detail.
{| class="wikitable"
{| class="wikitable"
|+22edo
|+22edo (patent 2.3.7.11.17 val)
!Degree
!Degree
!Primary interval name
!Primary interval name
Line 533: Line 652:
!Short-form
!Short-form
!Cents
!Cents
!Approximate 2.3.7.11.17
!Approximate Ratios
Ratios
|-
|-
|0
|0
Line 629: Line 747:
|cA4
|cA4
Cd5
Cd5
|super classic augmented fourth
|fourth-fifth
sub classic diminished fifth
|4-5
|ScA4
sCd5
|600
|600
|7/5, 17/12, 45/32
|7/5, 17/12, 45/32
Line 724: Line 840:
|1200
|1200
|2/1, 63/32
|2/1, 63/32
|}This interval names in this table tell us what the 7-limit ratios do, that 64/63 in tempered out, meaning it is a superpythagorean tuning and that 81/80 and 25/24 are represented by a single degree. They also show us that the chromatic semitone or apotome is 3 degrees wide.
|}2.3.5.7.11.17 patent val means using the best approximations to the 2nd, 3rd, 5th, 7th, 11th and 17th partials/harmonics for the interval names and approximated ratios. Only the 7-limit ratios are needed in this case for the interval names. This interval names in this table tell us what the 7-limit ratios do, that 64/63 in tempered out, meaning it is a superpythagorean tuning and that 81/80 and 25/24 are represented by a single degree. They also show us that the chromatic semitone or apotome is 3 degrees wide.
{| class="wikitable"
{| class="wikitable"
|+41edo
|+41edo (11-limit patent val)
!Degrees
!Degrees
!Interval names
!Interval names
Line 986: Line 1,102:
|}
|}


Secondary interval names are not available for every note without going into double and triple augmented and diminished intervals. I include only up to singly augmented and diminished and leave most secondary interval names out. Accordingly I do not write them in a separate column. We can see from the interval names that 64/63 and 81/80 are represented both by a single degree and the augmented unison by three, that it is a neutral tuning, and that it is a [[Schismatic]] tuning, where the diminished fourth approximates 5/4.
Secondary interval names are not available for every note without going into double and triple augmented and diminished intervals. I include only up to singly augmented and diminished and leave most secondary interval names out. Accordingly I do not write them in a separate column. We can see from the interval names that 64/63 and 81/80 are represented both by a single degree and the augmented unison by three, that it is a neutral tuning, and that it is a [[Schismatic]] tuning, where the diminished fourth approximates 5/4, and that it is at least an 11-limit tuning.
 
{| class="wikitable"
|+53edo (13-limit patent val)
!Degree
!Interval names
!Short-form
!Cents
!Approximate Ratios
|-
|0
|perfect unison
|P1
|0.00
|1/1
|-
|1
|comma-sharp unison/super unison
|C1/S1
|22.64
|81/80, 64/63, 65/64, 50/49
|-
|2
|unison-second
|1-2
|45.28
|49/48, 36/35, 33/32, 128/125
|-
|3
|sub minor second
|sm2
|67.92
|28/27, 27/26, 26/25, 25/24, 22/21
|-
|4
|minor second
|m2
|90.57
|21/20, 256/243
|-
|5
|classic minor second, augmented unison
|Cm2, A1
|113.21
|16/15, 15/14
|-
|6
|super classic minor second
|SCm2
|135.85
|14/13, 13/12, 27/25
|-
|7
|sub classic minor second
|scM2
|158.49
|12/11, 11/10, 800/729
|-
|8
|classic minor second, diminished third
|cM2, d3
|181.13
|10/9
|-
|9
|major second
|M2
|203.77
|9/8
|-
|10
|super major second
|SM2
|226.42
|8/7, 256/225
|-
|11
|second-third
|2-3
|249.06
|15/13, 144/125
|-
|12
|sub minor third
|sm3
|271.70
|7/6, 75/64
|-
|13
|minor third
|m3
|294.34
|13/11, 32/27
|-
|14
|classic minor third, augmented second
|Cm3, A2
|316.98
|6/5
|-
|15
|super classic minor third
|SCm3
|339.62
|11/9, 243/200
|-
|16
|sub classic major third
|scM3
|362.26
|16/13, 100/81
|-
|17
|classic major third, diminished fourth
|cM3, d4
|384.91
|5/4
|-
|18
|major third
|M3
|407.55
|81/64
|-
|19
|super major third
|SM3
|430.19
|9/7, 14/11
|-
|20
|third-fourth
|3-4
|452.83
|13/10, 125/96
|-
|21
|sub fourth
|s4
|475.47
|21/16, 675/512, 320/243
|-
|22
|perfect fourth
|P4
|498.11
|4/3
|-
|23
|comma-sharp fourth, augmented third
|C4, A3
|520.75
|27/20
|-
|24
|super comma-sharp fourth
|SC4
|543.40
|11/8, 15/11
|-
|25
|sub comma-flat augmented fourth
|scA4
|566.04
|18/13
|-
|26
|classic augmented fourth, diminished fifth
|cA4, d5
|588.68
|7/5, 45/32
|-
|27
|classic diminished fifth, augmented fourth
|Cd5, A4
|611.32
|10/7, 64/45
|-
|28
|super classic diminished fifth
|SCd5
|633.96
|13/9
|-
|29
|sub comma-flat fifth
|sc5
|656.60
|16/11, 22/15
|-
|30
|comma-flat fifth
|c5
|679.25
|40/27
|-
|31
|perfect fifth
|P5
|701.89
|3/2
|-
|32
|super fifth
|S5
|724.53
|32/21, 243/160, 1024/675
|-
|33
|fifth-sixth
|5-6
|747.17
|20/13, 192/125
|-
|34
|sub minor sixth
|sm6
|769.81
|14/9, 25/16, 11/7
|-
|35
|minor sixth
|m6
|792.45
|128/81
|-
|36
|classic minor sixth, augmented fifth
|Cm6, A5
|815.09
|8/5
|-
|37
|super classic minor sixth
|SCm6
|837.74
|13/8, 81/50
|-
|38
|sub classic major sixth
|scM6
|860.38
|18/11, 400/243
|-
|39
|classic major sixth, diminished seventh
|cM6, d7
|883.02
|5/3
|-
|40
|major sixth
|M6
|905.66
|22/13, 27/16
|-
|41
|super major sixth
|SM6
|928.30
|12/7
|-
|42
|sixth-seventh
|6-7
|950.94
|26/15, 125/72
|-
|43
|sub minor seventh
|sm7
|973.58
|7/4
|-
|44
|minor seventh
|m7
|996.23
|16/9
|-
|45
|classic minor seventh, augmented sixth
|Cm7, A6
|1018.87
|9/5
|-
|46
|super classic minor seventh
|SCm7
|1041.51
|11/6, 20/11, 729/400
|-
|47
|sub classic major seventh
|scM7
|1064.15
|13/7, 24/13, 50/27
|-
|48
|classic major seventh, diminished octave
|cM7, d8
|1086.79
|15/8
|-
|49
|major seventh
|M7
|1109.43
|40/21, 243/128
|-
|50
|super major seventh
|SM7
|1132.08
|48/25, 27/14
|-
|51
|seventh-octave
|7-8
|1154.72
|125/64
|-
|52
|comma-flat octave/sub octave
|c8/S8
|1177.36
|160/81, 63/32, 128/65
|-
|53
|perfect octave
|P8
|1200
|2/1
|}
We can see from the interval names that 64/63, 81/80 and 65/64 are represented all by a single degree and the augmented unison by five, that it is a barbados tuning, and that it is a [[Schismatic]] tuning, where the diminished fourth approximates 5/4 and that it is at least a 2.3.5.7.13 tuning.
 
=== 5''n-''edos ===
On the limit for a diatonic scale of 5 large and 2 small steps is 5edo, where the small steps, the diatonic semitones are reduced to unisons. The diatonic interval names for 5edo are as follows:
 
P1/m2 M2/m3 M3/P4 P5/m6 M6/m7 M7/P8
 
Our primary interval names are P1 M2/m3 P4 P5 M6/m7 P8.
 
Where the M6/m7 represents both 7/4 and 12/7, we know that 5edo is a superpythagorean tuning, tempering out 64/63, and a semaphore tuning, tempering out 49/48. It is therefore also a barbados tuning, tempering out 676/675. We may write 5edo then as
 
P1 SM2/sm3 P4 P5 SM6/m7 P8 to express it as a semaphore tuning, or
 
P1 2-3 P4 P5 6-7 P8 to express it as a barbados tuning, where secondary names for P1 are sm2 and 1-2 respectively, etc.
 
Up to 35edo, for all 5''n''-edos the 3\5 fifth (3 degrees of 5edo) is the best fifth. 10edo and 15edo may be easily named:
 
10edo: P1 N2 M2/m3 N3 P4 N4/N5 P5 N6 M6/m7 N7 P8
 
15edo: P1 C1/Cm2 cM2 M2/m3 Cm3 cM3 P4 C4 c5 P5 Cm6 cM6 M6/m7 Cm7 cM7/c8 P8
 
20edo, however, is difficult, as in the 13-limit, it's patent val maps only to notes of 10edo, so only half the notes are available. One option would be to use ups and downs:
 
20edo: P1 ^P1/^m2 N2 vM2 M2/m3 ^m3 N3 vM3/v4 P4 ^4 N4/N5 v5 P5 ^5/vN6 N6 vM6 M6/m7 ^m7 N7 vM7/v8 P8
 
Another would be to use a different mapping for primes in 20edo; a non-patent val. No-limit TOP error, arguably the simplest way to find the 'best' val for an edo gives
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