User:Moremajorthanmajor/5L 3s (15/7-equivalent)
| ↖ 4L 2s⟨15/7⟩ | ↑ 5L 2s⟨15/7⟩ | 6L 2s⟨15/7⟩ ↗ |
| ← 4L 3s⟨15/7⟩ | 5L 3s<15/7> | 6L 3s⟨15/7⟩ → |
| ↙ 4L 4s⟨15/7⟩ | ↓ 5L 4s⟨15/7⟩ | 6L 4s⟨15/7⟩ ↘ |
┌╥╥┬╥╥┬╥┬┐ │║║│║║│║││ ││││││││││ └┴┴┴┴┴┴┴┴┘
sLsLLsLL
5L 3s⟨15/7⟩ is a 15/7-equivalent (non-octave) moment of symmetry scale containing 5 large steps and 3 small steps, repeating every interval of 15/7 (1319.4¢). Generators that produce this scale range from 494.8¢ to 527.8¢, or from 791.7¢ to 824.7¢.
Any ed15/7 with an interval between 494.8¢ and 527.8¢ has a 5L 3s scale. 13ed15/7 is the smallest ed15/7 with a (non-degenerate) 5L 3s scale and thus is the most commonly used 5L 3s tuning.
Standing assumptions
The TAMNAMS system is used in this article to name 5L 3s<15/7> intervals and step size ratios and step ratio ranges.
The notation used in this article is G Mixolydian Mediant (LLsLLsLs) = GABCQDEFG, unless specified otherwise. We denote raising and lowering by a chroma (L − s) by # and f "flat (F molle)".
The chain of perfect 4ths becomes: ... Ef Af Qf Ff Bf Df G C E A Q F B D ...
Thus the 13ed15/7 gamut is as follows:
G/F# G#/Af A A#/Bf B/Cf C/B# C#/Qf Q Q#/Df D/Ef E/D# E#/Ff F/Gf G
The 18ed15/7 gamut is notated as follows:
G F#/Af G# A Bf A#/Cf B C B#/Qf C# Q Df Q#/Ef D E D#/Ff E#/Gf F G
The 21ed15/7 gamut:
G G# Af A A# Bf B B#/Cf C C# Qf Q Q# Df D D#/Ef E E# Ff F F#/Gf G
Names
The author suggests the name Neapolitan-oneirotonic.
Intervals
The table of Neapolitan-oneirotonic intervals below takes the fourth as the generator. Given the size of the fourth generator g, any oneirotonic interval can easily be found by noting what multiple of g it is, and multiplying the size by the number k of generators it takes to reach the interval.
| Notation (1/1 = G) | name | In L's and s's | # generators up | Notation of 15/7 inverse | name | In L's and s's | |
|---|---|---|---|---|---|---|---|
| The 8-note MOS has the following intervals (from some root): | |||||||
| 0 | G | perfect unison | 0L + 0s | 0 | G | “perfect” minor 9th | 5L + 3s |
| 1 | C | natural 4th | 2L + 1s | -1 | Df | minor 6th | 3L + 2s |
| 2 | E | major 7th | 4L + 2s | -2 | Bf | minor 3rd | 1L + 1s |
| 3 | A | major 2nd | 1L + 0s | -3 | Ff | diminished octave | 4L + 3s |
| 4 | Q | perfect 5th | 3L + 1s | -4 | Qf | diminished 5th | 2L + 2s |
| 5 | F | perfect octave | 5L + 2s | -5 | Af | minor 2nd | 0L + 1s |
| 6 | B | major 3rd | 2L + 0s | -6 | Ef | minor 7th | 3L + 3s |
| 7 | D | major 6th | 4L + 1s | -7 | Cf | diminished 4th | 1L + 2s |
| The chromatic 13-note MOS (either 5L 8s, 8L 5s, or 13ed15/7) also has the following intervals (from some root): | |||||||
| 8 | G# | augmented unison | 1L - 1s | -8 | Gf | diminished 9th | 4L + 4s |
| 9 | C# | augmented 4th | 3L + 0s | -9 | Dff | diminished 6th | 2L + 3s |
| 10 | E# | augmented 7th | 5L + 1s | -10 | Bff | diminished 3rd | 0L + 2s |
| 11 | A# | augmented 2nd | 2L - 1s | -11 | Fff | doubly diminished octave | 3L + 4s |
| 12 | Q# | augmented 5th | 4L + 0s | -12 | Qff | doubly diminished 5th | 1L + 3s |
Tuning ranges
Simple tunings
Table of intervals in the simplest Neapolitan-oneirotonic tunings:
| Degree | Size in 13ed15/7 (basic) | Size in 18ed15/7 (hard) | Size in 21ed15/7 (soft) | Note name on G | #Gens up |
|---|---|---|---|---|---|
| unison | 0\13, 0.00 | 0\18, 0.00 | 0\21, 0.00 | G | 0 |
| minor 2nd | 1\13, 101.496 | 1\18, 73.302 | 2\21, 125.661 | Af | -5 |
| major 2nd | 2\13, 202.991 | 3\18, 219.907 | 3\21, 188.492 | A | +3 |
| minor 3rd | 3\13, 304.487 | 4\18, 293.210 | 5\21, 314.153 | Bf | -2 |
| major 3rd | 4\13, 405.982 | 6\18, 439.814 | 6\21, 376.984 | B | +6 |
| diminished 4th | 5\18, 366.511 | 7\21, 439.814 | Cf | -7 | |
| natural 4th | 5\13, 507.478 | 7\18, 513.117 | 8\21, 502.645 | C | +1 |
| diminished 5th | 6\13, 608.974 | 8\18, 586.419 | 10\21, 628.306 | Qf | -4 |
| perfect 5th | 7\13, 710.469 | 10\18, 733.024 | 11\31, 691.137 | Q | +4 |
| minor 6th | 8\13, 811.965 | 11\18, 806.326 | 13\21, 816.798 | Df | -1 |
| major 6th | 9\13, 913.460 | 13\18, 952.931 | 14\21, 879.629 | D | +7 |
| minor 7th | 12\18, 879.629 | 15\21, 942.459 | Ef | -6 | |
| major 7th | 10\13, 1014.956 | 14\18, 1026.233 | 16\21, 1005.290 | E | +2 |
| diminished octave | 11\13, 1116.452 | 15\18, 1099.536 | 18\21, 1130.951 | Ff | -3 |
| perfect octave | 12\13, 1217.942 | 17\18, 1246.140 | 19\21, 1193.782 | F | +5 |
Hypohard
Hypohard Neapolitan-oneirotonic tunings (with generator between 5\13 and 7\18) have step ratios between 2/1 and 3/1.
Hypohard Neapolitan-oneirotonic can be considered " superpythagorean Neapolitan-oneirotonic". This is because these tunings share the following features with superpythagorean diatonic tunings:
- The large step is near the Pythagorean 9/8 whole tone, somewhere between as in 12edo and as in 17edo.
- The major 3rd (made of two large steps) is a near-Pythagorean to Neogothic major third.
EDIXs that are in the hypohard range include 13ed15/7, 18ed15/7, and 31ed15/7.
The sizes of the generator, large step and small step of Neapolitan-oneirotonic are as follows in various hypohard Neapolitan-oneiro tunings.
| 13ed15/7 (basic) | 18ed15/7 (hard) | 31ed15/7 (semihard) | |
|---|---|---|---|
| generator (g) | 5\13, 507.478 | 7\18, 513.117 | 12\31, 510.752 |
| L (3g - minor 9th) | 2\13, 202.991 | 3\18, 219.907 | 5\31, 212.813 |
| s (-5g + 2 minor 9ths) | 1\13, 101.496 | 1\18, 73.302 | 2\31, 85.125 |
Intervals
Sortable table of major and minor intervals in hypohard Neapolitan-oneiro tunings:
| Degree | Size in 13ed15/7 (basic) | Size in 18ed15/7 (hard) | Size in 31ed15/7 (semihard) | Note name on G | Approximate ratios | #Gens up |
|---|---|---|---|---|---|---|
| unison | 0\13, 0.00 | 0\18, 0.00 | 0\31, 0.00 | G | 1/1 | 0 |
| minor 2nd | 1\13, 101.496 | 1\18, 73.302 | 2\31, 85.125 | Af | 21/20, 22/21 | -5 |
| major 2nd | 2\13, 202.991 | 3\18, 219.907 | 5\31, 212.813 | A | 9/8 | +3 |
| minor 3rd | 3\13, 304.487 | 4\18, 293.210 | 7\31, 297.939 | Bf | 13/11, 33/28 | -2 |
| major 3rd | 4\13, 405.982 | 6\18, 439.814 | 10\31, 425.626 | B | 14/11, 33/26 | +6 |
| diminished 4th | 5\18, 366.511 | 9\31, 383.064 | Cf | 5/4, 11/9 | -7 | |
| natural 4th | 5\13, 507.478 | 7\18, 513.117 | 12\31, 510.752 | C | 4/3 | +1 |
| diminished 5th | 6\13, 608.974 | 8\18, 586.419 | 14\31, 595.877 | Qf | 7/5, 13/9, 16/11 | -4 |
| perfect 5th | 7\13, 710.469 | 10\18, 733.024 | 17\31, 723.565 | Q | 3/2 | +4 |
| minor 6th | 8\13, 811.965 | 11\18, 806.326 | 19\31, 808.691 | Df | 52/33, 11/7 | -1 |
| major 6th | 9\13, 913.460 | 13\18, 952.931 | 22\31, 936.379 | D | 56/33, 22/17 | +7 |
| minor 7th | 12\18, 879.629 | 21\31, 893.816 | Ef | 5/3, 18/11 | -6 | |
| major 7th | 10\13, 1014.956 | 14\18, 1026.233 | 24\31, 1021.04 | E | 16/9 | +2 |
| diminished octave | 11\13, 1116.452 | 15\18, 1099.536 | 26\31, 1106.6295 | Ff | 11/6, 13/7, 15/8 | -3 |
| perfect octave | 12\13, 1217.942 | 17\18, 1246.140 | 29\31, 1234.3175 | F | 2/1 | +5 |
Hyposoft
Hyposoft Neapolitan-oneirotonic tunings (with generator between 8\21 and 5\13) have step ratios between 3/2 and 2/1. The 8\21-to-5\13 range of Neapolitan-oneirotonic tunings can be considered "meantone Neapolitan-oneirotonic”. This is because these tunings share the following features with meantone diatonic tunings:
- The large step is between near the meantone and near the Pythagorean 9/8 whole tone, somewhere between as in 19edo and as in 12edo.
- The major 3rd (made of two large steps) is a near-just to near-Pythagorean major third.
The sizes of the generator, large step and small step of Neapolitan-oneirotonic are as follows in various hyposoft Neapolitan-oneiro tunings (13ed15/7 not shown).
| 21ed15/7 (soft) | 34ed15/7 (semisoft) | |
|---|---|---|
| generator (g) | 8\21, 502.645 | 13\34, 504.493 |
| L (3g - minor 9th) | 3\21, 188.492 | 5\34, 194.036 |
| s (-5g + 2 minor 9ths) | 2\21, 125.661 | 3\34, 116.421 |
Intervals
Sortable table of major and minor intervals in hyposoft tunings (13ed15/7 not shown):
| Degree | Size in 21ed15/7 (soft) | Size in 34ed15/7 (semisoft) | Note name on G | Approximate ratios | #Gens up |
|---|---|---|---|---|---|
| unison | 0\21, 0.00 | 0\34, 0.00 | G | 1/1 | 0 |
| minor 2nd | 2\21, 125.661 | 3\34, 116.421 | Af | 16/15 | -5 |
| major 2nd | 3\21, 188.492 | 5\34, 194.036 | A | 10/9, 9/8 | +3 |
| minor 3rd | 5\21, 314.153 | 8\34, 310.457 | Bf | 6/5 | -2 |
| major 3rd | 6\21, 376.984 | 10\34, 388.071 | B | 5/4 | +6 |
| diminished 4th | 7\21, 439.814 | 11\34, 426.879 | Cf | 9/7 | -7 |
| natural 4th | 8\21, 502.645 | 13\34, 504.493 | C | 4/3 | +1 |
| diminished 5th | 10\21, 628.306 | 16\34, 620.914 | Qf | 10/6 | -4 |
| perfect 5th | 11\31, 691.137 | 18\34, 698.529 | Q | 3/2 | +4 |
| minor 6th | 13\21, 816.798 | 21\34, 814.950 | Df | 8/5 | -1 |
| major 6th | 14\21, 879.629 | 23\34, 892.564 | D | 5/3 | +7 |
| minor 7th | 15\21, 942.459 | 24\34, 931.371 | Ef | 12/7 | -6 |
| major 7th | 16\21, 1005.290 | 26\34, 1008.986 | E | 9/5, 16/9 | +2 |
| diminished octave | 18\21, 1130.951 | 29\34, 1125.407 | Ff | 27/14, 48/25 | -3 |
| perfect octave | 19\21, 1193.782 | 31\34, 1203.021 | F | 2/1 | +5 |
Parasoft to ultrasoft tunings
The range of Neapolitan-oneirotonic tunings of step ratio between 6/5 and 3/2 (thus in the parasoft to ultrasoft range) may be of interest because it is closely related to flattone temperament.
The sizes of the generator, large step and small step of Neapolitan-oneirotonic are as follows in various tunings in this range.
| 29ed15/7 (supersoft) | 37ed15/7 | |
|---|---|---|
| generator (g) | 11\29, 500.478 | 14\37, 499.249 |
| L (3g - minor 9th) | 4\29, 181.992 | 5\37, 178.303 |
| s (-5g + 2 minor 9ths) | 3\29, 136.494 | 4\37, 142.642 |
Intervals
The intervals of the extended generator chain (-15 to +15 generators) are as follows in various softer-than-soft Neapolitan-oneirotonic tunings.
| Degree | Size in 29e15/7 (supersoft) | Note name on G | Approximate ratios | #Gens up |
|---|---|---|---|---|
| unison | 0\29, 0.00 | G | 1/1 | 0 |
| chroma | 1\29, 45.498 | G# | 33/32, 49/48, 36/35, 25/24 | +8 |
| diminished 2nd | 2\29, 90.996 | Aff | 21/20, 22/21, 26/25 | -13 |
| minor 2nd | 3\29, 136.494 | Af | 12/11, 13/12, 14/13, 16/15 | -5 |
| major 2nd | 4\29, 181.992 | A | 9/8, 10/9, 11/10 | +3 |
| augmented 2nd | 5\29, 227.490 | A# | 8/7, 15/13 | +11 |
| diminished 3rd | 6\29, 272.988 | Bff | 7/6, 13/11, 33/28 | -10 |
| minor 3rd | 7\29, 318.486 | Bf | 135/112, 6/5 | -2 |
| major 3rd | 8\29, 363.984 | B | 5/4, 11/9, 16/13 | +6 |
| augmented 3rd | 9\29, 409.482 | B# | 9/7, 14/11, 33/26 | +14 |
| diminished 4th | 10\29, 454.980 | Cf | 21/16, 13/10 | -7 |
| natural 4th | 11\29, 500.478 | C | 75/56, 4/3 | +1 |
| augmented 4th | 12\29, 545.976 | C# | 11/8, 18/13 | +9 |
| doubly augmented 4th, doubly diminished 5th | 13\29, 591.474 | Cx, Qff | 7/5, 10/7 | -12 |
| diminished 5th | 14\29, 636.972 | Qf | 16/11, 13/9 | -4 |
| perfect 5th | 15\29, 682.470 | Q | 112/75, 3/2 | +4 |
| augmented 5th | 16\29, 727.968 | Q# | 32/21, 20/13 | +12 |
| diminished 6th | 17\29, 773.466 | Dff | 11/7, 14/9 | -9 |
| minor 6th | 18\29, 818.965 | Df | 13/8, 8/5 | -1 |
| major 6th | 19\29, 864.463 | D | 5/3, 224/135 | +7 |
| augmented 6th | 20\29, 909.961 | D# | 12/7, 22/13 | -14 |
| minor 7th | 21\29, 955.459 | Ef | 7/4, 26/15 | -6 |
| major 7th | 22\29, 1000.956 | E | 9/5, 16/9, 20/11 | +2 |
| augmented 7th | 23\29, 1046.455 | E# | 11/6, 13/7, 15/8, 24/13 | +10 |
| doubly augmented 7th, doubly diminished octave | 24\29, 1091.953 | Ex, Fff | 21/11, 25/13, 40/21 | -11 |
| diminished octave | 25\29, 1137.451 | Ff | 64/33, 96/49, 35/18, 48/25 | -3 |
| perfect octave | 26\29, 1182.949 | F | 2/1 | +5 |
| augmented octave | 27\29, 1228.447 | F# | 33/16, 49/24, 72/35, 25/12 | +13 |
| doubly augmented octave, diminished 9th | 28\29, 1273.945 | Fx, Gf | 21/10, 44/21, 52/25 | -8 |
Parahard
23ed15/7 Neapolitan-oneiro combines the sound of the 15/7 minor ninth and the 8/7 whole tone. This is because 23ed15/7 Neapolitan-oneirotonic has a large step of 229.5¢.
Intervals
The intervals of the extended generator chain (-12 to +12 generators) are as follows in various Neapolitan-oneirotonic tunings close to 23edIX.
| Degree | Size in 23ed15/7
(superhard) |
Note name on G | Approximate ratios (23edIX) | #Gens up |
|---|---|---|---|---|
| unison | 0\23, 0.00 | G | 1/1 | 0 |
| chroma | 3\23, 172.101 | G# | 12/11, 11/10, 10/9 | +8 |
| minor 2nd | 1\23, 57.367 | Af | 36/35, 34/33, 33/32, 32/31 | -5 |
| major 2nd | 4\23, 229.468 | A | 9/8, 17/15, 8/7 | +3 |
| aug. 2nd | 7\23, 401.570 | A# | 5/4 | +11 |
| dim. 3rd | 2\23, 114.734 | Bf | 16/15 | -10 |
| minor 3rd | 5\23, 286.835 | B | 7/6 | -2 |
| major 3rd | 8\23, 458.937 | B# | 9/7, 14/11 | +6 |
| dim. 4th | 6\23, 344.202 | Cf | 6/5 | -7 |
| nat. 4th | 9\23, 516.304 | C | 4/3 | +1 |
| aug. 4th | 12\23, 688.405 | C# | 16/11, 22/15 | +9 |
| double dim. 5th | 7\23, 401.570 | Qff | 5/4 | -12 |
| dim. 5th | 10\23, 573.671 | Qf | 15/11, 11/8 | -4 |
| perf. 5th | 13\23, 745.772 | Q | 3/2 | +4 |
| aug. 5th | 16\23, 917.873 | Q# | 5/3 | +12 |
| dim. 6th | 11\23, 631.038 | Dff | 7/5, 24/17, 17/12, 10/7 | -9 |
| minor 6th | 14\23, 803.139 | Df | 14/9, 11/7 | -1 |
| major 6th | 17\23, 975.240 | D | 12/7 | +7 |
| minor 7th | 15\23, 860.560 | Ef | 8/5 | -6 |
| major 7th | 18\23, 1032.607 | E | 7/4, 30/17, 16/9 | +2 |
| aug. 7th | 21\23, 1204.709 | E# | 31/16, 64/33, 33/17, 35/18 | +10 |
| dim. octave | 19\23, 1089.9745 | Ff | 11/6, 20/11, 9/5 | -11 |
| perf. octave | 22\23, 1262.076 | F | 2/1 | -3 |
| aug. octave | 25\23, 1434.177 | F# | 24/11, 11/5, 20/9 | +5 |
| dim. ninth | 20\23, 1147.342 | Gf | 15/8 | -8 |
Ultrahard
Ultrapythagorean Neapolitan-oneirotonic is a rank-2 temperament in the pseudopaucitonic range. It represents the harmonic entropy minimum of the Neapolitan-oneirotonic spectrum where 7/4 is the major seventh.
In the broad sense, Ultrapyth can be viewed as any tuning that divides a 16/7 into 2 equal parts. 23ed15/7, 28ed15/7 and 33ed15/7 can nominally be viewed as supporting it, but are still very flat and in an ambiguous zone between 18ed15/7 and true Buzzard in terms of harmonies. 38ed15/7 & 43ed15/7 are good compromises between melodic utility and harmonic accuracy, as the small step is still large enough to be obvious to the untrained ear, but 48edIX is where it really comes into its own in terms of harmonies, providing not only excellent 7:8:9 melodies, but also 5/4 and archipelago harmonies, as by shifting one whole tone done a comma, it shifts from archipelago to septimal harmonies.
Beyond that, it's a question of which intervals you want to favor. 53ed15/7 has an essentially perfect 7/4, 58edIX also gives three essentially perfect chains of third-comma meantone, while 63ed15/7 has a double chain of essentially perfect quarter-comma meantone and gives about as low overall error as 83ed15/7 does for the basic 4:6:7 triad. But beyond 83edIX, general accuracy drops off rapidly and you might as well be playing equal pentatonic.
The sizes of the generator, large step and small step of Neapolitan-oneirotonic are as follows in various ultrapyth tunings.
| 38ed15/7 | 53ed15/7 | 63ed15/7 | Optimal (PNTE) Ultrapyth tuning | JI intervals represented (2.3.5.7.13 subgroup) | |
|---|---|---|---|---|---|
| generator (g) | 15\38, 520.833 | 21\53, 522.798 | 25\63, 523.58’ | 484.07 | 4/3 |
| L (3g - minor 9th) | 7/38, 243.055 | 10/53, 248.951 | 12/63, 251.322 | 231.51 | 8/7 |
| s (-5g + 2 minor 9ths) | 1/38, 34.722 | 1/53, 24.895 | 1/63, 20.934 | 21.05 | 50/49 81/80 91/90 |
Intervals
Sortable table of intervals in the Neapolitan-Dylathian mode and their Ultrapyth interpretations:
| Degree | Size in 38ed15/7 | Size in 53ed15/7 | Size in 63ed15/7 | Size in PNTE tuning | Note name on G | Approximate ratios | #Gens up |
|---|---|---|---|---|---|---|---|
| 1 | 0\38, 0.00 | 0\53, 0.00 | 0\63, 0.00 | 0.00 | G | 1/1 | 0 |
| 2 | 7/38, 243.055 | 10/53, 248.951 | 12/63, 251.322 | 231.51 | A | 8/7 | +3 |
| 3 | 14\38, 486.111 | 20\53, 497.903 | 24\63, 502.645 | 463.03 | B | 13/10, 21/16 | +6 |
| 4 | 15\38, 520.833 | 21\53, 522.798 | 25\63, 523.588 | 484.07 | C | 4/3 | +1 |
| 5 | 22\38, 763.888 | 31\53, 771.750 | 37\63, 774.991 | 715.59 | Q | 3/2 | +4 |
| 6 | 29\38, 1006.943 | 41\53, 1020.701 | 49\63, 1026.233 | 947.10 | D | 26/15 | +7 |
| 7 | 30\38, 1041.665 | 42\53, 1045.596 | 50\63, 1047.177 | 968.15 | E | 7/4 | +2 |
| 8 | 37\38, 1284.721 | 52\53, 1294.548 | 62\63, 1298.499 | 1199.66 | F | 2/1 | +5 |
Modes
Neapolitan-Oneirotonic modes are named after cities in the Dreamlands.
| Mode | UDP | Name |
| LLsLLsLs | 7|0 | Neapolitan-Dylathian (də-LA(H)TH-iən) |
| LLsLsLLs | 6|1 | Neapolitan-Illarnekian (ill-ar-NEK-iən) |
| LsLLsLLs | 5|2 | Neapolitan-Celephaïsian (kel-ə-FAY-zhən) |
| LsLLsLsL | 4|3 | Neapolitan-Ultharian (ul-THA(I)R-iən) |
| LsLsLLsL | 3|4 | Neapolitan-Mnarian (mə-NA(I)R-iən) |
| sLLsLLsL | 2|5 | Neapolitan-Kadathian (kə-DA(H)TH-iən) |
| sLLsLsLL | 1|6 | Neapolitan-Hlanithian (lə-NITH-iən) |
| sLsLLsLL | 0|7 | Neapolitan-Sarnathian (sar-NA(H)TH-iən), can be shortened to "Sarn" |
Scale tree
| Generator(ed15/7) | Cents | Step ratio | Comments | |||||||
|---|---|---|---|---|---|---|---|---|---|---|
| Bright | Dark | L:s | Hardness | |||||||
| 3\8 | 494.791 | 824.652 | 1:1 | 1.000 | Equalized 5L 3s⟨15/7⟩ | |||||
| 17\45 | 498.456 | 820.987 | 6:5 | 1.200 | ||||||
| 14\37 | 499.249 | 820.194 | 5:4 | 1.250 | ||||||
| 25\66 | 499.789 | 819.654 | 9:7 | 1.286 | ||||||
| 11\29 | 500.478 | 818.965 | 4:3 | 1.333 | Supersoft 5L 3s⟨15/7⟩ | |||||
| 30\79 | 501.054 | 818.389 | 11:8 | 1.375 | ||||||
| 19\50 | 501.388 | 818.055 | 7:5 | 1.400 | ||||||
| 27\71 | 501.760 | 817.683 | 10:7 | 1.429 | ||||||
| 8\21 | 502.645 | 816.798 | 3:2 | 1.500 | Soft 5L 3s⟨15/7⟩ | |||||
| 29\76 | 503.472 | 815.971 | 11:7 | 1.571 | ||||||
| 21\55 | 503.787 | 815.656 | 8:5 | 1.600 | ||||||
| 34\89 | 504.057 | 815.386 | 13:8 | 1.625 | ||||||
| 13\34 | 504.493 | 814.950 | 5:3 | 1.667 | Semisoft 5L 3s⟨15/7⟩ | |||||
| 31\81 | 504.972 | 814.471 | 12:7 | 1.714 | ||||||
| 18\47 | 505.319 | 814.124 | 7:4 | 1.750 | ||||||
| 23\60 | 505.786 | 813.656 | 9:5 | 1.800 | ||||||
| 5\13 | 507.478 | 811.965 | 2:1 | 2.000 | Basic 5L 3s⟨15/7⟩ Scales with tunings softer than this are proper | |||||
| 22\57 | 509.259 | 810.184 | 9:4 | 2.250 | ||||||
| 17\44 | 509.785 | 809.658 | 7:3 | 2.333 | ||||||
| 29\75 | 510.185 | 809.258 | 12:5 | 2.400 | ||||||
| 12\31 | 510.752 | 808.691 | 5:2 | 2.500 | Semihard 5L 3s⟨15/7⟩ | |||||
| 31\80 | 511.284 | 808.159 | 13:5 | 2.600 | ||||||
| 19\49 | 511.621 | 807.822 | 8:3 | 2.667 | ||||||
| 26\67 | 512.023 | 807.420 | 11:4 | 2.750 | ||||||
| 7\18 | 513.117 | 806.326 | 3:1 | 3.000 | Hard 5L 3s⟨15/7⟩ | |||||
| 23\59 | 514.359 | 805.084 | 10:3 | 3.333 | ||||||
| 16\41 | 514.905 | 804.538 | 7:2 | 3.500 | ||||||
| 25\64 | 515.407 | 804.035 | 11:3 | 3.667 | ||||||
| 9\23 | 516.304 | 803.139 | 4:1 | 4.000 | Superhard 5L 3s⟨15/7⟩ | |||||
| 20\51 | 517.429 | 802.014 | 9:2 | 4.500 | ||||||
| 11\28 | 518.353 | 801.090 | 5:1 | 5.000 | ||||||
| 13\33 | 519.781 | 799.662 | 6:1 | 6.000 | ||||||
| 2\5 | 527.777 | 791.666 | 1:0 | → ∞ | Collapsed 5L 3s⟨15/7⟩ | |||||