User:Moremajorthanmajor/5L 3s (15/7-equivalent): Difference between revisions

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{{Infobox MOS
{{Infobox MOS
| Name =
|Tuning=5L 3s<15/7>}}The minor ninth of a diatonic scale has a '''5L 3s''' [[MOS]] structure with generators ranging from 2\5 (two degrees of [[5ed15/7]] = 527.8¢) to 3\8 (three degrees of [[8ed15/7]] = 494.8¢). In the case of 8edo, L and s are the same size; in the case of 5ed15/7, s becomes so small it disappears (and all that remains are the five equal L's).
| Equave = 15/7
| Periods = 1
| nLargeSteps = 5
| nSmallSteps = 3
| Equalized = 3
| Collapsed = 2
| Pattern = LLsLLsLs
| Neutral = 2L 6s
}}The minor ninth of a diatonic scale has a '''5L 3s''' [[MOS]] structure with generators ranging from 2\5 (two degrees of 5ed8\7 = 548.6¢) to 3\8 (three degrees of [[8edo]] = 450¢). In the case of 8edo, L and s are the same size; in the case of 5ed8\7, s becomes so small it disappears (and all that remains are the five equal L's).


Any edIX of an interval up to 8\7 with an interval between 450¢ and 548.6¢ has a 5L 3s scale. [[13edIX]] is the smallest edIX with a (non-degenerate) 5L 3s scale and thus is the most commonly used 5L 3s tuning.
Any ed15/7 with an interval between 494.8¢ and 527.8¢ has a 5L 3s scale. [[13ed15/7]] is the smallest edIX with a (non-degenerate) 5L 3s scale and thus is the most commonly used 5L 3s tuning.
==Standing assumptions==
==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 [[TAMNAMS]] system is used in this article to name 5L 3s<15/7> intervals and step size ratios and step ratio ranges.
Line 19: Line 10:
The chain of perfect 4ths becomes: ... Ef Af Qf Ff Bf Df G C E A Q F B D ...
The chain of perfect 4ths becomes: ... Ef Af Qf Ff Bf Df G C E A Q F B D ...


Thus the [[13edIX]] gamut is as follows:
Thus the [[13edIX|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'''
'''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 18edIX gamut is notated as follows:
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'''
'''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 21edIX gamut:
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'''
'''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'''
Line 33: Line 24:
The author suggests the name '''Neapolitan'''-'''oneirotonic'''.
The author suggests the name '''Neapolitan'''-'''oneirotonic'''.
==Intervals==
==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 and reducing mod 1300 (for relative cents) if necessary (so you can use "''k''*''g'' % 1300" for search engines, for plugged-in values of ''k'' and ''g''). For example, since the major third is reached by six fourth generators, 18edIX's major third is 6*505.56 mod 1300 = 3033.33 mod 1300 = 433.33r¢.
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.
{| class="wikitable center-all"
{| class="wikitable center-all"
|-
|-
Line 41: Line 32:
!In L's and s's
!In L's and s's
!# generators up
!# generators up
!Notation of 2/1 inverse
!Notation of 15/7 inverse
!name
!name
!In L's and s's
!In L's and s's
Line 119: Line 110:
|1L + 2s
|1L + 2s
|-
|-
| colspan="8" |The chromatic 13-note MOS (either [[5L 8s (minor ninth equivalent)|5L 8s]], [[8L 5s (minor ninth equivalent)|8L 5s]], or [[13edIX]]) also has the following intervals (from some root):
| colspan="8" |The chromatic 13-note MOS (either [[5L 8s (15/7-equivalent)|5L 8s]], [[8L 5s (15/7-equivalent)|8L 5s]], or [[13edIX|13ed15/7]]) also has the following intervals (from some root):
|-
|-
|8
|8
Line 172: Line 163:
|-
|-
! class="unsortable" |Degree
! class="unsortable" |Degree
!Size in 13edIX (basic)
!Size in [[13edIX|13ed15/7]] (basic)
!Size in 18edIX (hard)
!Size in [[18ed15/7]] (hard)
!Size in 21edIX (soft)
!Size in [[21ed15/7]] (soft)
! class="unsortable" |Note name on G
! class="unsortable" |Note name on G
!#Gens up
!#Gens up
Line 186: Line 177:
|-
|-
|minor 2nd
|minor 2nd
|1\13, 100.00
|1\13, 101.496
|1\18, 70.59 (72.22)
|1\18, 73.302
|2\21, 126.32 (121.81)
|2\21, 125.661
|Af
|Af
| -5
| -5
|-
|-
|major 2nd
|major 2nd
|2\13, 200.00
|2\13, 202.991
|3\18, 211.765 (216.67)
|3\18, 219.907
|3\21, 189.47 (185.71)
|3\21, 188.492
|A
|A
| +3
| +3
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|minor 3rd
|minor 3rd
|3\13, 300.00
|3\13, 304.487
|4\18, 282.35 (288.89)
|4\18, 293.2095
|5\21, 315.79 (309.52)
|5\21, 314.153
|Bf
|Bf
| -2
| -2
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 3rd
|major 3rd
| rowspan="2" |4\13, 400.00
| rowspan="2" |4\13, 405.982
|6\18, 423.53 (433.33)
|6\18, 439.814
|6\21, 378.95 (371.43)
|6\21, 376.984
|B
|B
| +6
| +6
|-
|-
|diminished 4th
|diminished 4th
|5\18, 352.94 (361.11)
|5\18, 366.511
|7\21, 442.105 (433.33)
|7\21, 439.814
|Cf
|Cf
| -7
| -7
|-
|-
|natural 4th
|natural 4th
|5\13, 500.00
|5\13, 507.478
|7\18, 494.12 (505.56)
|7\18, 513.117
|8\21, 505.26 (495.24)
|8\21, 502.645
|C
|C
| +1
| +1
|-
|augmented 4th
| rowspan="2" |6\13, 600.00
|9\18, 635.29 (650.00)
|9\21, 568.42 (557.14)
|C#
| +9
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|diminished 5th
|diminished 5th
|8\18, 564.71 (577.78)
|6\13, 608.974
|10\21, 631.58 (619.05)
|8\18, 586.419
|10\21, 628.306
|Qf
|Qf
| -4
| -4
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|perfect 5th
|perfect 5th
|7\13, 700.00
|7\13, 710.469
|10\18, 705.88 (722.22)
|10\18, 733.024
|11\31, 694.74 (680.95)
|11\31, 691.137
|Q
|Q
| +4
| +4
|-
|-
|minor 6th
|minor 6th
|8\13, 800.00
|8\13, 811.965
|11\18, 776.47 (794.44)
|11\18, 806.326
|13\21, 821.05 (802.76)
|13\21, 816.798
|Df
|Df
| -1
| -1
|-
|-
|major 6th
|major 6th
| rowspan="2" |9\13, 900.00
| rowspan="2" |9\13, 913.460
|13\18, 917.65 (938.89)
|13\18, 952.931
|14\21, 884.21 (866.67)
|14\21, 879.6285
|D
|D
| +7
| +7
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|minor 7th
|minor 7th
|12\18, 847.06 (866.67)
|12\18, 879.6285
|15\21, 947.37 (928.57)
|15\21, 942.459
|Ef
|Ef
| -6
| -6
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 7th
|major 7th
|10\13, 1000.00
|10\13, 1014.956
|14\18, 988.235 (1011.11)
|14\18, 1026.233
|16\21, 1017.53 (990.48)
|16\21, 1005.290
|E
|E
| +2
| +2
|-
|-
|diminished octave
|diminished octave
|11\13, 1100.00
|11\13, 1116.452
|15\18, 1052.82 (1083.33)
|15\18, 1099.536
|18\21, 1136.84 (1114.29)
|18\21, 1130.951
|Ff
|Ff
| -3
| -3
|-
|-
|perfect octave
|perfect octave
|12\13, 1200.00
|12\13, 1217.942
|17\18, 1200.00 (1227.78)
|17\18, 1246.140
|19\21, 1200.00 (1178.19)
|19\21, 1193.782
|F
|F
| +5
| +5
|}
|}
===Hypohard===
===Hypohard===
Line 293: Line 278:
**The large step is near the Pythagorean 9/8 whole tone, somewhere between as in [[12edo]] and as in [[17edo]].
**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.
**The major 3rd (made of two large steps) is a near-[[Pythagorean]] to [[Neogothic]] major third.
EDIXs that are in the hypohard range include [[13edIX]], 18edIX, and 31edIX.
EDIXs that are in the hypohard range include [[13edIX|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.
The sizes of the generator, large step and small step of Neapolitan-oneirotonic are as follows in various hypohard Neapolitan-oneiro tunings.
Line 299: Line 284:
|-
|-
!
!
![[13edIX]] (basic)
![[13edIX|13ed15/7]] (basic)
!18edIX (hard)
![[18ed15/7]] (hard)
!31edIX (semihard)
!31ed15/7 (semihard)
|-
|-
|generator (g)
|generator (g)
|5\13, 500.00
|5\13, 507.478
|7\18, 494.12 (505.56)
|7\18, 513.117
|12\31, 496.55 (503.23)
|12\31, 510.752
|-
|-
|L (3g - minor 9th)
|L (3g - minor 9th)
|2\13, 200.00
|2\13, 202.991
|3\18, 211.765 (216.67)
|3\18, 219.907
|5\31, 206.87 (209.68)
|5\31, 212.813
|-
|-
|s (-5g + 2 minor 9ths)
|s (-5g + 2 minor 9ths)
|1\13, 100.00
|1\13, 101.496
|1\18, 70.59 (72.22)
|1\18, 73.302
|2\31, 82.76 (83.87)
|2\31, 85.125
|}
|}
====Intervals====
====Intervals====
Line 323: Line 308:
|-
|-
! class="unsortable" |Degree
! class="unsortable" |Degree
!Size in 13edIX (basic)
!Size in [[13edIX|13ed15/7]] (basic)
!Size in 18edIX (hard)
!Size in [[18ed15/7]] (hard)
!Size in 31edIX (semihard)
!Size in 31ed15/7 (semihard)
! class="unsortable" |Note name on G
! class="unsortable" |Note name on G
! class="unsortable" |Approximate ratios<ref>The ratio interpretations that are not valid for 18edo are italicized.</ref>
! class="unsortable" |Approximate ratios
!#Gens up
!#Gens up
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
Line 339: Line 324:
|-
|-
|minor 2nd
|minor 2nd
|1\13, 100.00
|1\13, 101.496
|1\18, 70.59 (72.22)
|1\18, 73.302
|2\31, 82.76 (83.87)
|2\31, 85.125
|Af
|Af
|21/20, ''22/21''
|21/20, ''22/21''
Line 347: Line 332:
|-
|-
|major 2nd
|major 2nd
|2\13, 200.00
|2\13, 202.991
|3\18, 211.765 (216.67)
|3\18, 219.907
|5\31, 206.87 (209.68)
|5\31, 212.813
|A
|A
|9/8
|9/8
Line 355: Line 340:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|minor 3rd
|minor 3rd
|3\13, 300.00
|3\13, 304.487
|4\18, 282.35 (288.89)
|4\18, 293.2095
|7\31, 289.655 (293.55)
|7\31, 297.939
|Bf
|Bf
|13/11, 33/28
|13/11, 33/28
Line 363: Line 348:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 3rd
|major 3rd
| rowspan="2" |4\13, 400.00
| rowspan="2" |4\13, 405.982
|6\18, 423.53 (433.33)
|6\18, 439.814
|10\31, 413.79 (419.355)
|10\31, 425.626
|B
|B
|14/11, 33/26
|14/11, 33/26
Line 371: Line 356:
|-
|-
|diminished 4th
|diminished 4th
|5\18, 352.94 (361.11)
|5\18, 366.511
|9\31, 372.41 (377.42)
|9\31, 383.064
|Cf
|Cf
|''5/4, 11/9''
|''5/4, 11/9''
Line 378: Line 363:
|-
|-
|natural 4th
|natural 4th
|5\13, 500.00
|5\13, 507.478
|7\18, 494.12 (505.56)
|7\18, 513.117
|12\31, 496.55 (503.23)
|12\31, 510.752
|C
|C
|4/3
|4/3
| +1
| +1
|- bgcolor="#eaeaff"
|augmented 4th
| rowspan="2" |6\13, 600.00
|9\18, 635.29 (650.00)
|15\31, 620.69 (629.03)
|C#
|''10/7, 18/13, 11/8''
| +9
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|diminished 5th
|diminished 5th
|8\18, 564.71 (577.78)
|6\13, 608.974
|14\31, 579.31 (587.10)
|8\18, 586.419
|14\31, 595.877
|Qf
|Qf
|''7/5, 13/9'', ''16/11''
|''7/5, 13/9'', ''16/11''
| -4
| -4
|-
|-
|perfect 5th
|perfect 5th
|7\13, 700.00
|7\13, 710.469
|10\18, 705.88 (722.22)
|10\18, 733.024
|17\31, 703.45 (712.90)
|17\31, 723.565
|Q
|Q
|3/2
|3/2
| +4
| +4
|-
|-
|minor 6th
|minor 6th
|8\13, 800.00
|8\13, 811.965
|11\18, 776.47 (794.44)
|11\18, 806.326
|19\31, 786.21 (796.77)
|19\31, 808.691
|Df
|Df
|52/33, 11/7
|52/33, 11/7
| -1
| -1
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 6th
|major 6th
| rowspan="2" |9\13, 900.00
| rowspan="2" |9\13, 913.460
|13\18, 917.65 (938.89)
|13\18, 952.931
|22\31, 910.345 (922.58)
|22\31, 936.379
|D
|D
|56/33, 22/17
|56/33, 22/17
| +7
| +7
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|minor 7th
|minor 7th
|12\18, 847.06 (866.67)
|12\18, 879.6285
|21\31, 868.97 (880.645)
|21\31, 893.816
|Ef
|Ef
|5/3, 18/11
|5/3, 18/11
| -6
| -6
|-
|-
|major 7th
|major 7th
|10\13, 1000.00
|10\13, 1014.956
|14\18, 988.235 (1011.11)
|14\18, 1026.233
|24\31, 993.13 (1007.45)
|24\31, 1021.04
|E
|E
|16/9
|16/9
| +2
| +2
|-
|-
|diminished octave
|diminished octave
|11\13, 1100.00
|11\13, 1116.452
|15\18, 1052.82 (1083.33)
|15\18, 1099.536
|26\31, 1075.86 (1090.32)
|26\31, 1106.6295
|Ff
|Ff
|11/6, 13/7, 15/8
|11/6, 13/7, 15/8
| -3
| -3
|-
|-
|perfect octave
|perfect octave
|12\13, 1200.00
|12\13, 1217.942
|17\18, 1200.00 (1227.78)
|17\18, 1246.140
|29\31, 1200.00 (1212.13)
|29\31, 1234.3175
|F
|F
|2/1
|2/1
| +5
| +5
|}
|}
<references />
===Hyposoft===
===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:
[[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 [[17edo|12edo]].
*The large step is between near the meantone and near the Pythagorean 9/8 whole tone, somewhere between as in [[19edo]] and as in [[17edo|12edo]].
*The major 3rd (made of two large steps) is a near-[[Just intonation|just]] to near-[[Pythagorean]] major third.
*The major 3rd (made of two large steps) is a near-[[Just intonation|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 (13edIX not shown).
The sizes of the generator, large step and small step of Neapolitan-oneirotonic are as follows in various hyposoft Neapolitan-oneiro tunings ([[13edIX|13ed15/7]] not shown).
{| class="wikitable right-2 right-3 right-4 right-5"
{| class="wikitable right-2 right-3 right-4 right-5"
|-
|-
!
!
!21edIX (soft)
![[21ed15/7]] (soft)
!34edIX (semisoft)
!34ed15/7 (semisoft)
|-
|-
|generator (g)
|generator (g)
|8\21, 505.26 (495.24)
|8\21, 502.645
|13\34, 458.82
|13\34, 504.493
|-
|-
|L (3g - minor 9th)
|L (3g - minor 9th)
|3\21, 189.47 (185.71)
|3\21, 188.492
|5\34, 193.55 (191.18)
|5\34, 194.036
|-
|-
|s (-5g + 2 minor 9ths)
|s (-5g + 2 minor 9ths)
|2\31, 82.76 (83.87)
|2\21, 125.661
|3\34, 116.19 (114.71)
|3\34, 116.421
|}
|}
====Intervals====
====Intervals====
Sortable table of major and minor intervals in hyposoft tunings (13edIX not shown):
Sortable table of major and minor intervals in hyposoft tunings ([[13edIX|13ed15/7]] not shown):
{| class="wikitable right-2 right-3 sortable"
{| class="wikitable right-2 right-3 sortable"
|-
|-
! class="unsortable" |Degree
! class="unsortable" |Degree
!Size in 21edIX (soft)
!Size in [[21ed15/7]] (soft)
!Size in 34edo (semisoft)
!Size in 34ed15/7 (semisoft)
! class="unsortable" |Note name on G
! class="unsortable" |Note name on G
! class="unsortable" |Approximate ratios
! class="unsortable" |Approximate ratios
Line 499: Line 475:
|-
|-
|minor 2nd
|minor 2nd
|2\21, 126.32 (121.81)
|2\21, 125.661
|3\34, 116.19 (114.71)
|3\34, 116.421
|Af
|Af
|16/15
|16/15
Line 506: Line 482:
|-
|-
|major 2nd
|major 2nd
|3\21, 189.47 (185.71)
|3\21, 188.492
|5\34, 193.55 (191.18)
|5\34, 194.036
|A
|A
|10/9, 9/8
|10/9, 9/8
Line 513: Line 489:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|minor 3rd
|minor 3rd
|5\21, 315.79 (309.52)
|5\21, 314.153
|8\34, 309.68 (305.88)
|8\34, 310.457
|Bf
|Bf
|6/5
|6/5
Line 520: Line 496:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 3rd
|major 3rd
|6\21, 378.95 (371.43)
|6\21, 376.984
|10\34, 387.10 (382.35)
|10\34, 388.071
|B
|B
|5/4
|5/4
Line 527: Line 503:
|-
|-
|diminished 4th
|diminished 4th
|7\21, 442.105 (433.33)
|7\21, 439.814
|11\34, 425.81 (420.59)
|11\34, 426.879
|Cf
|Cf
|9/7
|9/7
Line 534: Line 510:
|-
|-
|natural 4th
|natural 4th
|8\21, 505.26 (495.24)
|8\21, 502.645
|13\34, 503.23 (497.06)
|13\34, 504.493
|C
|C
|4/3
|4/3
| +1
| +1
|- bgcolor="#eaeaff"
|augmented 4th
|9\21, 568.42 (557.14)
|15\34, 580.645 (573.53)
|C#
|7/5
| +9
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|diminished 5th
|diminished 5th
|10\21, 631.58 (619.05)
|10\21, 628.306
|16\34, 619.355 (611.765)
|16\34, 620.914
|Qf
|Qf
|10/6
|10/6
| -4
| -4
|-
|-
|perfect 5th
|perfect 5th
|11\31, 694.74 (680.95)
|11\31, 691.137
|18\34, 696.77 (688.235)
|18\34, 698.5285
|Q
|Q
|3/2
|3/2
| +4
| +4
|-
|-
|minor 6th
|minor 6th
|13\21, 821.05 (802.76)
|13\21, 816.798
|21\34, 812.90 (802.94)
|21\34, 814.950
|Df
|Df
|8/5
|8/5
| -1
| -1
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 6th
|major 6th
|14\21, 884.21 (866.67)
|14\21, 879.6285
|23\34, 890.32 (879.41)
|23\34, 892.564
|D
|D
|5/3
|5/3
| +7
| +7
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|minor 7th
|minor 7th
|15\21, 947.37 (928.57)
|15\21, 942.459
|24\34, 929.03 (917.65)
|24\34, 931.3713
|Ef
|Ef
|12/7
|12/7
| -6
| -6
|-
|-
|major 7th
|major 7th
|16\21, 1017.53 (990.48)
|16\21, 1005.290
|26\34, 1006.45 (995.12)
|26\34, 1008.986
|E
|E
|9/5, 16/9
|9/5, 16/9
| +2
| +2
|-
|-
|diminished octave
|diminished octave
|18\21, 1136.84 (1114.29)
|18\21, 1130.951
|29\34, 1122.58 (1109.88)
|29\34, 1125.407
|Ff
|Ff
|27/14, 48/25
|27/14, 48/25
| -3
| -3
|-
|-
|perfect octave
|perfect octave
|19\21, 1200.00 (1178.19)
|19\21, 1193.782
|31\34, 1200.00 (1185.29)
|31\34, 1203.021
|F
|F
|2/1
|2/1
| +5
| +5
|}
|}
===Parasoft to ultrasoft tunings===
===Parasoft to ultrasoft tunings===
Line 610: Line 579:
|-
|-
!
!
!29edIX (supersoft)
!29ed15/7 (supersoft)
!37edIX
!37ed15/7
|-
|-
|generator (g)
|generator (g)
|11\29, 507.69 (493.10)
|11\29, 500.478
|14\37, 509.09 (491.89)
|14\37, 499.249
|-
|-
|L (3g - minor 9th)
|L (3g - minor 9th)
|4\29, 184.615 (179.31)
|4\29, 181.992
|5\37, 181.82 (175.68)
|5\37, 178.303
|-
|-
|s (-5g + 2 minor 9ths)
|s (-5g + 2 minor 9ths)
|3\29, 138.46 (134.49)
|3\29, 136.494
|4\37, 145.455 (140.54)
|4\37, 142.6425
|}
|}
====Intervals====
====Intervals====
Line 630: Line 599:
|-
|-
! class="unsortable" |Degree
! class="unsortable" |Degree
!Size in 29edIX (supersoft)
!Size in 29e15/7 (supersoft)
! class="unsortable" |Note name on G
! class="unsortable" |Note name on G
! class="unsortable" |Approximate ratios
! class="unsortable" |Approximate ratios
Line 642: Line 611:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|chroma
|chroma
|1\29, 46.15 (44.83)
|1\29, 45.498
|G#
|G#
|[[33/32]], [[49/48]], [[36/35]], [[25/24]]
|[[33/32]], [[49/48]], [[36/35]], [[25/24]]
Line 648: Line 617:
|-
|-
|diminished 2nd
|diminished 2nd
|2\29, 92.31 (89.655)
|2\29, 90.996
|Aff
|Aff
|[[21/20]], [[22/21]], [[26/25]]
|[[21/20]], [[22/21]], [[26/25]]
Line 654: Line 623:
|-
|-
|minor 2nd
|minor 2nd
|3\29, 138.46 (134.49)
|3\29, 136.494
|Af
|Af
|[[12/11]], [[13/12]], [[14/13]], [[16/15]]
|[[12/11]], [[13/12]], [[14/13]], [[16/15]]
Line 660: Line 629:
|-
|-
|major 2nd
|major 2nd
|4\29, 184.615 (179.31)
|4\29, 181.992
|A
|A
|[[9/8]], [[10/9]], [[11/10]]
|[[9/8]], [[10/9]], [[11/10]]
Line 666: Line 635:
|-
|-
|augmented 2nd
|augmented 2nd
|5\29, 230.77 (224.14)
|5\29, 227.490
|A#
|A#
|[[8/7]], [[15/13]]
|[[8/7]], [[15/13]]
Line 672: Line 641:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|diminished 3rd
|diminished 3rd
|6\29, 276.92 (268.97)
|6\29, 272.988
|Bff
|Bff
|[[7/6]], [[13/11]], [[33/28]]
|[[7/6]], [[13/11]], [[33/28]]
Line 678: Line 647:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|minor 3rd
|minor 3rd
|7\29, 323.08 (313.79)
|7\29, 318.486
|Bf
|Bf
|[[135/112]], [[6/5]]
|[[135/112]], [[6/5]]
Line 684: Line 653:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 3rd
|major 3rd
|8\29, 369.23 (358.21)
|8\29, 363.984
|B
|B
|[[5/4]], [[11/9]], [[16/13]]
|[[5/4]], [[11/9]], [[16/13]]
Line 690: Line 659:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|augmented 3rd
|augmented 3rd
|9\29, 415.385 (403.45)
|9\29, 409.482
|B#
|B#
|[[9/7]], [[14/11]], [[33/26]]
|[[9/7]], [[14/11]], [[33/26]]
Line 696: Line 665:
|-
|-
|diminished 4th
|diminished 4th
|10\29, 461.54 (448.28)
|10\29, 454.980
|Cf
|Cf
|[[21/16]], [[13/10]]
|[[21/16]], [[13/10]]
Line 702: Line 671:
|-
|-
|natural 4th
|natural 4th
|11\29, 507.69 (493.10)
|11\29, 500.478
|C
|C
|[[75/56]], [[4/3]]
|[[75/56]], [[4/3]]
Line 708: Line 677:
|-
|-
|augmented 4th
|augmented 4th
|12\29, 553.85 (537.93)
|12\29, 545.976
|C#
|C#
|[[11/8]], [[18/13]]
|[[11/8]], [[18/13]]
Line 714: Line 683:
|-
|-
|doubly augmented 4th, doubly diminished 5th
|doubly augmented 4th, doubly diminished 5th
|13\29, 600.00 (582.76)
|13\29, 591.474
|Cx, Qff
|Cx, Qff
|[[7/5]], [[10/7]]
|[[7/5]], [[10/7]]
Line 720: Line 689:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|diminished 5th
|diminished 5th
|14\29, 646.15 (627.59)
|14\29, 636.972
|Qf
|Qf
|[[16/11]], [[13/9]]
|[[16/11]], [[13/9]]
Line 726: Line 695:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|perfect 5th
|perfect 5th
|15\29, 692.31 (672.41)
|15\29, 682.470
|Q
|Q
|[[112/75]], [[3/2]]
|[[112/75]], [[3/2]]
Line 732: Line 701:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|augmented 5th
|augmented 5th
|16\29, 738.46 (717.24)
|16\29, 727.968
|Q#
|Q#
|[[32/21]], [[20/13]]
|[[32/21]], [[20/13]]
Line 738: Line 707:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|diminished 6th
|diminished 6th
|17\29, 784.615 (762.07)
|17\29, 773.4665
|Dff
|Dff
|[[11/7]], [[14/9]]
|[[11/7]], [[14/9]]
Line 744: Line 713:
|-
|-
|minor 6th
|minor 6th
|18\29, 830.77 (806.90)
|18\29, 818.9645
|Df
|Df
|[[13/8]], [[8/5]]
|[[13/8]], [[8/5]]
Line 750: Line 719:
|-
|-
|major 6th
|major 6th
|19\29, 876.92 (851.725)
|19\29, 864.4625
|D
|D
|[[5/3]], [[224/135]]
|[[5/3]], [[224/135]]
Line 756: Line 725:
|-
|-
|augmented 6th
|augmented 6th
|20\29, 923.08 (896.55)
|20\29, 909.961
|D#
|D#
|[[12/7]], [[22/13]]
|[[12/7]], [[22/13]]
Line 762: Line 731:
|-
|-
|minor 7th
|minor 7th
|21\29, 969.23 (941.38)
|21\29, 955.459
|Ef
|Ef
|[[7/4]], [[26/15]]
|[[7/4]], [[26/15]]
Line 768: Line 737:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 7th
|major 7th
|22\29, 1015.385 (982.21)
|22\29, 1000.956
|E
|E
|[[9/5]], [[16/9]], [[20/11]]
|[[9/5]], [[16/9]], [[20/11]]
Line 774: Line 743:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|augmented 7th
|augmented 7th
|23\29, 1061.54 (1031.03)
|23\29, 1046.455
|E#
|E#
|[[11/6]], [[13/7]], [[15/8]], [[24/13]]
|[[11/6]], [[13/7]], [[15/8]], [[24/13]]
Line 780: Line 749:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|doubly augmented 7th, doubly diminished octave
|doubly augmented 7th, doubly diminished octave
|24\29, 1107.69 (1075.86)
|24\29, 1091.953
|Ex, Fff
|Ex, Fff
|[[21/11]], [[25/13]], [[40/21]]
|[[21/11]], [[25/13]], [[40/21]]
Line 786: Line 755:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|diminished octave
|diminished octave
|25\29, 1153.85 (1120.69)
|25\29, 1137.451
|Ff
|Ff
|[[64/33]], [[96/49]], [[35/18]], [[48/25]]
|[[64/33]], [[96/49]], [[35/18]], [[48/25]]
Line 792: Line 761:
|-
|-
|perfect octave
|perfect octave
|26\29, 1200.00 (1165.52)
|26\29, 1182.949
|F
|F
|2/1
|2/1
Line 798: Line 767:
|-
|-
|augmented octave
|augmented octave
|27\29, 1246.15 (1210.345)
|27\29, 1228.447
|F#
|F#
|33/16, 49/24, 72/35, 25/12
|33/16, 49/24, 72/35, 25/12
Line 804: Line 773:
|-
|-
|doubly augmented octave, diminished 9th
|doubly augmented octave, diminished 9th
|28\29, 1292.31 (1255.13)
|28\29, 1273.945
|Fx, Gf
|Fx, Gf
|21/10, 44/21, 52/25
|21/10, 44/21, 52/25
Line 810: Line 779:
|}
|}
===Parahard===
===Parahard===
23edIX Neapolitan-oneiro combines the sound of the 32/15 minor ninth and the [[8/7]] whole tone. This is because 23edIX Neapolitan-oneirotonic a large step of 218.2¢, 22edIX's superpythagorean major second, which is both a warped Pythagorean [[9/8]] whole tone and a warped [[8/7]] septimal whole tone.  
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====
====Intervals====
The intervals of the extended generator chain (-12 to +12 generators) are as follows in various Neapolitan-oneirotonic tunings close to 23edIX.
The intervals of the extended generator chain (-12 to +12 generators) are as follows in various Neapolitan-oneirotonic tunings close to 23edIX.
Line 816: Line 785:
|-
|-
! class="unsortable" |Degree
! class="unsortable" |Degree
!Size in 23edIX
!Size in 23ed15/7
(superhard)
(superhard)
! class="unsortable" |Note name on G
! class="unsortable" |Note name on G
Line 829: Line 798:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|chroma
|chroma
|3\23, 163.63 (169.565)
|3\23, 172.101
|G#
|G#
|12/11, 11/10, 10/9
|12/11, 11/10, 10/9
Line 835: Line 804:
|-
|-
|minor 2nd
|minor 2nd
|1\23, 54.545 (56.52)
|1\23, 57.367
|Af
|Af
|[[36/35]], [[34/33]], [[33/32]], [[32/31]]
|[[36/35]], [[34/33]], [[33/32]], [[32/31]]
Line 841: Line 810:
|-
|-
|major 2nd
|major 2nd
|4\23, 218.18 (226.09)
|4\23, 229.468
|A
|A
|[[9/8]], [[17/15]], [[8/7]]
|[[9/8]], [[17/15]], [[8/7]]
Line 847: Line 816:
|-
|-
|aug. 2nd
|aug. 2nd
|7\23, 381.82 (395.65)
|7\23, 401.570
|A#
|A#
|5/4
|5/4
Line 853: Line 822:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|dim. 3rd
|dim. 3rd
|2\23, 109.09 (113.04)
|2\23, 114.734
|Bf
|Bf
|16/15
|16/15
Line 859: Line 828:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|minor 3rd
|minor 3rd
|5\23, 272.73 (282.61)
|5\23, 286.835
|B
|B
|7/6
|7/6
Line 865: Line 834:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 3rd
|major 3rd
|8\23, 436.36 (452.17)
|8\23, 458.937
|B#
|B#
|9/7, 14/11
|9/7, 14/11
Line 871: Line 840:
|-
|-
|dim. 4th
|dim. 4th
|6\23, 327.27 (339.13)
|6\23, 344.2025
|Cf
|Cf
|6/5
|6/5
Line 877: Line 846:
|-
|-
|nat. 4th
|nat. 4th
|9\23, 490.91 (508.70)
|9\23, 516.304
|C
|C
|4/3
|4/3
Line 883: Line 852:
|-
|-
|aug. 4th
|aug. 4th
|12\23, 654.545 (678.26)
|12\23, 688.405
|C#
|C#
|[[16/11]], [[22/15]]
|[[16/11]], [[22/15]]
Line 889: Line 858:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|double dim. 5th
|double dim. 5th
|7\23, 381.82 (395.65)
|7\23, 401.570
|Qff
|Qff
|5/4
|5/4
Line 895: Line 864:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|dim. 5th
|dim. 5th
|10\23, 545.455 (565.22)
|10\23, 573.671
|Qf
|Qf
|[[15/11]], [[11/8]]
|[[15/11]], [[11/8]]
Line 901: Line 870:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|perf. 5th
|perf. 5th
|13\23, 709.09 (734.78)
|13\23, 745.772
|Q
|Q
|3/2
|3/2
Line 907: Line 876:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|aug. 5th
|aug. 5th
|16\23, 872.73 (904.35)
|16\23, 917.873
|Q#
|Q#
|5/3
|5/3
Line 913: Line 882:
|-
|-
|dim. 6th
|dim. 6th
|11\23, 600.00 (621.74)
|11\23, 631.038
|Dff
|Dff
|[[7/5]], [[24/17]], [[17/12]], [[10/7]]
|[[7/5]], [[24/17]], [[17/12]], [[10/7]]
Line 919: Line 888:
|-
|-
|minor 6th
|minor 6th
|14\23, 763.64 (791.30)
|14\23, 803.139
|Df
|Df
|14/9, 11/7
|14/9, 11/7
Line 925: Line 894:
|-
|-
|major 6th
|major 6th
|17\23, 927.27 (960.87)
|17\23, 975.240
|D
|D
|12/7
|12/7
Line 932: Line 901:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|minor 7th
|minor 7th
|15\23, 818.18  (847.83)
|15\23, 860.560
|Ef
|Ef
|8/5
|8/5
Line 938: Line 907:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 7th
|major 7th
|18\23, 981.82 (1017.39)
|18\23, 1032.607
|E
|E
|[[7/4]], [[30/17]], [[16/9]]
|[[7/4]], [[30/17]], [[16/9]]
Line 944: Line 913:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|aug. 7th
|aug. 7th
|21\23, 1145.455 (1186.96)
|21\23, 1204.709
|E#
|E#
|[[31/16]], [[64/33]], [[33/17]], [[35/18]]
|[[31/16]], [[64/33]], [[33/17]], [[35/18]]
Line 950: Line 919:
|-
|-
|dim. octave
|dim. octave
|19\23, 1036.36 (1073.91)
|19\23, 1089.9745
|Ff
|Ff
|11/6, 20/11, 9/5
|11/6, 20/11, 9/5
Line 956: Line 925:
|-
|-
|perf. octave
|perf. octave
|22\23, 1200.00 (1243.48)
|22\23, 1262.076
|F
|F
|2/1
|2/1
Line 962: Line 931:
|-
|-
|aug. octave
|aug. octave
|25\23, 1363.64 (1413.04)
|25\23, 1434.177
|F#
|F#
|24/11, 11/5, 20/9
|24/11, 11/5, 20/9
Line 968: Line 937:
|-
|-
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|dim. 8-step
|dim. ninth
|20\23, 1090.91 (1130.435)
|20\23, 1147.342
|J@
|Gf
|15/8
|15/8
| -8
| -8
Line 977: Line 946:
[[Archytas clan#Ultrapyth|Ultrapythagorean]] Neapolitan-oneirotonic is a rank-2 temperament in the [[Step ratio|pseudopaucitonic]] range. It represents the [[harmonic entropy]] minimum of the Neapolitan-oneirotonic spectrum where [[7/4]] is the major seventh.
[[Archytas clan#Ultrapyth|Ultrapythagorean]] Neapolitan-oneirotonic is a rank-2 temperament in the [[Step ratio|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. 23edIX, 28edIX and 33edIX can nominally be viewed as supporting it, but are still very flat and in an ambiguous zone between 18edIX and true Buzzard in terms of harmonies. 38edIX & 43edIX 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 [[The_Archipelago|archipelago]] harmonies, as by shifting one whole tone done a comma, it shifts from [[The Archipelago|archipelago]] to septimal harmonies.
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 [[The_Archipelago|archipelago]] harmonies, as by shifting one whole tone done a comma, it shifts from [[The Archipelago|archipelago]] to septimal harmonies.


Beyond that, it's a question of which intervals you want to favor. 53edIX has an essentially perfect [[7/4]], 58edIX also gives three essentially perfect chains of third-comma meantone, while 63edIX has a double chain of essentially perfect quarter-comma meantone and gives about as low overall error as 83edIX 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.
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.
The sizes of the generator, large step and small step of Neapolitan-oneirotonic are as follows in various ultrapyth tunings.
Line 985: Line 954:
|-
|-
!
!
!38edIX
!38ed15/7
!53edIX
!53ed15/7
!63edIX
!63ed15/7
!Optimal ([[POTE|PNTE]]) Ultrapyth tuning
!Optimal ([[POTE|PNTE]]) Ultrapyth tuning
!JI intervals represented (2.3.5.7.13 subgroup)
!JI intervals represented (2.3.5.7.13 subgroup)
|-
|-
|generator (g)
|generator (g)
|15\38, 486.49  (513.16)
|15\38, 520.833
|21\53, 484.615 (515.09)
|21\53, 522.798
|25\63, 487.87 (515.87)
|25\63, 523.58’
|484.07 (515.52)
|484.07
|4/3
|4/3
|-
|-
|L (3g - minor 9th)
|L (3g - minor 9th)
|7/38, 227.03 (239,47)
|7/38, 243.055
|10/53, 230.77 (245.28)
|10/53, 248.9515
|12/63, 232.26 (247.62)
|12/63, 251.322
|231.51 (246.55)
|231.51
|8/7
|8/7
|-
|-
|s (-5g + 2 minor 9ths)
|s (-5g + 2 minor 9ths)
|1/38, 32.43 (34.21)
|1/38, 34.722
|1/53, 23.08 (24.53)
|1/53, 24.895
|1/63, 19.355 (20.635)
|1/63, 20.9345
|21.05 (22.41)
|21.05  
|50/49 81/80 91/90
|50/49 81/80 91/90
|}
|}
Line 1,017: Line 986:
|-
|-
!Degree
!Degree
!Size in 38edo
!Size in 38ed15/7
!Size in 53edo
!Size in 53ed15/7
!Size in 63edo
!Size in 63ed15/7
!Size in PNTE tuning
!Size in PNTE tuning
!Note name on G
!Note name on G
Line 1,035: Line 1,004:
|-
|-
|2
|2
|7/38, 227.03 (239.47)
|7/38, 243.055
|10/53, 230.77 (245.28)
|10/53, 248.9515
|12/63, 232.26 (247.62)
|12/63, 251.322
|231.51 (246.55)
|231.51
|A
|A
|8/7
|8/7
Line 1,044: Line 1,013:
|-
|-
|3
|3
|14\38, 454.05 (478.95)
|14\38, 486.1105
|20\53, 461.54 (490.57)
|20\53, 497.903
|24\63, 464.52 (495.24)
|24\63, 502.645
|463.03 (293.10)
|463.03  
|B
|B
|13/10, 21/16
|13/10, 21/16
Line 1,053: Line 1,022:
|-
|-
|4
|4
|15\38, 486.49  (513.16)
|15\38, 520.833
|21\53, 484.615 (515.09)
|21\53, 522.798
|25\63, 483.87 (515.87)
|25\63, 523.58’
|484.07 (515.52)
|484.07
|C
|C
|4/3
|4/3
Line 1,062: Line 1,031:
|-
|-
|5
|5
|22\38, 713.51 (752.63)
|22\38, 763.888
|31\53, 715.385 (760.38)
|31\53, 771.750
|37\63, 716.13 (763.49)
|37\63, 774.991
|715.59 (762.07)
|715.59  
|Q
|Q
|3/2
|3/2
Line 1,071: Line 1,040:
|-
|-
|6
|6
|29\38, 940.54 (992.105)
|29\38, 1006.943
|41\53, 946.15 (1005.66)
|41\53, 1020.701
|49\63, 948.39 (1011.11)
|49\63, 1026.233
|947.10 (1008.63)
|947.10
|D
|D
|26/15
|26/15
Line 1,080: Line 1,049:
|-
|-
|7
|7
|30\38, 972.97 (1026.32)
|30\38, 1041.665
|42\53, 969.23 (1030.19)
|42\53, 1045.596
|50\63, 967.74 (1031.75)
|50\63, 1047.177
|968.15 (1031.03)
|968.15  
|E
|E
|7/4
|7/4
Line 1,089: Line 1,058:
|-
|-
|8
|8
|37\38, 1200.00 (1265.57)
|37\38, 1284.721
|52\53, 1200.00 (1275.47)
|52\53, 1294.548
|62\63, 1200.00 (1280.645)
|62\63, 1298.499
|1199.66 (1277.59)
|1199.66
|F
|F
|2/1
|2/1

Revision as of 04:57, 12 June 2023

↖ 4L 2s⟨15/7⟩ ↑ 5L 2s⟨15/7⟩ 6L 2s⟨15/7⟩ ↗
← 4L 3s⟨15/7⟩ 5L 3s (15/7-equivalent) 6L 3s⟨15/7⟩ →
↙ 4L 4s⟨15/7⟩ ↓ 5L 4s⟨15/7⟩ 6L 4s⟨15/7⟩ ↘ 
┌╥╥┬╥╥┬╥┬┐
│║║│║║│║││
││││││││││
└┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern LLsLLsLs
sLsLLsLL
Equave 15/7 (1319.4 ¢)
Period 15/7 (1319.4 ¢)
Generator size(ed15/7)
Bright 3\8 to 2\5 (494.8 ¢ to 527.8 ¢)
Dark 3\5 to 5\8 (791.7 ¢ to 824.7 ¢)
Related MOS scales
Parent 3L 2s⟨15/7⟩
Sister 3L 5s⟨15/7⟩
Daughters 8L 5s⟨15/7⟩, 5L 8s⟨15/7⟩
Neutralized 2L 6s⟨15/7⟩
2-Flought 13L 3s⟨15/7⟩, 5L 11s⟨15/7⟩
Equal tunings(ed15/7)
Equalized (L:s = 1:1) 3\8 (494.8 ¢)
Supersoft (L:s = 4:3) 11\29 (500.5 ¢)
Soft (L:s = 3:2) 8\21 (502.6 ¢)
Semisoft (L:s = 5:3) 13\34 (504.5 ¢)
Basic (L:s = 2:1) 5\13 (507.5 ¢)
Semihard (L:s = 5:2) 12\31 (510.8 ¢)
Hard (L:s = 3:1) 7\18 (513.1 ¢)
Superhard (L:s = 4:1) 9\23 (516.3 ¢)
Collapsed (L:s = 1:0) 2\5 (527.8 ¢)

The minor ninth of a diatonic scale has a 5L 3s MOS structure with generators ranging from 2\5 (two degrees of 5ed15/7 = 527.8¢) to 3\8 (three degrees of 8ed15/7 = 494.8¢). In the case of 8edo, L and s are the same size; in the case of 5ed15/7, s becomes so small it disappears (and all that remains are the five equal L's).

Any ed15/7 with an interval between 494.8¢ and 527.8¢ has a 5L 3s scale. 13ed15/7 is the smallest edIX 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, 293.2095 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, 879.6285 D +7
minor 7th 12\18, 879.6285 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, 293.2095 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, 879.6285 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, 698.5285 Q 3/2 +4
minor 6th 13\21, 816.798 21\34, 814.950 Df 8/5 -1
major 6th 14\21, 879.6285 23\34, 892.564 D 5/3 +7
minor 7th 15\21, 942.459 24\34, 931.3713 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, 142.6425

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, 773.4665 Dff 11/7, 14/9 -9
minor 6th 18\29, 818.9645 Df 13/8, 8/5 -1
major 6th 19\29, 864.4625 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, 344.2025 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, 248.9515 12/63, 251.322 231.51 8/7
s (-5g + 2 minor 9ths) 1/38, 34.722 1/53, 24.895 1/63, 20.9345 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, 248.9515 12/63, 251.322 231.51 A 8/7 +3
3 14\38, 486.1105 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, 523.58’ 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"

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