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{{Infobox MOS
{{Infobox MOS
| Name = Neapolitan-Oneirotonic
|Tuning=5L 3s<15/7>
| Equave = 15/7
}}
| Periods = 1
{{MOS intro|Scale Signature=5L 3s<15/7>}}
| nLargeSteps = 5
| nSmallSteps = 3
| Equalized = 3
| Paucitonic = 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).


5L 3s is a [[Warped diatonic|warped diatonic scale]], because it has one extra small step compared to diatonic ([[5L 2s]]): for example, the Ionian diatonic mode LLsLLLs can be distorted to the Dylathian mode LLsLLsLs (a Mixolydian Mediant scale generated by ~4/3).
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.


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.
5L 3s has a pentatonic MOS subset [[3L 2s]] (SLSLL). (Note: [[3L 5s]] scales also have 3L 2s subsets.)
==Standing assumptions==
==Standing assumptions==
The [[TAMNAMS]] system is used in this article to name 5L 3s 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.


The notation used in this article is G Mixolydian Mediant (LLsLLsLs) = GABCQDEFG, unless specified otherwise. We denote raising and lowering by a chroma (L &minus; s) by # and f "flat (F molle)".  
The notation used in this article is G Mixolydian Mediant (LLsLLsLs) = GABCQDEFG, unless specified otherwise. We denote raising and lowering by a chroma (L &minus; s) by # and f "flat (F molle)".  
Line 23: Line 13:
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 [[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'''
==Names==
==Names==
The [[TAMNAMS]] system suggests the name '''Neapolitan'''-'''oneirotonic''' (/oʊnaɪrəˈtɒnɪk/ ''oh-ny-rə-TON-ik'' or /ənaɪrə-/ ''ə-ny-rə-'') or 'oneiro' for short. The name ''oneirotonic'' (from Greek ''oneiros'' 'dream') is coined after the Dreamlands in H.P. Lovecraft's Dream Cycle mythos.
The author suggests the name '''Neapolitan'''-'''oneirotonic'''.
 
'Father' is sometimes also used to denote 5L 3s, but it's a misnomer, as [[father]] is technically an abstract [[regular temperament]] (although a very inaccurate one), not a generator range. A more correct name would be 'father[8]' or 'father octatonic'. "Father" is also vague regarding the number of notes, because optimal generators for it also generate [[3L 2s]].
==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 47: Line 35:
!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 125: Line 113:
|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 [[13ed15/7]]) also has the following intervals (from some root):
|-
|-
|8
|8
Line 178: Line 166:
|-
|-
! class="unsortable" |Degree
! class="unsortable" |Degree
!Size in 13edIX (basic)
!Size in [[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 192: Line 180:
|-
|-
|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, 293.210
|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, 879.629
|D
|D
| +7
| +7
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|minor 7th
|minor 7th
|12\18, 847.06 (866.67)
|12\18, 879.629
|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 299: Line 281:
**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 [[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 305: Line 287:
|-
|-
!
!
![[13edIX]] (basic)
![[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 329: Line 311:
|-
|-
! class="unsortable" |Degree
! class="unsortable" |Degree
!Size in 13edIX (basic)
!Size in [[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 345: Line 327:
|-
|-
|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 353: Line 335:
|-
|-
|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 361: Line 343:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|minor 3rd
|minor 3rd
|3\13, 300.00
|3\13, 304.487
|4\18, 282.35 (288.89)
|4\18, 293.210
|7\31, 289.655 (293.55)
|7\31, 297.939
|Bf
|Bf
|13/11, 33/28
|13/11, 33/28
Line 369: Line 351:
|- 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 377: Line 359:
|-
|-
|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 384: Line 366:
|-
|-
|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, 879.629
|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, 1106.629
|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, 1234.317
|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 [[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 ([[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 505: Line 478:
|-
|-
|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 512: Line 485:
|-
|-
|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 519: Line 492:
|- 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 526: Line 499:
|- 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 533: Line 506:
|-
|-
|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 540: Line 513:
|-
|-
|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, 698.529
|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, 879.629
|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, 931.371
|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 616: Line 582:
|-
|-
!
!
!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, 142.642
|}
|}
====Intervals====
====Intervals====
Line 636: Line 602:
|-
|-
! 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 648: Line 614:
|- 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 654: Line 620:
|-
|-
|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 660: Line 626:
|-
|-
|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 666: Line 632:
|-
|-
|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 672: Line 638:
|-
|-
|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 678: Line 644:
|- 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 684: Line 650:
|- 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 690: Line 656:
|- 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 696: Line 662:
|- 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 702: Line 668:
|-
|-
|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 708: Line 674:
|-
|-
|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 714: Line 680:
|-
|-
|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 720: Line 686:
|-
|-
|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 726: Line 692:
|- 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 732: Line 698:
|- 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 738: Line 704:
|- 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 744: Line 710:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|diminished 6th
|diminished 6th
|17\29, 784.615 (762.07)
|17\29, 773.466
|Dff
|Dff
|[[11/7]], [[14/9]]
|[[11/7]], [[14/9]]
Line 750: Line 716:
|-
|-
|minor 6th
|minor 6th
|18\29, 830.77 (806.90)
|18\29, 818.965
|Df
|Df
|[[13/8]], [[8/5]]
|[[13/8]], [[8/5]]
Line 756: Line 722:
|-
|-
|major 6th
|major 6th
|19\29, 876.92 (851.725)
|19\29, 864.463
|D
|D
|[[5/3]], [[224/135]]
|[[5/3]], [[224/135]]
Line 762: Line 728:
|-
|-
|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 768: Line 734:
|-
|-
|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 774: Line 740:
|- 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 780: Line 746:
|- 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 786: Line 752:
|- 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 792: Line 758:
|- 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 798: Line 764:
|-
|-
|perfect octave
|perfect octave
|26\29, 1200.00 (1165.52)
|26\29, 1182.949
|F
|F
|2/1
|2/1
Line 804: Line 770:
|-
|-
|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 810: Line 776:
|-
|-
|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
| -8
| -8
|}
|}
===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 has 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 822: Line 789:
|-
|-
! 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 835: Line 802:
|- 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 841: Line 808:
|-
|-
|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 847: Line 814:
|-
|-
|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 853: Line 820:
|-
|-
|aug. 2nd
|aug. 2nd
|7\23, 381.82 (395.65)
|7\23, 401.570
|A#
|A#
|5/4
|5/4
Line 859: Line 826:
|- 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 865: Line 832:
|- 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 871: Line 838:
|- 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 877: Line 844:
|-
|-
|dim. 4th
|dim. 4th
|6\23, 327.27 (339.13)
|6\23, 344.202
|Cf
|Cf
|6/5
|6/5
Line 883: Line 850:
|-
|-
|nat. 4th
|nat. 4th
|9\23, 490.91 (508.70)
|9\23, 516.304
|C
|C
|4/3
|4/3
Line 889: Line 856:
|-
|-
|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 895: Line 862:
|- 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 901: Line 868:
|- 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 907: Line 874:
|- 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 913: Line 880:
|- 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 919: Line 886:
|-
|-
|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 925: Line 892:
|-
|-
|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 931: Line 898:
|-
|-
|major 6th
|major 6th
|17\23, 927.27 (960.87)
|17\23, 975.240
|D
|D
|12/7
|12/7
Line 938: Line 905:
|- 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 944: Line 911:
|- 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 950: Line 917:
|- 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 956: Line 923:
|-
|-
|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 962: Line 929:
|-
|-
|perf. octave
|perf. octave
|22\23, 1200.00 (1243.48)
|22\23, 1262.076
|F
|F
|2/1
|2/1
Line 968: Line 935:
|-
|-
|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 974: Line 941:
|-
|-
|- 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 983: Line 950:
[[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 991: Line 958:
|-
|-
!
!
!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, 248.951
|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, 20.934
|21.05 (22.41)
|21.05  
|[[Tel:50/49 81/80|50/49 81/80]] 91/90
|50/49 81/80 91/90
|}
|}
====Intervals====
====Intervals====
Line 1,023: Line 990:
|-
|-
!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 1,041: Line 1,008:
|-
|-
|2
|2
|7/38, 227.03 (239.47)
|7/38, 243.055
|10/53, 230.77 (245.28)
|10/53, 248.951
|12/63, 232.26 (247.62)
|12/63, 251.322
|231.51 (246.55)
|231.51
|A
|A
|8/7
|8/7
Line 1,050: Line 1,017:
|-
|-
|3
|3
|14\38, 454.05 (478.95)
|14\38, 486.111
|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 1,059: Line 1,026:
|-
|-
|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, 523.588
|484.07 (515.52)
|484.07
|C
|C
|4/3
|4/3
Line 1,068: Line 1,035:
|-
|-
|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 1,077: Line 1,044:
|-
|-
|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 1,086: Line 1,053:
|-
|-
|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 1,095: Line 1,062:
|-
|-
|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
Line 1,144: Line 1,111:
|}
|}
==Scale tree==
==Scale tree==
{| class="wikitable center-all"
{{MOS tuning spectrum|Scale Signature=5L 3s<15/7>}}
! colspan="12" |Normalized Generator
!Cents
!Relative Cents
!L
!s
!L/s
!Comments
|-
| colspan="3" |3\(8\7)|| || || || ||
|
|
|
| ||514.285…
|487.5||1||1||1.000||
|-
|
|
| || || || colspan="3" |17\(45\40)
|
|
|
| ||510
|491.1̄||6||5||1.200||
|-
|
|
| || || colspan="3" |14\(37\33)||
|
|
|
| ||509.0̄9̄
|491.8̄9̄1̄||5||4||1.250||
|-
|
|
| || || || colspan="3" |25\(66\59)
|
|
|
| ||508.474…
|492.4̄2̄||9||7||1.286||
|-
|
|
| || colspan="3" |11\(29\26)|| ||
|
|
|
| ||507.692…
|493.103…||4||3||1.333||
|-
|
|
| || || || colspan="3" |30\(79\71)
|
|
|
| ||507.042…
|493.670…||11||8||1.375||
|-
|
|
| || || colspan="3" |19\(50\45)||
|
|
|
| ||506.6̄
|494||7||5||1.400||
|-
|
|
| || || || colspan="3" |27\(71\64)
|
|
|
| ||506.25
|494.366…||10||7||1.429||
|-
|
|
| colspan="3" |8\(21\19)|| || ||
|
|
|
| ||505.263…
|495.238…||3||2||1.500||L/s = 3/2
|-
|
|
| || || || colspan="3" |29\(76\69)
|
|
|
| ||504.347…
|496.052…||11||7||1.571||
|-
|
|
| || || colspan="3" |21\(55\50)||
|
|
|
| ||504
|496.3̄6̄||8||5||1.600||
|-
|
|
| || || || colspan="3" |34\(89\81)
|
|
|
| ||503.7̄0̄3̄
|496.629…||13||8||1.625||Golden Neapolitan-oneirotonic
|-
|
|
| || colspan="3" |13\(34\31)|| ||
|
|
|
| ||503.225…
|497.058…||5||3||1.667||
|-
|
|
| || || || colspan="3" |31\(81\74)
|
|
|
| ||502.7̄0̄2̄
|497.530…||12||7||1.714||
|-
|
|
| || || colspan="3" |18\(47\43)||
|
|
|
| ||502.326…
|497.872…||7||4||1.750||
|-
|
|
| || || || colspan="3" |23\(60\55)
|
|
|
| ||501.8̄1̄
|498.3̄||9||5||1.800||
|-
|
|
|
|
|
|
| colspan="3" |28\(73/67)
|
|
|
|501.492…
|498.630…
|11
|6
|1.833
|
|-
|
|
|
|
|
|
|
| colspan="3" |33\(86\79)
|
|
|501.265…
|498.837…
|13
|7
|1.857
|
|-
|
|
|
|
|
|
|
|
| colspan="3" |38\(99\91)
|
|501.098…
|498.9̄8̄
|15
|8
|1.875
|
|-
|
|
|
|
|
|
|
|
|
| colspan="3" |43\(112\103)
|500.970…
|499.107…
|17
|9
|1.889
|
|-
|
| colspan="3" |5\(13\12)|| || || ||
|
|
|
| ||500
|500||2||1||2.000||Basic Neapolitan-oneirotonic
(generators smaller than this are proper)
|-
|
|
|
|
|
|
|
|
|
| colspan="3" |42\(109\101)
|499.009…
|500.916̄
|17
|8
|2.125
|
|-
|
|
|
|
|
|
|
|
| colspan="3" |37\(96\89)
|
|498.876…
|501.041…
|15
|7
|2.143
|
|-
|
|
|
|
|
|
|
| colspan="3" |32\(83\77)
|
|
|498.701…
|501.204…
|13
|6
|2.167
|
|-
|
|
|
|
|
|
| colspan="3" |27\(70\65)
|
|
|
|498.461…
|501.429…
|11
|5
|2.200
|
|-
|
|
| || || || colspan="3" |22\(57\53)
|
|
|
| ||498.113…
|501.754…||9||4||2.250||
|-
|
|
| || || colspan="3" |17\(44\41)||
|
|
|
| ||497.560…
|502.2̄7̄||7||3||2.333||
|-
|
|
| || || || colspan="3" |29\(75\70)
|
|
|
| ||497.142…
|502.6̄||12||5||2.400||
|-
|
|
| || colspan="3" |12\(31\29)|| ||
|
|
|
| ||496.551…
|503.225…||5||2||2.500||
|-
|
|
| || || || colspan="3" |31\(80\75)
|
|
|
| ||496
|503.75||13||5||2.600||
|-
|
|
| || || colspan="3" |19\(49\46)||
|
|
|
| ||495.652…
|504.081…||8||3||2.667||
|-
|
|
| || || || colspan="3" |26\(67\63)
|
|
|
| ||495.238…
|504.477…||11||4||2.750||
|-
|
|
| colspan="3" |7\(18\17)|| || ||
|
|
|
| ||494.117…
|505.5̄||3||1||3.000||L/s = 3/1
|-
|
|
|
|
|
|
| colspan="3" |30\(77\73)
|
|
|
|493.150…
|506.493…
|13
|4
|3.250
|
|-
|
|
| || || || colspan="3" |23\(59\56)
|
|
|
| ||492.857…
|506.779…||10||3||3.333||
|-
|
|
| || || colspan="3" |16\(41\39)||
|
|
|
| ||492.307…
|507.317…||7||2||3.500||
|-
|
|
| || || || colspan="3" |25\(64\61)
|
|
|
| ||491.803…
|[[Tel:507.8125|507.8125]]||11||3||3.667||
|-
|
|
| || colspan="3" |9\(23\22)|| ||
|
|
|
| ||490.9̄0̄
|508.695…||4||1||4.000||
|-
|
|
| || || || colspan="3" |20\(51\49)
|
|
|
| ||489.795…
|509.803…||9||2||4.500||
|-
|
|
| || || colspan="3" |11\(28\27)||
|
|
|
| ||488.8̄
|510.714…||5||1||5.000||
|-
|
|
|
|
|
|
| colspan="3" |24\(61\59)
|
|
|
|488.135…
|511.475…
|11
|2
|5.500
|
|-
|
|
| || || || colspan="3" |13\(33\32)
|
|
|
| ||487.5
|512.1̄2̄||6||1||6.000||
|-
| colspan="3" |2\5|| || || || ||
|
|
|
| ||480.000
|520||1||0||→ inf||
|}
Retrieved from "https://en.xen.wiki/w/User:Moremajorthanmajor/5L_3s_(15/7-equivalent)"