User:Moremajorthanmajor/5L 3s (11/5-equivalent): Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
mNo edit summary
ArrowHead294 (talk | contribs)
14,512 edits
mNo edit summary
 
(9 intermediate revisions by 2 users not shown)
Line 1: Line 1:
{{Infobox MOS
{{Infobox MOS
| Name = Neapolitan-oneirotonic
|Tuning=5L 3s<11/5>}}{{MOS intro|Scale Signature=5L 3s<11/5>}}
| Equave = 11/5
| 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.
[[13ed11/5]] is the smallest ed11/5 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<11/5> intervals and step size ratios and step ratio ranges.
The [[TAMNAMS]] system is used in this article to name 5L 3s<11/5><ref>the rare simplest tuning for a diatonic minor ninth </ref> 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 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 [[13ed11/5]] 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 [[18ed11/5]] 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 [[21ed11/5]] 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 and reducing mod 11/5.
{| class="wikitable center-all"
{| class="wikitable center-all"
|-
|-
Line 43: Line 34:
!Notation of 2/1 inverse
!Notation of 2/1 inverse
!name
!name
!In L's and s's
! In L's and s's
|-
|-
| colspan="8" |The 8-note MOS has the following intervals (from some root):
| colspan="8" |The 8-note MOS has the following intervals (from some root):
Line 54: Line 45:
|G
|G
|“perfect” minor 9th
|“perfect” minor 9th
|5L + 3s
| 5L + 3s
|-
|-
|1
|1
Line 78: Line 69:
|major 2nd
|major 2nd
|1L + 0s
|1L + 0s
| -3
| -3
|Ff
|Ff
|diminished octave
|diminished octave
Line 102: Line 93:
|-
|-
|6
|6
|B
| B
|major 3rd
|major 3rd
|2L + 0s
|2L + 0s
Line 112: Line 103:
|7
|7
|D
|D
|major 6th
| major 6th
|4L + 1s
|4L + 1s
| -7
| -7
|Cf
|Cf
|diminished 4th
| diminished 4th
|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 (11/5-equivalent)|5L 8s]], [[8L 5s (11/5-equivalent)|8L 5s]], or [[13ed11/5]]) also has the following intervals (from some root):
|-
|-
|8
|8
Line 125: Line 116:
|augmented unison
|augmented unison
|1L - 1s
|1L - 1s
| -8
| -8
|Gf
|Gf
|diminished 9th
|diminished 9th
|4L + 4s
|4L + 4s
|-
|-
|9
| 9
|C#
|C#
|augmented 4th
| augmented 4th
|3L + 0s
|3L + 0s
| -9
| -9
|Dff
|Dff
|diminished 6th
|diminished 6th
Line 142: Line 133:
|E#
|E#
|augmented 7th
|augmented 7th
|5L + 1s
| 5L + 1s
| -10
| -10
|Bff
|Bff
Line 153: Line 144:
|2L - 1s
|2L - 1s
| -11
| -11
|Fff
| Fff
|doubly diminished octave
|doubly diminished octave
|3L + 4s
|3L + 4s
Line 162: Line 153:
|4L + 0s
|4L + 0s
| -12
| -12
|Qff
| Qff
|doubly diminished 5th
|doubly diminished 5th
|1L + 3s
|1L + 3s
|}
|}
==Tuning ranges==
==Tuning ranges==
===Simple tunings===
=== Simple tunings===
Table of intervals in the simplest Neapolitan-oneirotonic tunings:
Table of intervals in the simplest Neapolitan-oneirotonic tunings:
{| class="wikitable right-2 right-3 right-4 sortable"
{| class="wikitable right-2 right-3 right-4 sortable"
|-
|-
! class="unsortable" |Degree
! class="unsortable" |Degree
!Size in 13edIX (basic)
!Size in 13ed11/5 (basic)
!Size in 18edIX (hard)
!Size in 18ed11/5 (hard)
!Size in 21edIX (soft)
!Size in 21ed11/5 (soft)
! class="unsortable" |Note name on G
! class="unsortable" |Note name on G
!#Gens up
!#Gens up
Line 185: Line 176:
|0
|0
|-
|-
|minor 2nd
| minor 2nd
|1\13, 100.00
|1\13, 105.000
|1\18, 70.59 (72.22)
|1\18, 75.834
|2\21, 126.32 (121.81)
|2\21, 130.000
|Af
|Af
| -5
| -5
|-
|-
|major 2nd
|major 2nd
|2\13, 200.00
|2\13, 210.001
|3\18, 211.765 (216.67)
|3\18, 227.501
|3\21, 189.47 (185.71)
|3\21, 195.001
|A
|A
| +3
| +3
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|minor 3rd
|minor 3rd
|3\13, 300.00
|3\13, 315.001
|4\18, 282.35 (288.89)
|4\18, 303.334
|5\21, 315.79 (309.52)
|5\21, 325.001
|Bf
|Bf
| -2
| -2
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 3rd
|major 3rd
| rowspan="2" |4\13, 400.00
| rowspan="2" |4\13, 420.001
|6\18, 423.53 (433.33)
|6\18, 455.001
|6\21, 378.95 (371.43)
|6\21, 390.001
|B
|B
| +6
| +6
|-
|-
|diminished 4th
| diminished 4th
|5\18, 352.94 (361.11)
|5\18, 379.168
|7\21, 442.105 (433.33)
|7\21, 455.001
|Cf
|Cf
| -7
| -7
|-
|-
|natural 4th
|natural 4th
|5\13, 500.00
| 5\13, 525.002
|7\18, 494.12 (505.56)
|7\18, 530.835
|8\21, 505.26 (495.24)
|8\21, 520.002
|C
|C
| +1
| +1
|-
|-
|augmented 4th
|augmented 4th
| rowspan="2" |6\13, 600.00
| rowspan="2" |6\13, 630.002
|9\18, 635.29 (650.00)
|9\18, 682.502
|9\21, 568.42 (557.14)
|9\21, 585.002
|C#
|C#
| +9
| +9
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|diminished 5th
|diminished 5th
|8\18, 564.71 (577.78)
|8\18, 606.669
|10\21, 631.58 (619.05)
|10\21, 650.002
|Qf
|Qf
| -4
| -4
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|perfect 5th
|perfect 5th
|7\13, 700.00
|7\13, 735.002
|10\18, 705.88 (722.22)
|10\18, 758.336
|11\31, 694.74 (680.95)
| 11\21, 715.002
|Q
|Q
| +4
| +4
|-
|-
|minor 6th
|minor 6th
|8\13, 800.00
|8\13, 840.003
|11\18, 776.47 (794.44)
|11\18, 834.169
|13\21, 821.05 (802.76)
|13\21, 780.002
|Df
|Df
| -1
| -1
|-
|-
|major 6th
|major 6th
| rowspan="2" |9\13, 900.00
| rowspan="2" |9\13, 945.003
|13\18, 917.65 (938.89)
|13\18, 985.836
|14\21, 884.21 (866.67)
|14\21, 845.003
|D
|D
| +7
| +7
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|minor 7th
| minor 7th
|12\18, 847.06 (866.67)
|12\18, 910.003
|15\21, 947.37 (928.57)
|15\21, 910.003
|Ef
|Ef
| -6
| -6
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 7th
|major 7th
|10\13, 1000.00
|10\13, 1050.003
|14\18, 988.235 (1011.11)
|14\18, 1061.670
|16\21, 1017.53 (990.48)
|16\21, 975.003
|E
|E
| +2
| +2
|-
|-
|diminished octave
|diminished octave
|11\13, 1100.00
|11\13, 1155.004
|15\18, 1052.82 (1083.33)
|15\18, 1137.504
|18\21, 1136.84 (1114.29)
|18\21, 1105.003
|Ff
| Ff
| -3
| -3
|-
|-
|perfect octave
|perfect octave
|12\13, 1200.00
|12\13, 1260.004
|17\18, 1200.00 (1227.78)
|17\18, 1289.171
|19\21, 1200.00 (1178.19)
|19\21, 1170.003
|F
|F
| +5
| +5
Line 290: Line 281:
[[Hypohard]] Neapolitan-oneirotonic tunings (with generator between 5\13 and 7\18) have step ratios between 2/1 and 3/1.
[[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 [[Superpyth|superpythagorean]] diatonic tunings:
Hypohard Neapolitan-oneirotonic can be considered "superpythagorean Neapolitan-oneirotonic". This is because these tunings share the following features with [[Superpyth|superpythagorean]] diatonic tunings:
**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 [[13ed11/5]], [[18ed11/5]], and [[31ed11/5]].


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 290:
|-
|-
!
!
![[13edIX]] (basic)
![[13ed11/5]] (basic)
!18edIX (hard)
![[18ed11/5]] (hard)
!31edIX (semihard)
![[31ed11/5]] (semihard)
|-
|-
|generator (g)
|generator (g)
|5\13, 500.00
|5\13, 525.002
|7\18, 494.12 (505.56)
|10\18, 758.336
|12\31, 496.55 (503.23)
|12\31, 496.55  
|-
|-
|L (3g - minor 9th)
|L (3g - minor 9th)
|2\13, 200.00
|2\13, 210.001
|3\18, 211.765 (216.67)
|3\18, 227.501
|5\31, 206.87 (209.68)
|5\31, 206.87
|-
|-
|s (-5g + 2 minor 9ths)
|s (-5g + 2 minor 9ths)
|1\13, 100.00
|1\13, 105.000
|1\18, 70.59 (72.22)
| 1\18, 75.834
|2\31, 82.76 (83.87)
|2\31, 82.76  
|}
|}
====Intervals====
====Intervals====
Line 323: Line 314:
|-
|-
! class="unsortable" |Degree
! class="unsortable" |Degree
!Size in 13edIX (basic)
!Size in 13ed11/5 (basic)
!Size in 18edIX (hard)
!Size in 18ed11/5 (hard)
!Size in 31edIX (semihard)
!Size in 31ed11/5 (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 335: Line 326:
|0\31, 0.00
|0\31, 0.00
|G
|G
|1/1
| 1/1
|0
|0
|-
|-
|minor 2nd
|minor 2nd
|1\13, 100.00
|1\13, 105.000
|1\18, 70.59 (72.22)
|1\18, 75.834
|2\31, 82.76 (83.87)
|2\31, 88.065
|Af
|Af
|21/20, ''22/21''
|21/20, ''22/21''
Line 347: Line 338:
|-
|-
|major 2nd
|major 2nd
|2\13, 200.00
|2\13, 210.001
|3\18, 211.765 (216.67)
|3\18, 227.501
|5\31, 206.87 (209.68)
|5\31, 220.162
|A
|A
|9/8
|9/8
| +3
| +3
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|minor 3rd
|minor 3rd
|3\13, 300.00
|3\13, 315.001
|4\18, 282.35 (288.89)
|4\18, 303.334
|7\31, 289.655 (293.55)
|7\31, 308.227
|Bf
|Bf
|13/11, 33/28
|13/11, 33/28
Line 363: Line 354:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 3rd
|major 3rd
| rowspan="2" |4\13, 400.00
| rowspan="2" |4\13, 420.001
|6\18, 423.53 (433.33)
|6\18, 455.001
|10\31, 413.79 (419.355)
|10\31, 440.324
|B
|B
|14/11, 33/26
|14/11, 33/26
Line 371: Line 362:
|-
|-
|diminished 4th
|diminished 4th
|5\18, 352.94 (361.11)
|5\18, 379.168
|9\31, 372.41 (377.42)
|9\31, 396.292
|Cf
|Cf
|''5/4, 11/9''
|''5/4, 11/9''
Line 378: Line 369:
|-
|-
|natural 4th
|natural 4th
|5\13, 500.00
|5\13, 525.002
|7\18, 494.12 (505.56)
|7\18, 530.835
|12\31, 496.55 (503.23)
|12\31, 528.389
|C
|C
|4/3
|4/3
| +1
| +1
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|augmented 4th
|augmented 4th
| rowspan="2" |6\13, 600.00
| rowspan="2" |6\13, 630.002
|9\18, 635.29 (650.00)
|9\18, 682.502
|15\31, 620.69 (629.03)
|15\31, 660.486
|C#
|C#
|''10/7, 18/13, 11/8''
|''10/7, 18/13, 11/8''
Line 394: Line 385:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|diminished 5th
|diminished 5th
|8\18, 564.71 (577.78)
| 8\18, 606.669
|14\31, 579.31 (587.10)
|14\31, 616.454
|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, 735.002
|10\18, 705.88 (722.22)
|10\18, 758.336
|17\31, 703.45 (712.90)
| 17\31, 748.551
|Q
|Q
|3/2
|3/2
Line 409: Line 400:
|-
|-
|minor 6th
|minor 6th
|8\13, 800.00
|8\13, 840.003
|11\18, 776.47 (794.44)
|11\18, 834.169
|19\31, 786.21 (796.77)
|19\31, 836.615
|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, 945.003
|13\18, 917.65 (938.89)
|13\18, 985.836
|22\31, 910.345 (922.58)
|22\31, 968.713
|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, 910.003
|21\31, 868.97 (880.645)
|21\31, 924.680
|Ef
|Ef
|5/3, 18/11
|5/3, 18/11
| -6
| -6
|-
|-
|major 7th
| major 7th
|10\13, 1000.00
|10\13, 1050.003
|14\18, 988.235 (1011.11)
|14\18, 1061.670
|24\31, 993.13 (1007.45)
|24\31, 1056.778
|E
|E
|16/9
|16/9
Line 440: Line 431:
|-
|-
|diminished octave
|diminished octave
|11\13, 1100.00
|11\13, 1155.004
|15\18, 1052.82 (1083.33)
|15\18, 1137.5035
|26\31, 1075.86 (1090.32)
|26\31, 1144.84
|Ff
|Ff
|11/6, 13/7, 15/8
|11/6, 13/7, 15/8
Line 448: Line 439:
|-
|-
|perfect octave
|perfect octave
|12\13, 1200.00
|12\13, 1260.004
|17\18, 1200.00 (1227.78)
|17\18, 1289.171
|29\31, 1200.00 (1212.13)
|29\31, 1276.939
|F
|F
|2/1
|2/1
Line 465: Line 456:
|-
|-
!
!
!21edIX (soft)
! 21ed11/5 (soft)
!34edIX (semisoft)
!34ed11/5 (semisoft)
|-
|-
|generator (g)
| generator (g)
|8\21, 505.26 (495.24)
|8\21, 520.002
|13\34, 458.82
|13\34, 521.913
|-
|-
|L (3g - minor 9th)
|L (3g - minor 9th)
|3\21, 189.47 (185.71)
| 3\21, 195.001
|5\34, 193.55 (191.18)
|5\34, 200.736
|-
|-
|s (-5g + 2 minor 9ths)
|s (-5g + 2 minor 9ths)
|2\31, 82.76 (83.87)
|2\21, 130.000
|3\34, 116.19 (114.71)
|3\34, 120.442
|}
|}
====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 (13ed11/5 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 21ed11/5 (soft)
!Size in 34edo (semisoft)
!Size in 34ed11/5 (semisoft)
! class="unsortable" |Note name on G
! class="unsortable" |Note name on G
! class="unsortable" |Approximate ratios
! class="unsortable" |Approximate ratios
Line 499: Line 490:
|-
|-
|minor 2nd
|minor 2nd
|2\21, 126.32 (121.81)
| 2\21, 130.000
|3\34, 116.19 (114.71)
|3\34, 120.442
|Af
|Af
|16/15
|16/15
Line 506: Line 497:
|-
|-
|major 2nd
|major 2nd
|3\21, 189.47 (185.71)
|3\21, 195.001
|5\34, 193.55 (191.18)
|5\34, 200.736
|A
|A
|10/9, 9/8
|10/9, 9/8
| +3
| +3
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|minor 3rd
|minor 3rd
|5\21, 315.79 (309.52)
|5\21, 325.001
|8\34, 309.68 (305.88)
|8\34, 321.177
|Bf
|Bf
|6/5
|6/5
Line 520: Line 511:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 3rd
|major 3rd
|6\21, 378.95 (371.43)
|6\21, 390.001
|10\34, 387.10 (382.35)
|10\34, 401.472
|B
|B
|5/4
|5/4
Line 527: Line 518:
|-
|-
|diminished 4th
|diminished 4th
|7\21, 442.105 (433.33)
|7\21, 455.001
|11\34, 425.81 (420.59)
|11\34, 441.619
|Cf
|Cf
|9/7
|9/7
Line 534: Line 525:
|-
|-
|natural 4th
|natural 4th
|8\21, 505.26 (495.24)
|8\21, 520.002
|13\34, 503.23 (497.06)
|13\34, 521.913
|C
|C
|4/3
| 4/3
| +1
| +1
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|augmented 4th
| augmented 4th
|9\21, 568.42 (557.14)
|9\21, 585.002
|15\34, 580.645 (573.53)
|15\34, 602.208
|C#
| C#
|7/5
|7/5
| +9
| +9
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|diminished 5th
|diminished 5th
|10\21, 631.58 (619.05)
|10\21, 650.002
|16\34, 619.355 (611.765)
| 16\34, 642.355
|Qf
|Qf
|10/6
|10/6
Line 555: Line 546:
|-
|-
|perfect 5th
|perfect 5th
|11\31, 694.74 (680.95)
|11\21, 715.002
|18\34, 696.77 (688.235)
| 18\34, 722.649
|Q
|Q
|3/2
|3/2
Line 562: Line 553:
|-
|-
|minor 6th
|minor 6th
|13\21, 821.05 (802.76)
|13\21, 780.002
|21\34, 812.90 (802.94)
|21\34, 843.091
|Df
|Df
|8/5
|8/5
Line 569: Line 560:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 6th
|major 6th
|14\21, 884.21 (866.67)
|14\21, 845.003
|23\34, 890.32 (879.41)
|23\34, 923.385
|D
|D
|5/3
|5/3
| +7
| +7
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|minor 7th
|minor 7th
|15\21, 947.37 (928.57)
|15\21, 910.003
|24\34, 929.03 (917.65)
|24\34, 963.532
|Ef
|Ef
|12/7
|12/7
Line 583: Line 574:
|-
|-
|major 7th
|major 7th
|16\21, 1017.53 (990.48)
|16\21, 975.003
|26\34, 1006.45 (995.12)
|26\34, 1043.826
|E
|E
|9/5, 16/9
|9/5, 16/9
Line 590: Line 581:
|-
|-
|diminished octave
|diminished octave
|18\21, 1136.84 (1114.29)
|18\21, 1105.003
|29\34, 1122.58 (1109.88)
|29\34, 1164.268
|Ff
|Ff
|27/14, 48/25
|27/14, 48/25
Line 597: Line 588:
|-
|-
|perfect octave
|perfect octave
|19\21, 1200.00 (1178.19)
|19\21, 1170.003
|31\34, 1200.00 (1185.29)
|31\34, 1244.563
|F
|F
|2/1
|2/1
Line 610: Line 601:
|-
|-
!
!
!29edIX (supersoft)
!29ed11/5 (supersoft)
!37edIX
!37ed11/5
|-
|-
|generator (g)
| generator (g)
|11\29, 507.69 (493.10)
|11\29, 517.760
|14\37, 509.09 (491.89)
|14\37, 516.488
|-
|-
|L (3g - minor 9th)
|L (3g - minor 9th)
|4\29, 184.615 (179.31)
|4\29, 188.276
|5\37, 181.82 (175.68)
| 5\37, 184.460
|-
|-
|s (-5g + 2 minor 9ths)
|s (-5g + 2 minor 9ths)
|3\29, 138.46 (134.49)
|3\29, 141.207
|4\37, 145.455 (140.54)
|4\37, 147.568
|}
|}
====Intervals====
====Intervals====
Line 630: Line 621:
|-
|-
! class="unsortable" |Degree
! class="unsortable" |Degree
!Size in 29edIX (supersoft)
!Size in 29ed11/5 (supersoft)
! class="unsortable" |Note name on G
! class="unsortable" |Note name on G
! class="unsortable" |Approximate ratios
! class="unsortable" |Approximate ratios
Line 642: Line 633:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|chroma
|chroma
|1\29, 46.15 (44.83)
|1\29, 47.069
|G#
|G#
|[[33/32]], [[49/48]], [[36/35]], [[25/24]]
|[[33/32]], [[49/48]], [[36/35]], [[25/24]]
Line 648: Line 639:
|-
|-
|diminished 2nd
|diminished 2nd
|2\29, 92.31 (89.655)
|2\29, 94.138
|Aff
|Aff
|[[21/20]], [[22/21]], [[26/25]]
|[[21/20]], [[22/21]], [[26/25]]
Line 654: Line 645:
|-
|-
|minor 2nd
|minor 2nd
|3\29, 138.46 (134.49)
|3\29, 141.207
|Af
|Af
|[[12/11]], [[13/12]], [[14/13]], [[16/15]]
|[[12/11]], [[13/12]], [[14/13]], [[16/15]]
| -5
| -5
|-
|-
|major 2nd
| major 2nd
|4\29, 184.615 (179.31)
|4\29, 188.276
|A
|A
|[[9/8]], [[10/9]], [[11/10]]
|[[9/8]], [[10/9]], [[11/10]]
Line 666: Line 657:
|-
|-
|augmented 2nd
|augmented 2nd
|5\29, 230.77 (224.14)
|5\29, 235.346
|A#
| A#
|[[8/7]], [[15/13]]
|[[8/7]], [[15/13]]
| +11
| +11
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|diminished 3rd
|diminished 3rd
|6\29, 276.92 (268.97)
|6\29, 282.415
|Bff
|Bff
|[[7/6]], [[13/11]], [[33/28]]
|[[7/6]], [[13/11]], [[33/28]]
Line 678: Line 669:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|minor 3rd
|minor 3rd
|7\29, 323.08 (313.79)
|7\29, 329.483
|Bf
|Bf
|[[135/112]], [[6/5]]
|[[135/112]], [[6/5]]
Line 684: Line 675:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 3rd
|major 3rd
|8\29, 369.23 (358.21)
|8\29, 376.553
|B
|B
|[[5/4]], [[11/9]], [[16/13]]
|[[5/4]], [[11/9]], [[16/13]]
| +6
| +6
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|augmented 3rd
| augmented 3rd
|9\29, 415.385 (403.45)
|9\29, 423.622
|B#
| B#
|[[9/7]], [[14/11]], [[33/26]]
|[[9/7]], [[14/11]], [[33/26]]
| +14
| +14
|-
|-
|diminished 4th
|diminished 4th
|10\29, 461.54 (448.28)
|10\29, 470.691
|Cf
|Cf
|[[21/16]], [[13/10]]
|[[21/16]], [[13/10]]
Line 702: Line 693:
|-
|-
|natural 4th
|natural 4th
|11\29, 507.69 (493.10)
|11\29, 517.760
|C
|C
|[[75/56]], [[4/3]]
|[[75/56]], [[4/3]]
Line 708: Line 699:
|-
|-
|augmented 4th
|augmented 4th
|12\29, 553.85 (537.93)
|12\29, 564.829
|C#
|C#
|[[11/8]], [[18/13]]
|[[11/8]], [[18/13]]
| +9
| +9
|-
|-
|doubly augmented 4th, doubly diminished 5th
|doubly augmented 4th, doubly diminished 5th
|13\29, 600.00 (582.76)
|13\29, 611.898
|Cx, Qff
|Cx, Qff
|[[7/5]], [[10/7]]
|[[7/5]], [[10/7]]
Line 720: Line 711:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|diminished 5th
|diminished 5th
|14\29, 646.15 (627.59)
|14\29, 658.968
|Qf
|Qf
|[[16/11]], [[13/9]]
|[[16/11]], [[13/9]]
Line 726: Line 717:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|perfect 5th
|perfect 5th
|15\29, 692.31 (672.41)
|15\29, 706.037
|Q
|Q
|[[112/75]], [[3/2]]
|[[112/75]], [[3/2]]
| +4
| +4
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|augmented 5th
|augmented 5th
|16\29, 738.46 (717.24)
|16\29, 753.106
|Q#
|Q#
|[[32/21]], [[20/13]]
|[[32/21]], [[20/13]]
Line 738: Line 729:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|diminished 6th
|diminished 6th
|17\29, 784.615 (762.07)
|17\29, 800.175
|Dff
|Dff
|[[11/7]], [[14/9]]
|[[11/7]], [[14/9]]
Line 744: Line 735:
|-
|-
|minor 6th
|minor 6th
|18\29, 830.77 (806.90)
|18\29, 847.244
|Df
|Df
|[[13/8]], [[8/5]]
|[[13/8]], [[8/5]]
Line 750: Line 741:
|-
|-
|major 6th
|major 6th
|19\29, 876.92 (851.725)
|19\29, 894.313
|D
|D
|[[5/3]], [[224/135]]
|[[5/3]], [[224/135]]
Line 756: Line 747:
|-
|-
|augmented 6th
|augmented 6th
|20\29, 923.08 (896.55)
| 20\29, 941.382
|D#
|D#
|[[12/7]], [[22/13]]
|[[12/7]], [[22/13]]
Line 762: Line 753:
|-
|-
|minor 7th
|minor 7th
|21\29, 969.23 (941.38)
|21\29, 988.451
|Ef
|Ef
|[[7/4]], [[26/15]]
|[[7/4]], [[26/15]]
Line 768: Line 759:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 7th
|major 7th
|22\29, 1015.385 (982.21)
|22\29, 1035.520
|E
|E
|[[9/5]], [[16/9]], [[20/11]]
|[[9/5]], [[16/9]], [[20/11]]
Line 774: Line 765:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|augmented 7th
|augmented 7th
|23\29, 1061.54 (1031.03)
|23\29, 1082.590
|E#
|E#
|[[11/6]], [[13/7]], [[15/8]], [[24/13]]
|[[11/6]], [[13/7]], [[15/8]], [[24/13]]
Line 780: Line 771:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|doubly augmented 7th, doubly diminished octave
|doubly augmented 7th, doubly diminished octave
|24\29, 1107.69 (1075.86)
|24\29, 1129.659
|Ex, Fff
|Ex, Fff
|[[21/11]], [[25/13]], [[40/21]]
|[[21/11]], [[25/13]], [[40/21]]
Line 786: Line 777:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|diminished octave
|diminished octave
|25\29, 1153.85 (1120.69)
|25\29, 1176.728
|Ff
| Ff
|[[64/33]], [[96/49]], [[35/18]], [[48/25]]
|[[64/33]], [[96/49]], [[35/18]], [[48/25]]
| -3
| -3
|-
|-
|perfect octave
|perfect octave
|26\29, 1200.00 (1165.52)
|26\29, 1223.797
|F
|F
|2/1
|2/1
Line 798: Line 789:
|-
|-
|augmented octave
|augmented octave
|27\29, 1246.15 (1210.345)
|27\29, 1270.866
|F#
|F#
|33/16, 49/24, 72/35, 25/12
|33/16, 49/24, 72/35, 25/12
Line 804: Line 795:
|-
|-
|doubly augmented octave, diminished 9th
|doubly augmented octave, diminished 9th
|28\29, 1292.31 (1255.13)
|28\29, 1317.935
|Fx, Gf
|Fx, Gf
|21/10, 44/21, 52/25
|21/10, 44/21, 52/25
Line 810: Line 801:
|}
|}
===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.  
23ed11/5 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.  
====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.
{| class="wikitable right-2 right-3 sortable"
{| class="wikitable right-2 right-3 sortable"
|-
|-
! class="unsortable" |Degree
! class="unsortable" |Degree
!Size in 23edIX
!Size in 23ed11/5
(superhard)
(superhard)
! class="unsortable" |Note name on G
! class="unsortable" |Note name on G
Line 825: Line 816:
|0\23, 0.00
|0\23, 0.00
|G
|G
|1/1
| 1/1
|0
|0
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|chroma
|chroma
|3\23, 163.63 (169.565)
|3\23, 178.044
|G#
|G#
|12/11, 11/10, 10/9
|12/11, 11/10, 10/9
Line 835: Line 826:
|-
|-
|minor 2nd
|minor 2nd
|1\23, 54.545 (56.52)
| 1\23, 59.348
|Af
|Af
|[[36/35]], [[34/33]], [[33/32]], [[32/31]]
|[[36/35]], [[34/33]], [[33/32]], [[32/31]]
| -5
| -5
|-
|-
|major 2nd
|major 2nd
|4\23, 218.18 (226.09)
| 4\23, 237.392
|A
|A
|[[9/8]], [[17/15]], [[8/7]]
|[[9/8]], [[17/15]], [[8/7]]
Line 847: Line 838:
|-
|-
|aug. 2nd
|aug. 2nd
|7\23, 381.82 (395.65)
|7\23, 415.436
|A#
|A#
|5/4
|5/4
Line 853: Line 844:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|dim. 3rd
|dim. 3rd
|2\23, 109.09 (113.04)
|2\23, 118.696
|Bf
|Bf
|16/15
|16/15
Line 859: Line 850:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|minor 3rd
|minor 3rd
|5\23, 272.73 (282.61)
|5\23, 296.740
|B
|B
|7/6
| 7/6
| -2
| -2
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 3rd
|major 3rd
|8\23, 436.36 (452.17)
|8\23, 474.784
|B#
|B#
|9/7, 14/11
|9/7, 14/11
Line 871: Line 862:
|-
|-
|dim. 4th
|dim. 4th
|6\23, 327.27 (339.13)
|6\23, 356.088
|Cf
|Cf
|6/5
|6/5
| -7
| -7
|-
|-
|nat. 4th
| nat. 4th
|9\23, 490.91 (508.70)
|9\23, 534.132
|C
|C
|4/3
|4/3
Line 883: Line 874:
|-
|-
|aug. 4th
|aug. 4th
|12\23, 654.545 (678.26)
|12\23, 712.176
|C#
|C#
|[[16/11]], [[22/15]]
|[[16/11]], [[22/15]]
Line 889: Line 880:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|double dim. 5th
|double dim. 5th
|7\23, 381.82 (395.65)
|7\23, 415.436
|Qff
|Qff
|5/4
|5/4
Line 895: Line 886:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|dim. 5th
|dim. 5th
|10\23, 545.455 (565.22)
|10\23, 593.480
|Qf
|Qf
|[[15/11]], [[11/8]]
|[[15/11]], [[11/8]]
Line 901: Line 892:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|perf. 5th
|perf. 5th
|13\23, 709.09 (734.78)
|13\23, 771.524
|Q
|Q
|3/2
| 3/2
| +4
| +4
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|aug. 5th
|aug. 5th
|16\23, 872.73 (904.35)
|16\23, 949.568
|Q#
|Q#
|5/3
|5/3
| +12
| +12
|-
|-
|dim. 6th
|dim. 6th
|11\23, 600.00 (621.74)
|11\23, 652.828
|Dff
|Dff
|[[7/5]], [[24/17]], [[17/12]], [[10/7]]
|[[7/5]], [[24/17]], [[17/12]], [[10/7]]
Line 919: Line 910:
|-
|-
|minor 6th
|minor 6th
|14\23, 763.64 (791.30)
|14\23, 830.872
|Df
|Df
|14/9, 11/7
|14/9, 11/7
| -1
| -1
|-
|-
|major 6th
| major 6th
|17\23, 927.27 (960.87)
|17\23, 1008.916
|D
|D
|12/7
| 12/7
| +7
| +7
|-
|-
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|minor 7th
|minor 7th
|15\23, 818.18  (847.83)
|15\23, 890.220
|Ef
|Ef
|8/5
|8/5
Line 938: Line 929:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 7th
|major 7th
|18\23, 981.82 (1017.39)
|18\23, 1068.264
|E
|E
|[[7/4]], [[30/17]], [[16/9]]
|[[7/4]], [[30/17]], [[16/9]]
Line 944: Line 935:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|aug. 7th
|aug. 7th
|21\23, 1145.455 (1186.96)
|21\23, 1246.308
|E#
|E#
|[[31/16]], [[64/33]], [[33/17]], [[35/18]]
|[[31/16]], [[64/33]], [[33/17]], [[35/18]]
| +10
| +10
|-
|-
|dim. octave
| dim. octave
|19\23, 1036.36 (1073.91)
|19\23, 1127.612
|Ff
|Ff
|11/6, 20/11, 9/5
| 11/6, 20/11, 9/5
| -11
| -11
|-
|-
|perf. octave
|perf. octave
|22\23, 1200.00 (1243.48)
|22\23, 1305.656
|F
|F
|2/1
|2/1
| -3
| -3
|-
|-
|aug. octave
| aug. octave
|25\23, 1363.64 (1413.04)
|25\23, 1483.700
|F#
|F#
|24/11, 11/5, 20/9
|24/11, 11/5, 20/9
Line 969: Line 960:
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|dim. 8-step
|dim. 8-step
|20\23, 1090.91 (1130.435)
|20\23, 1186.960
|Gf
|Gf
|15/8
|15/8
| -8
| -8
|}
|}
===Ultrahard===
===Ultrahard===
[[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. 23ed11/5, 28ed11/5 and 33ed11/5 can nominally be viewed as supporting it, but are still very flat and in an ambiguous zone between 18ed11/5 and true Buzzard in terms of harmonies. 38ed11/5 & 43ed11/5 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 48ed11/5 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. 53ed11/5 has an essentially perfect [[7/4]], 58ed11/5 also gives three essentially perfect chains of third-comma meantone, while 63ed11/5 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 83ed11/5, 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 976:
|-
|-
!
!
!38edIX
!38ed11/5
!53edIX
!53ed11/5
!63edIX
!63ed11/5
!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, 538.817
|21\53, 484.615 (515.09)
| 21\53, 540.850
|25\63, 487.87 (515.87)
|25\63, 541.668
|484.07 (515.52)
|484.07
|4/3
|4/3
|-
|-
|L (3g - minor 9th)
|L (3g - 11/5)
|7/38, 227.03 (239,47)
|7/38, 251.448
|10/53, 230.77 (245.28)
|10/53, 257.548
|12/63, 232.26 (247.62)
|12/63, 260.001
|231.51 (246.55)
|231.51
|8/7
|8/7
|-
|-
|s (-5g + 2 minor 9ths)
|s (-5g + 2 11/5s)
|1/38, 32.43 (34.21)
|1/38, 35.921
|1/53, 23.08 (24.53)
|1/53, 25.755
|1/63, 19.355 (20.635)
|1/63, 21.667
|21.05 (22.41)
|21.05
|50/49 81/80 91/90
|50/49 81/80 91/90
|}
|}
Line 1,017: Line 1,008:
|-
|-
!Degree
!Degree
!Size in 38edIX
!Size in 38ed11/5
!Size in 53edIX
!Size in 53ed11/5
!Size in 63edIX
!Size in 63ed11/5
!Size in PNTE tuning
!Size in PNTE tuning
!Note name on G
!Note name on G
Line 1,034: Line 1,025:
|0
|0
|-
|-
|2
| 2
|7/38, 227.03 (239.47)
|7/38, 251.448
|10/53, 230.77 (245.28)
|10/53, 257.548
|12/63, 232.26 (247.62)
|12/63, 260.001
|231.51 (246.55)
|231.51
|A
|A
|8/7
|8/7
| +3
| +3
|-
|-
|3
|3
|14\38, 454.05 (478.95)
|14\38, 502.896
|20\53, 461.54 (490.57)
|20\53, 515.096
|24\63, 464.52 (495.24)
|24\63, 520.002
|463.03 (293.10)
|463.03
|B
|B
|13/10, 21/16
|13/10, 21/16
Line 1,053: Line 1,044:
|-
|-
|4
|4
|15\38, 486.49  (513.16)
|15\38, 538.817
|21\53, 484.615 (515.09)
|21\53, 540.850
|25\63, 483.87 (515.87)
|25\63, 541.668
|484.07 (515.52)
|484.07
|C
|C
|4/3
|4/3
Line 1,062: Line 1,053:
|-
|-
|5
|5
|22\38, 713.51 (752.63)
|22\38, 754.344
|31\53, 715.385 (760.38)
|31\53, 798.399
|37\63, 716.13 (763.49)
|37\63, 801.669
|715.59 (762.07)
|715.59
|Q
|Q
|3/2
|3/2
Line 1,071: Line 1,062:
|-
|-
|6
|6
|29\38, 940.54 (992.105)
|29\38, 1005.793
|41\53, 946.15 (1005.66)
|41\53, 1055.947
|49\63, 948.39 (1011.11)
|49\63, 1061.670
|947.10 (1008.63)
|947.10
|D
|D
|26/15
|26/15
Line 1,080: Line 1,071:
|-
|-
|7
|7
|30\38, 972.97 (1026.32)
| 30\38, 1077.635
|42\53, 969.23 (1030.19)
| 42\53, 1081.701
|50\63, 967.74 (1031.75)
|50\63, 1083.337
|968.15 (1031.03)
|968.15
|E
|E
|7/4
|7/4
| +2
| +2
|-
|-
|8
| 8
|37\38, 1200.00 (1265.57)
|37\38, 1329.083
|52\53, 1200.00 (1275.47)
|52\53, 1339.249
|62\63, 1200.00 (1280.645)
|62\63, 1343.337
|1199.66 (1277.59)
|1199.66
|F
|F
|2/1
|2/1
Line 1,138: Line 1,129:
|}
|}
==Scale tree==
==Scale tree==
{| class="wikitable center-all"
{{MOS tuning spectrum|Scale Signature=5L 3s<11/5>}}
! colspan="12" |Normalized Generator
!Cents
!Relative Cents
!L
!s
!L/s
!Comments
|-
| colspan="3" |3\(8\7)|| || || || ||
|
|
|
| ||514.286
|487.5||1||1||1.000||
|-
|
|
| || || || colspan="3" |17\(45\40)
|
|
|
| ||510
|491.111||6||5||1.200||
|-
|
|
| || || colspan="3" |14\(37\33)||
|
|
|
| ||509.091
|491.892||5||4||1.250||
|-
|
|
| || || || colspan="3" |25\(66\59)
|
|
|
| ||508.475
|492.424||9||7||1.286||
|-
|
|
| || colspan="3" |11\(29\26)|| ||
|
|
|
| ||507.692
|493.103||4||3||1.333||
|-
|
|
| || || || colspan="3" |30\(79\71)
|
|
|
| ||507.042
|493.671||11||8||1.375||
|-
|
|
| || || colspan="3" |19\(50\45)||
|
|
|
| ||506.667
|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)
|
|
|
| ||504.348
|496.053||11||7||1.571||
|-
|
|
| || || colspan="3" |21\(55\50)||
|
|
|
| ||504
|496.364||8||5||1.600||
|-
|
|
| || || || colspan="3" |34\(89\81)
|
|
|
| ||503.704
|496.629||13||8||1.625||Golden Neapolitan-oneirotonic
|-
|
|
| || colspan="3" |13\(34\31)|| ||
|
|
|
| ||503.226
|497.059||5||3||1.667||
|-
|
|
| || || || colspan="3" |31\(81\74)
|
|
|
| ||502.703
|497.531||12||7||1.714||
|-
|
|
| || || colspan="3" |18\(47\43)||
|
|
|
| ||502.326
|497.872||7||4||1.750||
|-
|
|
| || || || colspan="3" |23\(60\55)
|
|
|
| ||501.818
|498.333||9||5||1.800||
|-
|
|
|
|
|
|
| colspan="3" |28\(73/67)
|
|
|
|501.4925
|498.63
|11
|6
|1.833
|
|-
|
|
|
|
|
|
|
| colspan="3" |33\(86\79)
|
|
|501.265
|498.837
|13
|7
|1.857
|
|-
|
|
|
|
|
|
|
|
| colspan="3" |38\(99\91)
|
|501.099
|498.99
|15
|8
|1.875
|
|-
|
|
|
|
|
|
|
|
|
| colspan="3" |43\(112\103)
|500.971
|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)
|499.01
|500.917
|17
|8
|2.125
|
|-
|
|
|
|
|
|
|
|
| colspan="3" |37\(96\89)
|
|498.876
|501.041
|15
|7
|2.143
|
|-
|
|
|
|
|
|
|
| colspan="3" |32\(83\77)
|
|
|498.701
|501.205
|13
|6
|2.167
|
|-
|
|
|
|
|
|
| colspan="3" |27\(70\65)
|
|
|
|498.4615
|501.429
|11
|5
|2.200
|
|-
|
|
| || || || colspan="3" |22\(57\53)
|
|
|
| ||498.113
|501.754||9||4||2.250||
|-
|
|
| || || colspan="3" |17\(44\41)||
|
|
|
| ||497.561
|502.273||7||3||2.333||
|-
|
|
| || || || colspan="3" |29\(75\70)
|
|
|
| ||497.143
|502.667||12||5||2.400||
|-
|
|
| || colspan="3" |12\(31\29)|| ||
|
|
|
| ||496.552
|503.226||5||2||2.500||
|-
|
|
| || || || colspan="3" |31\(80\75)
|
|
|
| ||496
|503.75||13||5||2.600||
|-
|
|
| || || colspan="3" |19\(49\46)||
|
|
|
| ||495.652
|504.082||8||3||2.667||
|-
|
|
| || || || colspan="3" |26\(67\63)
|
|
|
| ||495.238
|504.478||11||4||2.750||
|-
|
|
| colspan="3" |7\(18\17)|| || ||
|
|
|
| ||494.118
|505.556||3||1||3.000||L/s = 3/1
|-
|
|
|
|
|
|
| colspan="3" |30\(77\73)
|
|
|
|493.151
|506.4935
|13
|4
|3.250
|
|-
|
|
| || || || colspan="3" |23\(59\56)
|
|
|
| ||492.857
|506.78||10||3||3.333||
|-
|
|
| || || colspan="3" |16\(41\39)||
|
|
|
| ||492.308
|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)|| ||
|
|
|
| ||490.909
|508.696||4||1||4.000||
|-
|
|
| || || || colspan="3" |20\(51\49)
|
|
|
| ||489.796
|509.804||9||2||4.500||
|-
|
|
| || || colspan="3" |11\(28\27)||
|
|
|
| ||488.889
|510.714||5||1||5.000||
|-
|
|
|
|
|
|
| colspan="3" |24\(61\59)
|
|
|
|488.136
|511.475
|11
|2
|5.500
|
|-
|
|
| || || || colspan="3" |13\(33\32)
|
|
|
| ||487.5
|512.121||6||1||6.000||
|-
| colspan="3" |2\5|| || || || ||
|
|
|
| ||480.000
|520||1||0||→ inf||
|}

Latest revision as of 17:59, 3 March 2025

↖ 4L 2s⟨11/5⟩ ↑ 5L 2s⟨11/5⟩ 6L 2s⟨11/5⟩ ↗
← 4L 3s⟨11/5⟩ 5L 3s (11/5-equivalent) 6L 3s⟨11/5⟩ →
↙ 4L 4s⟨11/5⟩ ↓ 5L 4s⟨11/5⟩ 6L 4s⟨11/5⟩ ↘ 
┌╥╥┬╥╥┬╥┬┐
│║║│║║│║││
││││││││││
└┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern LLsLLsLs
sLsLLsLL
Equave 11/5 (1365.0 ¢)
Period 11/5 (1365.0 ¢)
Generator size(ed11/5)
Bright 3\8 to 2\5 (511.9 ¢ to 546.0 ¢)
Dark 3\5 to 5\8 (819.0 ¢ to 853.1 ¢)
Related MOS scales
Parent 3L 2s⟨11/5⟩
Sister 3L 5s⟨11/5⟩
Daughters 8L 5s⟨11/5⟩, 5L 8s⟨11/5⟩
Neutralized 2L 6s⟨11/5⟩
2-Flought 13L 3s⟨11/5⟩, 5L 11s⟨11/5⟩
Equal tunings(ed11/5)
Equalized (L:s = 1:1) 3\8 (511.9 ¢)
Supersoft (L:s = 4:3) 11\29 (517.8 ¢)
Soft (L:s = 3:2) 8\21 (520.0 ¢)
Semisoft (L:s = 5:3) 13\34 (521.9 ¢)
Basic (L:s = 2:1) 5\13 (525.0 ¢)
Semihard (L:s = 5:2) 12\31 (528.4 ¢)
Hard (L:s = 3:1) 7\18 (530.8 ¢)
Superhard (L:s = 4:1) 9\23 (534.1 ¢)
Collapsed (L:s = 1:0) 2\5 (546.0 ¢)

5L 3s⟨11/5⟩ is a 11/5-equivalent (non-octave) moment of symmetry scale containing 5 large steps and 3 small steps, repeating every interval of 11/5 (1365.0 ¢). Generators that produce this scale range from 511.9 ¢ to 546 ¢, or from 819 ¢ to 853.1 ¢.

13ed11/5 is the smallest ed11/5 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<11/5>[1] 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 13ed11/5 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 18ed11/5 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 21ed11/5 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 and reducing mod 11/5.

Notation (1/1 = G) name In L's and s's # generators up Notation of 2/1 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 13ed11/5) 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 13ed11/5 (basic) Size in 18ed11/5 (hard) Size in 21ed11/5 (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, 105.000 1\18, 75.834 2\21, 130.000 Af -5
major 2nd 2\13, 210.001 3\18, 227.501 3\21, 195.001 A +3
minor 3rd 3\13, 315.001 4\18, 303.334 5\21, 325.001 Bf -2
major 3rd 4\13, 420.001 6\18, 455.001 6\21, 390.001 B +6
diminished 4th 5\18, 379.168 7\21, 455.001 Cf -7
natural 4th 5\13, 525.002 7\18, 530.835 8\21, 520.002 C +1
augmented 4th 6\13, 630.002 9\18, 682.502 9\21, 585.002 C# +9
diminished 5th 8\18, 606.669 10\21, 650.002 Qf -4
perfect 5th 7\13, 735.002 10\18, 758.336 11\21, 715.002 Q +4
minor 6th 8\13, 840.003 11\18, 834.169 13\21, 780.002 Df -1
major 6th 9\13, 945.003 13\18, 985.836 14\21, 845.003 D +7
minor 7th 12\18, 910.003 15\21, 910.003 Ef -6
major 7th 10\13, 1050.003 14\18, 1061.670 16\21, 975.003 E +2
diminished octave 11\13, 1155.004 15\18, 1137.504 18\21, 1105.003 Ff -3
perfect octave 12\13, 1260.004 17\18, 1289.171 19\21, 1170.003 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 13ed11/5, 18ed11/5, and 31ed11/5.

The sizes of the generator, large step and small step of Neapolitan-oneirotonic are as follows in various hypohard Neapolitan-oneiro tunings.

13ed11/5 (basic) 18ed11/5 (hard) 31ed11/5 (semihard)
generator (g) 5\13, 525.002 10\18, 758.336 12\31, 496.55
L (3g - minor 9th) 2\13, 210.001 3\18, 227.501 5\31, 206.87
s (-5g + 2 minor 9ths) 1\13, 105.000 1\18, 75.834 2\31, 82.76

Intervals

Sortable table of major and minor intervals in hypohard Neapolitan-oneiro tunings:

Degree Size in 13ed11/5 (basic) Size in 18ed11/5 (hard) Size in 31ed11/5 (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, 105.000 1\18, 75.834 2\31, 88.065 Af 21/20, 22/21 -5
major 2nd 2\13, 210.001 3\18, 227.501 5\31, 220.162 A 9/8 +3
minor 3rd 3\13, 315.001 4\18, 303.334 7\31, 308.227 Bf 13/11, 33/28 -2
major 3rd 4\13, 420.001 6\18, 455.001 10\31, 440.324 B 14/11, 33/26 +6
diminished 4th 5\18, 379.168 9\31, 396.292 Cf 5/4, 11/9 -7
natural 4th 5\13, 525.002 7\18, 530.835 12\31, 528.389 C 4/3 +1
augmented 4th 6\13, 630.002 9\18, 682.502 15\31, 660.486 C# 10/7, 18/13, 11/8 +9
diminished 5th 8\18, 606.669 14\31, 616.454 Qf 7/5, 13/9, 16/11 -4
perfect 5th 7\13, 735.002 10\18, 758.336 17\31, 748.551 Q 3/2 +4
minor 6th 8\13, 840.003 11\18, 834.169 19\31, 836.615 Df 52/33, 11/7 -1
major 6th 9\13, 945.003 13\18, 985.836 22\31, 968.713 D 56/33, 22/17 +7
minor 7th 12\18, 910.003 21\31, 924.680 Ef 5/3, 18/11 -6
major 7th 10\13, 1050.003 14\18, 1061.670 24\31, 1056.778 E 16/9 +2
diminished octave 11\13, 1155.004 15\18, 1137.5035 26\31, 1144.84 Ff 11/6, 13/7, 15/8 -3
perfect octave 12\13, 1260.004 17\18, 1289.171 29\31, 1276.939 F 2/1 +5
  1. ↑ the rare simplest tuning for a diatonic minor ninth

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 (13edIX not shown).

21ed11/5 (soft) 34ed11/5 (semisoft)
generator (g) 8\21, 520.002 13\34, 521.913
L (3g - minor 9th) 3\21, 195.001 5\34, 200.736
s (-5g + 2 minor 9ths) 2\21, 130.000 3\34, 120.442

Intervals

Sortable table of major and minor intervals in hyposoft tunings (13ed11/5 not shown):

Degree Size in 21ed11/5 (soft) Size in 34ed11/5 (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, 130.000 3\34, 120.442 Af 16/15 -5
major 2nd 3\21, 195.001 5\34, 200.736 A 10/9, 9/8 +3
minor 3rd 5\21, 325.001 8\34, 321.177 Bf 6/5 -2
major 3rd 6\21, 390.001 10\34, 401.472 B 5/4 +6
diminished 4th 7\21, 455.001 11\34, 441.619 Cf 9/7 -7
natural 4th 8\21, 520.002 13\34, 521.913 C 4/3 +1
augmented 4th 9\21, 585.002 15\34, 602.208 C# 7/5 +9
diminished 5th 10\21, 650.002 16\34, 642.355 Qf 10/6 -4
perfect 5th 11\21, 715.002 18\34, 722.649 Q 3/2 +4
minor 6th 13\21, 780.002 21\34, 843.091 Df 8/5 -1
major 6th 14\21, 845.003 23\34, 923.385 D 5/3 +7
minor 7th 15\21, 910.003 24\34, 963.532 Ef 12/7 -6
major 7th 16\21, 975.003 26\34, 1043.826 E 9/5, 16/9 +2
diminished octave 18\21, 1105.003 29\34, 1164.268 Ff 27/14, 48/25 -3
perfect octave 19\21, 1170.003 31\34, 1244.563 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.

29ed11/5 (supersoft) 37ed11/5
generator (g) 11\29, 517.760 14\37, 516.488
L (3g - minor 9th) 4\29, 188.276 5\37, 184.460
s (-5g + 2 minor 9ths) 3\29, 141.207 4\37, 147.568

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 29ed11/5 (supersoft) Note name on G Approximate ratios #Gens up
unison 0\29, 0.00 G 1/1 0
chroma 1\29, 47.069 G# 33/32, 49/48, 36/35, 25/24 +8
diminished 2nd 2\29, 94.138 Aff 21/20, 22/21, 26/25 -13
minor 2nd 3\29, 141.207 Af 12/11, 13/12, 14/13, 16/15 -5
major 2nd 4\29, 188.276 A 9/8, 10/9, 11/10 +3
augmented 2nd 5\29, 235.346 A# 8/7, 15/13 +11
diminished 3rd 6\29, 282.415 Bff 7/6, 13/11, 33/28 -10
minor 3rd 7\29, 329.483 Bf 135/112, 6/5 -2
major 3rd 8\29, 376.553 B 5/4, 11/9, 16/13 +6
augmented 3rd 9\29, 423.622 B# 9/7, 14/11, 33/26 +14
diminished 4th 10\29, 470.691 Cf 21/16, 13/10 -7
natural 4th 11\29, 517.760 C 75/56, 4/3 +1
augmented 4th 12\29, 564.829 C# 11/8, 18/13 +9
doubly augmented 4th, doubly diminished 5th 13\29, 611.898 Cx, Qff 7/5, 10/7 -12
diminished 5th 14\29, 658.968 Qf 16/11, 13/9 -4
perfect 5th 15\29, 706.037 Q 112/75, 3/2 +4
augmented 5th 16\29, 753.106 Q# 32/21, 20/13 +12
diminished 6th 17\29, 800.175 Dff 11/7, 14/9 -9
minor 6th 18\29, 847.244 Df 13/8, 8/5 -1
major 6th 19\29, 894.313 D 5/3, 224/135 +7
augmented 6th 20\29, 941.382 D# 12/7, 22/13 -14
minor 7th 21\29, 988.451 Ef 7/4, 26/15 -6
major 7th 22\29, 1035.520 E 9/5, 16/9, 20/11 +2
augmented 7th 23\29, 1082.590 E# 11/6, 13/7, 15/8, 24/13 +10
doubly augmented 7th, doubly diminished octave 24\29, 1129.659 Ex, Fff 21/11, 25/13, 40/21 -11
diminished octave 25\29, 1176.728 Ff 64/33, 96/49, 35/18, 48/25 -3
perfect octave 26\29, 1223.797 F 2/1 +5
augmented octave 27\29, 1270.866 F# 33/16, 49/24, 72/35, 25/12 +13
doubly augmented octave, diminished 9th 28\29, 1317.935 Fx, Gf 21/10, 44/21, 52/25 -8

Parahard

23ed11/5 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.

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 23ed11/5

(superhard)

Note name on G Approximate ratios (23edIX) #Gens up
unison 0\23, 0.00 G 1/1 0
chroma 3\23, 178.044 G# 12/11, 11/10, 10/9 +8
minor 2nd 1\23, 59.348 Af 36/35, 34/33, 33/32, 32/31 -5
major 2nd 4\23, 237.392 A 9/8, 17/15, 8/7 +3
aug. 2nd 7\23, 415.436 A# 5/4 +11
dim. 3rd 2\23, 118.696 Bf 16/15 -10
minor 3rd 5\23, 296.740 B 7/6 -2
major 3rd 8\23, 474.784 B# 9/7, 14/11 +6
dim. 4th 6\23, 356.088 Cf 6/5 -7
nat. 4th 9\23, 534.132 C 4/3 +1
aug. 4th 12\23, 712.176 C# 16/11, 22/15 +9
double dim. 5th 7\23, 415.436 Qff 5/4 -12
dim. 5th 10\23, 593.480 Qf 15/11, 11/8 -4
perf. 5th 13\23, 771.524 Q 3/2 +4
aug. 5th 16\23, 949.568 Q# 5/3 +12
dim. 6th 11\23, 652.828 Dff 7/5, 24/17, 17/12, 10/7 -9
minor 6th 14\23, 830.872 Df 14/9, 11/7 -1
major 6th 17\23, 1008.916 D 12/7 +7
minor 7th 15\23, 890.220 Ef 8/5 -6
major 7th 18\23, 1068.264 E 7/4, 30/17, 16/9 +2
aug. 7th 21\23, 1246.308 E# 31/16, 64/33, 33/17, 35/18 +10
dim. octave 19\23, 1127.612 Ff 11/6, 20/11, 9/5 -11
perf. octave 22\23, 1305.656 F 2/1 -3
aug. octave 25\23, 1483.700 F# 24/11, 11/5, 20/9 +5
dim. 8-step 20\23, 1186.960 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. 23ed11/5, 28ed11/5 and 33ed11/5 can nominally be viewed as supporting it, but are still very flat and in an ambiguous zone between 18ed11/5 and true Buzzard in terms of harmonies. 38ed11/5 & 43ed11/5 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 48ed11/5 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. 53ed11/5 has an essentially perfect 7/4, 58ed11/5 also gives three essentially perfect chains of third-comma meantone, while 63ed11/5 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 83ed11/5, 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.

38ed11/5 53ed11/5 63ed11/5 Optimal (PNTE) Ultrapyth tuning JI intervals represented (2.3.5.7.13 subgroup)
generator (g) 15\38, 538.817 21\53, 540.850 25\63, 541.668 484.07 4/3
L (3g - 11/5) 7/38, 251.448 10/53, 257.548 12/63, 260.001 231.51 8/7
s (-5g + 2 11/5s) 1/38, 35.921 1/53, 25.755 1/63, 21.667 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 38ed11/5 Size in 53ed11/5 Size in 63ed11/5 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, 251.448 10/53, 257.548 12/63, 260.001 231.51 A 8/7 +3
3 14\38, 502.896 20\53, 515.096 24\63, 520.002 463.03 B 13/10, 21/16 +6
4 15\38, 538.817 21\53, 540.850 25\63, 541.668 484.07 C 4/3 +1
5 22\38, 754.344 31\53, 798.399 37\63, 801.669 715.59 Q 3/2 +4
6 29\38, 1005.793 41\53, 1055.947 49\63, 1061.670 947.10 D 26/15 +7
7 30\38, 1077.635 42\53, 1081.701 50\63, 1083.337 968.15 E 7/4 +2
8 37\38, 1329.083 52\53, 1339.249 62\63, 1343.337 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

Scale tree and tuning spectrum of 5L 3s⟨11/5⟩
Generator(ed11/5) Cents Step ratio Comments
Bright Dark L:s Hardness
3\8 511.877 853.128 1:1 1.000 Equalized 5L 3s⟨11/5⟩
17\45 515.668 849.336 6:5 1.200
14\37 516.488 848.516 5:4 1.250
25\66 517.047 847.957 9:7 1.286
11\29 517.760 847.244 4:3 1.333 Supersoft 5L 3s⟨11/5⟩
30\79 518.356 846.648 11:8 1.375
19\50 518.702 846.303 7:5 1.400
27\71 519.086 845.918 10:7 1.429
8\21 520.002 845.003 3:2 1.500 Soft 5L 3s⟨11/5⟩
29\76 520.857 844.147 11:7 1.571
21\55 521.183 843.821 8:5 1.600
34\89 521.462 843.542 13:8 1.625
13\34 521.913 843.091 5:3 1.667 Semisoft 5L 3s⟨11/5⟩
31\81 522.409 842.595 12:7 1.714
18\47 522.768 842.237 7:4 1.750
23\60 523.252 841.753 9:5 1.800
5\13 525.002 840.003 2:1 2.000 Basic 5L 3s⟨11/5⟩
Scales with tunings softer than this are proper
22\57 526.844 838.160 9:4 2.250
17\44 527.388 837.616 7:3 2.333
29\75 527.802 837.203 12:5 2.400
12\31 528.389 836.615 5:2 2.500 Semihard 5L 3s⟨11/5⟩
31\80 528.939 836.065 13:5 2.600
19\49 529.287 835.717 8:3 2.667
26\67 529.703 835.301 11:4 2.750
7\18 530.835 834.169 3:1 3.000 Hard 5L 3s⟨11/5⟩
23\59 532.120 832.884 10:3 3.333
16\41 532.685 832.320 7:2 3.500
25\64 533.205 831.799 11:3 3.667
9\23 534.132 830.872 4:1 4.000 Superhard 5L 3s⟨11/5⟩
20\51 535.296 829.708 9:2 4.500
11\28 536.252 828.753 5:1 5.000
13\33 537.729 827.275 6:1 6.000
2\5 546.002 819.003 1:0 → ∞ Collapsed 5L 3s⟨11/5⟩
Retrieved from "https://en.xen.wiki/index.php?title=User:Moremajorthanmajor/5L_3s_(11/5-equivalent)&oldid=184226"