View source for User:Moremajorthanmajor/5L 3s (minor ninth-equivalent)

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.
==Standing assumptions==
The [[TAMNAMS]] system is used in this article to name 5L 3s<minor ninth> 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 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:

'''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:

'''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:

'''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 if necessary (so you can use "''k''*''g'' % ''x''" 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*494.12 mod 1270.59 = 2964.71 mod 1270.59 = 423.53¢.
{| class="wikitable center-all"
|-
!
!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
|-
| colspan="8" |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
|-
| 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):
|-
|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:
{| class="wikitable right-2 right-3 right-4 sortable"
|-
! class="unsortable" |Degree
!Size in 13edIX (basic)
!Size in 18edIX (hard)
!Size in 21edIX (soft)
! class="unsortable" |Note name on G
!#Gens up
|- bgcolor="#eaeaff"
|unison
|0\13, 0.00
|0\18, 0.00
|0\21, 0.00
|G
|0
|-
|minor 2nd
|1\13, 100.00
|1\18, 70.59
|2\21, 126.32
|Af
| -5
|-
|major 2nd
|2\13, 200.00
|3\18, 211.76
|3\21, 189.47
|A
| +3
|- bgcolor="#eaeaff"
|minor 3rd
|3\13, 300.00
|4\18, 282.35
|5\21, 315.79
|Bf
| -2
|- bgcolor="#eaeaff"
|major 3rd
| rowspan="2" |4\13, 400.00
|6\18, 423.53
|6\21, 378.95
|B
| +6
|-
|diminished 4th
|5\18, 352.94
|7\21, 442.105
|Cf
| -7
|-
|natural 4th
|5\13, 500.00
|7\18, 494.12
|8\21, 505.24
|C
| +1
|-
|augmented 4th
| rowspan="2" |6\13, 600.00
|9\18, 635.29
|9\21, 568.42
|C#
| +9
|- bgcolor="#eaeaff"
|diminished 5th
|8\18, 564.71
|10\21, 631.58
|Qf
| -4
|- bgcolor="#eaeaff"
|perfect 5th
|7\13, 700.00
|10\18, 705.88
|11\31, 694.74
|Q
| +4
|-
|minor 6th
|8\13, 800.00
|11\18, 776.47
|13\21, 821.05
|Df
| -1
|-
|major 6th
| rowspan="2" |9\13, 900.00
|13\18, 917.65
|14\21, 884.21
|D
| +7
|- bgcolor="#eaeaff"
|minor 7th
|12\18, 847.06
|15\21, 947.37
|Ef
| -6
|- bgcolor="#eaeaff"
|major 7th
|10\13, 1000.00
|14\18, 988.24
|16\21, 1017.53
|E
| +2
|-
|diminished octave
|11\13, 1100.00
|15\18, 1052.82
|18\21, 1136.84
|Ff
| -3
|-
|perfect octave
|12\13, 1200.00
|17\18, 1200.00
|19\21, 1200.00
|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 [[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 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.

The sizes of the generator, large step and small step of Neapolitan-oneirotonic are as follows in various hypohard Neapolitan-oneiro tunings.
{| class="wikitable right-2 right-3 right-4 right-5"
|-
!
![[13edIX]] (basic)
!18edIX (hard)
!31edIX (semihard)
|-
|generator (g)
|5\13, 500.00
|7\18, 494.12
|12\31, 496.55
|-
|L (3g - minor 9th)
|2\13, 200.00
|3\18, 211.765
|5\31, 206.87
|-
|s (-5g + 2 minor 9ths)
|1\13, 100.00
|1\18, 70.59
|2\31, 82.76
|}
====Intervals====
Sortable table of major and minor intervals in hypohard Neapolitan-oneiro tunings:
{| class="wikitable right-2 right-3 right-4 sortable"
|-
! class="unsortable" |Degree
!Size in 13edIX (basic)
!Size in 18edIX (hard)
!Size in 31edIX (semihard)
! class="unsortable" |Note name on G
! class="unsortable" |Approximate ratios<ref>The ratio interpretations that are not valid for 18edo are italicized.</ref>
!#Gens up
|- bgcolor="#eaeaff"
|unison
|0\13, 0.00
|0\18, 0.00
|0\31, 0.00
|G
|1/1
|0
|-
|minor 2nd
|1\13, 100.00
|1\18, 70.59
|2\31, 82.76
|Af
|21/20, ''22/21''
| -5
|-
|major 2nd
|2\13, 200.00
|3\18, 211.76
|5\31, 206.87
|A
|9/8
| +3
|- bgcolor="#eaeaff"
|minor 3rd
|3\13, 300.00
|4\18, 282.35
|7\31, 289.66
|Bf
|13/11, 33/28
| -2
|- bgcolor="#eaeaff"
|major 3rd
| rowspan="2" |4\13, 400.00
|6\18, 423.53
|10\31, 413.79 
|B
|14/11, 33/26
| +6
|-
|diminished 4th
|5\18, 352.94
|9\31, 372.41 
|Cf
|''5/4, 11/9''
| -7
|-
|natural 4th
|5\13, 500.00
|7\18, 494.12
|12\31, 496.55
|C
|4/3
| +1
|- bgcolor="#eaeaff"
|augmented 4th
| rowspan="2" |6\13, 600.00
|9\18, 635.29
|15\31, 620.69 
|C#
|''10/7, 18/13, 11/8''
| +9
|- bgcolor="#eaeaff"
|diminished 5th
|8\18, 564.71
|14\31, 579.31 
|Qf
|''7/5, 13/9'', ''16/11''
| -4
|-
|perfect 5th
|7\13, 700.00
|10\18, 705.88
|17\31, 703.45 
|Q
|3/2
| +4
|-
|minor 6th
|8\13, 800.00
|11\18, 776.47
|19\31, 786.21 
|Df
|52/33, 11/7
| -1
|- bgcolor="#eaeaff"
|major 6th
| rowspan="2" |9\13, 900.00
|13\18, 917.65
|22\31, 910.34
|D
|56/33, 22/17
| +7
|- bgcolor="#eaeaff"
|minor 7th
|12\18, 847.06
|21\31, 868.97 
|Ef
|5/3, 18/11
| -6
|-
|major 7th
|10\13, 1000.00
|14\18, 988.24
|24\31, 993.13 
|E
|16/9
| +2
|-
|diminished octave
|11\13, 1100.00
|15\18, 1052.82
|26\31, 1075.86 
|Ff
|11/6, 13/7, 15/8
| -3
|-
|perfect octave
|12\13, 1200.00
|17\18, 1200.00
|29\31, 1200.00 
|F
|2/1
| +5
|}<references />
===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 [[17edo|12edo]].
*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).
{| class="wikitable right-2 right-3 right-4 right-5"
|-
!
!21edIX (soft)
!34edIX (semisoft)
|-
|generator (g)
|8\21, 505.26
|13\34, 503.23
|-
|L (3g - minor 9th)
|3\21, 189.47
|5\34, 193.55
|-
|s (-5g + 2 minor 9ths)
|2\21, 126.32
|3\34, 116.19
|}
====Intervals====
Sortable table of major and minor intervals in hyposoft tunings (13edIX not shown):
{| class="wikitable right-2 right-3 sortable"
|-
! class="unsortable" |Degree
!Size in 21edIX (soft)
!Size in 34edIX (semisoft)
! class="unsortable" |Note name on G
! class="unsortable" |Approximate ratios
!#Gens up
|- bgcolor="#eaeaff"
|unison
|0\21, 0.00
|0\34, 0.00
|G
|1/1
|0
|-
|minor 2nd
|2\21, 126.32
|3\34, 116.19
|Af
|16/15
| -5
|-
|major 2nd
|3\21, 189.47
|5\34, 193.55 
|A
|10/9, 9/8
| +3
|- bgcolor="#eaeaff"
|minor 3rd
|5\21, 315.79
|8\34, 309.68 
|Bf
|6/5
| -2
|- bgcolor="#eaeaff"
|major 3rd
|6\21, 378.95
|10\34, 387.10 
|B
|5/4
| +6
|-
|diminished 4th
|7\21, 442.105
|11\34, 425.81 
|Cf
|9/7
| -7
|-
|natural 4th
|8\21, 505.24
|13\34, 503.23
|C
|4/3
| +1
|- bgcolor="#eaeaff"
|augmented 4th
|9\21, 568.42
|15\34, 580.645 
|C#
|7/5
| +9
|- bgcolor="#eaeaff"
|diminished 5th
|10\21, 631.58
|16\34, 619.355 
|Qf
|10/6
| -4
|-
|perfect 5th
|11\31, 694.74
|18\34, 696.77 
|Q
|3/2
| +4
|-
|minor 6th
|13\21, 821.05
|21\34, 812.90 
|Df
|8/5
| -1
|- bgcolor="#eaeaff"
|major 6th
|14\21, 884.21
|23\34, 890.32 
|D
|5/3
| +7
|- bgcolor="#eaeaff"
|minor 7th
|15\21, 947.37
|24\34, 929.03 
|Ef
|12/7
| -6
|-
|major 7th
|16\21, 1017.53
|26\34, 1006.45 
|E
|9/5, 16/9
| +2
|-
|diminished octave
|18\21, 1136.84
|29\34, 1122.58 
|Ff
|27/14, 48/25
| -3
|-
|perfect octave
|19\21, 1200.00
|31\34, 1200.00
|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 [[Meantone family|flattone]] temperament.

The sizes of the generator, large step and small step of Neapolitan-oneirotonic are as follows in various tunings in this range.
{| class="wikitable right-2 right-3 right-4 right-5"
|-
!
!29edIX (supersoft)
!37edIX
|-
|generator (g)
|11\29, 507.69
|14\37, 509.09
|-
|L (3g - minor 9th)
|4\29, 184.615
|5\37, 181.82
|-
|s (-5g + 2 minor 9ths)
|3\29, 138.46
|4\37, 145.455
|}
====Intervals====
The intervals of the extended generator chain (-15 to +15 generators) are as follows in various softer-than-soft Neapolitan-oneirotonic tunings.
{| class="wikitable right-2 right-3 sortable"
|-
! class="unsortable" |Degree
!Size in 29edIX (supersoft)
! class="unsortable" |Note name on G
! class="unsortable" |Approximate ratios
!#Gens up
|- bgcolor="#eaeaff"
|unison
|0\29, 0.00
|G
|1/1
|0
|- bgcolor="#eaeaff"
|chroma
|1\29, 46.15
|G#
|[[33/32]], [[49/48]], [[36/35]], [[25/24]]
| +8
|-
|diminished 2nd
|2\29, 92.31
|Aff
|[[21/20]], [[22/21]], [[26/25]]
| -13
|-
|minor 2nd
|3\29, 138.46
|Af
|[[12/11]], [[13/12]], [[14/13]], [[16/15]]
| -5
|-
|major 2nd
|4\29, 184.615
|A
|[[9/8]], [[10/9]], [[11/10]]
| +3
|-
|augmented 2nd
|5\29, 230.77
|A#
|[[8/7]], [[15/13]]
| +11
|- bgcolor="#eaeaff"
|diminished 3rd
|6\29, 276.92
|Bff
|[[7/6]], [[13/11]], [[33/28]]
| -10
|- bgcolor="#eaeaff"
|minor 3rd
|7\29, 323.08
|Bf
|[[135/112]], [[6/5]]
| -2
|- bgcolor="#eaeaff"
|major 3rd
|8\29, 369.23
|B
|[[5/4]], [[11/9]], [[16/13]]
| +6
|- bgcolor="#eaeaff"
|augmented 3rd
|9\29, 415.385
|B#
|[[9/7]], [[14/11]], [[33/26]]
| +14
|-
|diminished 4th
|10\29, 461.54
|Cf
|[[21/16]], [[13/10]]
| -7
|-
|natural 4th
|11\29, 507.69
|C
|[[75/56]], [[4/3]]
| +1
|-
|augmented 4th
|12\29, 553.85
|C#
|[[11/8]], [[18/13]]
| +9
|-
|doubly augmented 4th, doubly diminished 5th
|13\29, 600.00
|Cx, Qff
|[[7/5]], [[10/7]]
| -12
|- bgcolor="#eaeaff"
|diminished 5th
|14\29, 646.15
|Qf
|[[16/11]], [[13/9]]
| -4
|- bgcolor="#eaeaff"
|perfect 5th
|15\29, 692.31
|Q
|[[112/75]], [[3/2]]
| +4
|- bgcolor="#eaeaff"
|augmented 5th
|16\29, 738.46
|Q#
|[[32/21]], [[20/13]]
| +12
|- bgcolor="#eaeaff"
|diminished 6th
|17\29, 784.615
|Dff
|[[11/7]], [[14/9]]
| -9
|-
|minor 6th
|18\29, 830.77
|Df
|[[13/8]], [[8/5]]
| -1
|-
|major 6th
|19\29, 876.92
|D
|[[5/3]], [[224/135]]
| +7
|-
|augmented 6th
|20\29, 923.08
|D#
|[[12/7]], [[22/13]]
| -14
|-
|minor 7th
|21\29, 969.23
|Ef
|[[7/4]], [[26/15]]
| -6
|- bgcolor="#eaeaff"
|major 7th
|22\29, 1015.385
|E
|[[9/5]], [[16/9]], [[20/11]]
| +2
|- bgcolor="#eaeaff"
|augmented 7th
|23\29, 1061.54
|E#
|[[11/6]], [[13/7]], [[15/8]], [[24/13]]
| +10
|- bgcolor="#eaeaff"
|doubly augmented 7th, doubly diminished octave
|24\29, 1107.69
|Ex, Fff
|[[21/11]], [[25/13]], [[40/21]]
|<nowiki>-11</nowiki>
|- bgcolor="#eaeaff"
|diminished octave
|25\29, 1153.85
|Ff
|[[64/33]], [[96/49]], [[35/18]], [[48/25]]
| -3
|-
|perfect octave
|26\29, 1200.00
|F
|2/1
| +5
|-
|augmented octave
|27\29, 1246.15
|F#
|33/16, 49/24, 72/35, 25/12
| +13
|-
|doubly augmented octave, diminished 9th
|28\29, 1292.31
|Fx, Gf
|21/10, 44/21, 52/25
| -8
|}
===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.
====Intervals====
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="unsortable" |Degree
!Size in 23edIX
(superhard)
! class="unsortable" |Note name on G
! class="unsortable" |Approximate ratios (23edIX)
!#Gens up
|- bgcolor="#eaeaff"
|unison
|0\23, 0.00
|G
|1/1
|0
|- bgcolor="#eaeaff"
|chroma
|3\23, 163.63
|G#
|12/11, 11/10, 10/9
| +8
|-
|minor 2nd
|1\23, 54.545 
|Af
|[[36/35]], [[34/33]], [[33/32]], [[32/31]]
| -5
|-
|major 2nd
|4\23, 218.18 
|A
|[[9/8]], [[17/15]], [[8/7]]
| +3
|-
|aug. 2nd
|7\23, 381.82
|A#
|5/4
| +11
|- bgcolor="#eaeaff"
|dim. 3rd
|2\23, 109.09
|Bf
|16/15
| -10
|- bgcolor="#eaeaff"
|minor 3rd
|5\23, 272.73
|B
|7/6
| -2
|- bgcolor="#eaeaff"
|major 3rd
|8\23, 436.36
|B#
|9/7, 14/11
| +6
|-
|dim. 4th
|6\23, 327.27
|Cf
|6/5
| -7
|-
|nat. 4th
|9\23, 490.91
|C
|4/3
| +1
|-
|aug. 4th
|12\23, 654.545
|C#
|[[16/11]], [[22/15]]
| +9
|- bgcolor="#eaeaff"
|double dim. 5th
|7\23, 381.82
|Qff
|5/4
| -12
|- bgcolor="#eaeaff"
|dim. 5th
|10\23, 545.455
|Qf
|[[15/11]], [[11/8]]
| -4
|- bgcolor="#eaeaff"
|perf. 5th
|13\23, 709.09
|Q
|3/2
| +4
|- bgcolor="#eaeaff"
|aug. 5th
|16\23, 872.73
|Q#
|5/3
| +12
|-
|dim. 6th
|11\23, 600.00
|Dff
|[[7/5]], [[24/17]], [[17/12]], [[10/7]]
| -9
|-
|minor 6th
|14\23, 763.64
|Df
|14/9, 11/7
| -1
|-
|major 6th
|17\23, 927.27
|D
|12/7
| +7
|-
|- bgcolor="#eaeaff"
|minor 7th
|15\23, 818.18
|Ef
|8/5
| -6
|- bgcolor="#eaeaff"
|major 7th
|18\23, 981.82 
|E
|[[7/4]], [[30/17]], [[16/9]]
| +2
|- bgcolor="#eaeaff"
|aug. 7th
|21\23, 1145.455
|E#
|[[31/16]], [[64/33]], [[33/17]], [[35/18]]
| +10
|-
|dim. octave
|19\23, 1036.36
|Ff
|11/6, 20/11, 9/5
| -11
|-
|perf. octave
|22\23, 1200.00
|F
|2/1
| -3
|-
|aug. octave
|25\23, 1363.64
|F#
|24/11, 11/5, 20/9
| +5
|-
|- bgcolor="#eaeaff"
|dim. ninth
|20\23, 1090.91
|J@
|15/8
| -8
|}
===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.

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.

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.

The sizes of the generator, large step and small step of Neapolitan-oneirotonic are as follows in various ultrapyth tunings.
{| class="wikitable right-2 right-3 right-4 right-5"
|-
!
!38edIX
!53edIX
!63edIX
!Optimal ([[POTE|PNTE]]) Ultrapyth tuning
!JI intervals represented (2.3.5.7.13 subgroup)
|-
|generator (g)
|15\38, 486.49  
|21\53, 484.615 
|25\63, 483.87 
|484.07
|4/3
|-
|L (3g - minor 9th)
|7/38, 227.03 
|10/53, 230.77 
|12/63, 232.26 
|231.51
|8/7
|-
|s (-5g + 2 minor 9ths)
|1/38, 32.43 
|1/53, 23.08 
|1/63, 19.355 
|21.05
|50/49 81/80 91/90
|}
====Intervals====
Sortable table of intervals in the Neapolitan-Dylathian mode and their Ultrapyth interpretations:
{| class="wikitable right-2 right-3 right-4 right-5 sortable"
|-
!Degree
!Size in 38edIX
!Size in 53edIX
!Size in 63edIX
!Size in PNTE tuning
!Note name on G
! class="unsortable" |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, 227.03 
|10/53, 230.77 
|12/63, 232.26 
|231.51 
|A
|8/7
| +3
|-
|3
|14\38, 454.05 
|20\53, 461.54 
|24\63, 464.52 
|463.03
|B
|13/10, 21/16
| +6
|-
|4
|15\38, 486.49  
|21\53, 484.615 
|25\63, 483.87 
|484.07
|C
|4/3
| +1
|-
|5
|22\38, 713.51 
|31\53, 715.385 
|37\63, 716.13 
|715.59
|Q
|3/2
| +4
|-
|6
|29\38, 940.54 
|41\53, 946.15 
|49\63, 948.39 
|947.10
|D
|26/15
| +7
|-
|7
|30\38, 972.97 
|42\53, 969.23 
|50\63, 967.74 
|968.15
|E
|7/4
| +2
|-
|8
|37\38, 1200.00 
|52\53, 1200.00 
|62\63, 1200.00 
|1199.66
|F
|2/1
| +5
|}
==Modes==
Neapolitan-Oneirotonic modes are named after cities in the Dreamlands.
{| class="wikitable"
|-
|'''Mode'''
|[[Modal UDP Notation|'''UDP''']]
|'''Name'''
|-
| |LLsLLsLs
|<nowiki>7|0</nowiki>
| |Neapolitan-Dylathian (də-LA(H)TH-iən)
|-
| |LLsLsLLs
|<nowiki>6|1</nowiki>
| |Neapolitan-Illarnekian (ill-ar-NEK-iən)
|-
| |LsLLsLLs
|<nowiki>5|2</nowiki>
| |Neapolitan-Celephaïsian (kel-ə-FAY-zhən)
|-
| |LsLLsLsL
|<nowiki>4|3</nowiki>
| |Neapolitan-Ultharian (ul-THA(I)R-iən)
|-
| |LsLsLLsL
|<nowiki>3|4</nowiki>
| |Neapolitan-Mnarian (mə-NA(I)R-iən)
|-
| |sLLsLLsL
|<nowiki>2|5</nowiki>
| |Neapolitan-Kadathian (kə-DA(H)TH-iən)
|-
| |sLLsLsLL
|<nowiki>1|6</nowiki>
| |Neapolitan-Hlanithian (lə-NITH-iən)
|-
| |sLsLLsLL
|<nowiki>0|7</nowiki>
| |Neapolitan-Sarnathian (sar-NA(H)TH-iən), can be shortened to "Sarn"
|}
==Scale tree==
{| class="wikitable center-all"
! colspan="3" |Normalized Generator
!Cents
!L
!s
!L/s
!Comments
|-
| colspan="3" |3\8||514.286||1||1||1.000||
|-
| colspan="3" |17\45||510.000||6||5||1.200||
|-
| colspan="3" |14\37||509.091||5||4||1.250||
|-
| colspan="3" |25\66||508.475||9||7||1.286||
|-
| colspan="3" |11\29||507.692||4||3||1.333||
|-
| colspan="3" |19\50||506.667||7||5||1.400||
|-
| colspan="3" |8\21||505.263||3||2||1.500||L/s = 3/2
|-
| colspan="3" |29\76||504.348||11||7||1.571||
|-
| colspan="3" |21\55||504.000||8||5||1.600||
|-
| colspan="3" |34\89||503.704||13||8||1.625||Golden Neapolitan-oneirotonic
|-
| colspan="3" |13\34||503.226||5||3||1.667||
|-
| colspan="3" |31\81||502.703||12||7||1.714||
|-
| colspan="3" |18\47||502.326||7||4||1.750||
|-
| colspan="3" |23\60||501.818||9||5||1.800||
|-
| colspan="3" |28\73
|501.493
|11
|6
|1.833
|
|-
| colspan="3" |33\86
|501.265
|13
|7
|1.857
|
|-
| colspan="3" |38\99
|501.099
|15
|8
|1.875
|
|-
| colspan="3" |43\112
|500.971
|17
|9
|1.889
|
|-
| colspan="3" |5\13||500||2||1||2.000||Basic Neapolitan-oneirotonic
(generators smaller than this are proper)
|-
| colspan="3" |42\109
|499.010
|17
|8
|2.125
|
|-
| colspan="3" |37\96
|498.876
|15
|7
|2.143
|
|-
| colspan="3" |32\83
|498.701
|13
|6
|2.167
|
|-
| colspan="3" |27\70
|498.462
|11
|5
|2.200
|
|-
| colspan="3" |22\57||498.113||9||4||2.250||
|-
| colspan="3" |17\44||497.561||7||3||2.333||
|-
| colspan="3" |29\75||497.143||12||5||2.400||
|-
| colspan="3" |12\31||496.552||5||2||2.500||
|-
| colspan="3" |31\80||496.000||13||5||2.600||
|-
| colspan="3" |19\49||495.652||8||3||2.667||
|-
| colspan="3" |26\67||495.238||11||4||2.750||
|-
| colspan="3" |7\18||494.118||3||1||3.000||L/s = 3/1
|-
| colspan="3" |30\77
|493.151
|13
|4
|3.250
|
|-
| colspan="3" |23\59||492.857||10||3||3.333||
|-
| colspan="3" |16\41||492.308||7||2||3.500||
|-
| colspan="3" |25\64||491.803||11||3||3.667||
|-
| colspan="3" |9\23||490.909||4||1||4.000||
|-
| colspan="3" |20\51||489.796||9||2||4.500||
|-
| colspan="3" |11\28||488.889||5||1||5.000||
|-
| colspan="3" |24\61
|488.136
|11
|2
|5.500
|
|-
| colspan="3" |13\33||487.500||6||1||6.000||
|-
| colspan="3" |2\5||480.000||1||0||→ inf||
|}

== See also ==
[[5L 3s (45/22-equivalent)]] - undecimal small diesis tuning 

[[5L 3s (33/16-equivalent)]] - harmonic subminor ninth tuning 

[[5L 3s (56/27-equivalent)]] - Archytas diatonic minor ninth tuning 

[[5L 3s (25/12-equivalent)]] - classical chromatic minor ninth tuning 

[[5L 3s (44/21-equivalent)]], [[5L 3s (208/99-equivalent)]] - Neogothic undecimal diatonic minor ninth tuning 

[[5L 3s (21/10-equivalent)]] - septimal chromatic minor ninth tuning

[[5L 3s (32/15-equivalent)]] - classical diatonic minor ninth tuning

[[5L 3s (891/416-equivalent)]], [[5L 3s (189/88-equivalent)]] - Neogothic chromatic minor ninth tuning

[[5L 3s (15/7-equivalent)]] - septimal diatonic minor ninth tuning

[[5L 3s (243/112-equivalent)]] - Archytas chromatic minor ninth tuning

[[5L 3s (11/5-equivalent)]] - undecimal neutral ninth tuning