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{{Infobox MOS
{{Infobox MOS
|Tuning=5L 3s<15/7>}}The minor ninth of a diatonic scale has a '''5L 3s''' [[MOS]] structure with generators ranging from 2\5 (two degrees of [[5ed15/7]] = 527.8¢) to 3\8 (three degrees of [[8ed15/7]] = 494.8¢). In the case of 8edo, L and s are the same size; in the case of 5ed15/7, s becomes so small it disappears (and all that remains are the five equal L's).
|Tuning=5L 3s<15/7>
}}
{{MOS intro|Scale Signature=5L 3s<15/7>}}
 
Any ed15/7 with an interval between 494.8¢ and 527.8¢ has a 5L 3s scale. [[13ed15/7]] is the smallest ed15/7 with a (non-degenerate) 5L 3s scale and thus is the most commonly used 5L 3s tuning.


Any ed15/7 with an interval between 494.8¢ and 527.8¢ has a 5L 3s scale. [[13ed15/7]] is the smallest edIX with a (non-degenerate) 5L 3s scale and thus is the most commonly used 5L 3s tuning.
==Standing assumptions==
==Standing assumptions==
The [[TAMNAMS]] system is used in this article to name 5L 3s<15/7> intervals and step size ratios and step ratio ranges.
The [[TAMNAMS]] system is used in this article to name 5L 3s<15/7> intervals and step size ratios and step ratio ranges.
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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|13ed15/7]] 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'''
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|1L + 2s
|1L + 2s
|-
|-
| colspan="8" |The chromatic 13-note MOS (either [[5L 8s (15/7-equivalent)|5L 8s]], [[8L 5s (15/7-equivalent)|8L 5s]], or [[13edIX|13ed15/7]]) also has the following intervals (from some root):
| 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
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|-
|-
! class="unsortable" |Degree
! class="unsortable" |Degree
!Size in [[13edIX|13ed15/7]] (basic)
!Size in [[13ed15/7]] (basic)
!Size in [[18ed15/7]] (hard)
!Size in [[18ed15/7]] (hard)
!Size in [[21ed15/7]] (soft)
!Size in [[21ed15/7]] (soft)
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|minor 3rd
|minor 3rd
|3\13, 304.487
|3\13, 304.487
|4\18, 293.2095
|4\18, 293.210
|5\21, 314.153
|5\21, 314.153
|Bf
|Bf
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| rowspan="2" |9\13, 913.460
| rowspan="2" |9\13, 913.460
|13\18, 952.931
|13\18, 952.931
|14\21, 879.6285
|14\21, 879.629
|D
|D
|  +7
|  +7
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|minor 7th
|minor 7th
|12\18, 879.6285
|12\18, 879.629
|15\21, 942.459
|15\21, 942.459
|Ef
|Ef
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**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|13ed15/7]], [[18ed15/7]], and 31ed15/7.
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.
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|-
|-
!
!
![[13edIX|13ed15/7]] (basic)
![[13ed15/7]] (basic)
![[18ed15/7]] (hard)
![[18ed15/7]] (hard)
!31ed15/7 (semihard)
!31ed15/7 (semihard)
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|-
|-
! class="unsortable" |Degree
! class="unsortable" |Degree
!Size in [[13edIX|13ed15/7]] (basic)
!Size in [[13ed15/7]] (basic)
!Size in [[18ed15/7]] (hard)
!Size in [[18ed15/7]] (hard)
!Size in 31ed15/7 (semihard)
!Size in 31ed15/7 (semihard)
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|minor 3rd
|minor 3rd
|3\13, 304.487
|3\13, 304.487
|4\18, 293.2095
|4\18, 293.210
|7\31, 297.939
|7\31, 297.939
|Bf
|Bf
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|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|minor 7th
|minor 7th
|12\18, 879.6285
|12\18, 879.629
|21\31, 893.816
|21\31, 893.816
|Ef
|Ef
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|11\13, 1116.452
|11\13, 1116.452
|15\18, 1099.536
|15\18, 1099.536
|26\31, 1106.6295
|26\31, 1106.629
|Ff
|Ff
|11/6, 13/7, 15/8
|11/6, 13/7, 15/8
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|12\13, 1217.942
|12\13, 1217.942
|17\18, 1246.140
|17\18, 1246.140
|29\31, 1234.3175
|29\31, 1234.317
|F
|F
|2/1
|2/1
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===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|13ed15/7]] 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"
|-
|-
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|perfect 5th
|perfect 5th
|11\31, 691.137
|11\31, 691.137
|18\34, 698.5285
|18\34, 698.529
|Q
|Q
|3/2
|3/2
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|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 6th
|major 6th
|14\21, 879.6285
|14\21, 879.629
|23\34, 892.564
|23\34, 892.564
|D
|D
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|minor 7th
|minor 7th
|15\21, 942.459
|15\21, 942.459
|24\34, 931.3713
|24\34, 931.371
|Ef
|Ef
|12/7
|12/7
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|s (-5g + 2 minor 9ths)
|s (-5g + 2 minor 9ths)
|3\29, 136.494
|3\29, 136.494
|4\37, 142.6425
|4\37, 142.642
|}
|}
====Intervals====
====Intervals====
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|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|diminished 6th
|diminished 6th
|17\29, 773.4665
|17\29, 773.466
|Dff
|Dff
|[[11/7]], [[14/9]]
|[[11/7]], [[14/9]]
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|-
|-
|minor 6th
|minor 6th
|18\29, 818.9645
|18\29, 818.965
|Df
|Df
|[[13/8]], [[8/5]]
|[[13/8]], [[8/5]]
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|-
|-
|major 6th
|major 6th
|19\29, 864.4625
|19\29, 864.463
|D
|D
|[[5/3]], [[224/135]]
|[[5/3]], [[224/135]]
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| -8
| -8
|}
|}
===Parahard===
===Parahard===
23ed15/7 Neapolitan-oneiro combines the sound of the 15/7 minor ninth and the [[8/7]] whole tone. This is because 23ed15/7 Neapolitan-oneirotonic has a large step of 229.5¢.  
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¢.  
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|-
|-
|dim. 4th
|dim. 4th
|6\23, 344.2025
|6\23, 344.202
|Cf
|Cf
|6/5
|6/5
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|L (3g - minor 9th)
|L (3g - minor 9th)
|7/38, 243.055
|7/38, 243.055
|10/53, 248.9515
|10/53, 248.951
|12/63, 251.322
|12/63, 251.322
|231.51
|231.51
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|1/38, 34.722
|1/38, 34.722
|1/53, 24.895
|1/53, 24.895
|1/63, 20.9345
|1/63, 20.934
|21.05  
|21.05  
|50/49 81/80 91/90
|50/49 81/80 91/90
Line 1,005: Line 1,009:
|2
|2
|7/38, 243.055
|7/38, 243.055
|10/53, 248.9515
|10/53, 248.951
|12/63, 251.322
|12/63, 251.322
|231.51
|231.51
Line 1,013: Line 1,017:
|-
|-
|3
|3
|14\38, 486.1105
|14\38, 486.111
|20\53, 497.903
|20\53, 497.903
|24\63, 502.645
|24\63, 502.645
Line 1,024: Line 1,028:
|15\38, 520.833
|15\38, 520.833
|21\53, 522.798
|21\53, 522.798
|25\63, 523.58’
|25\63, 523.588
|484.07
|484.07
|C
|C
Line 1,107: Line 1,111:
|}
|}
==Scale tree==
==Scale tree==
{{Scale tree|5L 3s<15/7>}}
{{MOS tuning spectrum|Scale Signature=5L 3s<15/7>}}
Retrieved from "https://en.xen.wiki/w/User:Moremajorthanmajor/5L_3s_(15/7-equivalent)"