7L 3s (8/3-equivalent): Difference between revisions

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| Collapsed = 7
| Collapsed = 7
| Pattern = LLLsLLsLLs
| Pattern = LLLsLLsLLs
|Equave=8/3|Neutral=4L 6s}}'''7L 3s<8/3>''' (sometimes called '''Bolivar''' or '''Choralic''') refers to a non-octave [[MOS scale]] family with a period of an [[8/3]] and which has 7 large and 3 small steps. These scales are the sister of '''[[7L 3s (4/1-equivalent)|diaquadic]]''' with the melodic spacing of [[5L 2s|diatonic scales]]. A [[pathological]] trait these scales exhibit is that normalization to [[edo]] collapses the range for the [[bright]] [[generator]] to the octave.
|Equave=8/3|Neutral=4L 6s}}
==Modes==
{{MOS intro}} A [[pathological]] trait these scales exhibit is that normalization to [[edo]] collapses the range for the [[bright]] [[generator]] to the octave.
The [[mode]]s contain fundamental chords with notes such that they convert a [[wikipedia:Tritone_substitution|tritone substitution]] into a [[Diatonic scale|diatonic]] chord substitution.


* LLLsLLsLLs 9|0 (Lydian ♮11)
== Scale properties ==
* LLsLLLsLLs 8|1 (Major, Ionian)
{{TAMNAMS use}}
* LLsLLsLLLs 7|2 (Mixolydian)
{{MOS data}}
* LLsLLsLLsL 6|3 (Mahur)
* LsLLLsLLsL 5|4 (Dorian)
* LsLLsLLLsL 4|5 (Minor, Aeolian)
* LsLLsLLsLL 3|6 (Aeolian b9)
* sLLLsLLsLL 2|7 (Phrygian)
* sLLsLLLsLL 1|8 (Locrian)
* sLLsLLsLLL 0|9 (Locrian b8)


{{todo|inline=1|clarify|text=What is a diatonic chord substitution?}}
== Scale tree ==
 
{{MOS tuning spectrum}}
==Intervals==
The generator (g) will fall between 480 [[cents]] (2\5 - two degrees of [[5edo]]) and 514 cents (2\5 - two degrees of [[5edo]]), hence a perfect fourth.
 
2g, then, will fall between 960 cents (4\5) and 1029 cents (6\7), the range of minor sevenths.
 
The "large step" will fall between 171 cents (1\7) and 240 cents (1\5), the range of major seconds.
 
The "small step" will fall between 0 cents and 171 cents, sometimes sounding like a [[submajor]] second, and sometimes sounding like a [[quartertone]] or smaller [[microtone]].
 
{| class="wikitable"
!# generators up
!Notation (1/1 = 0)
!name
!In L's and s's
!# generators up
!Notation of 8/3 inverse
!name
!In L's and s's
|-
| colspan="8" style="text-align:center" |The 10-note MOS has the following intervals (from some root):
|-
|0
|0
|perfect unison
|0
|0
|0
|perfect eleventh
|7L+3s
|-
|1
|7
|perfect octave
|5L+2s
| -1
|3
|perfect fourth
|2L+1s
|-
|2
|4
|just fifth
|3L+1s
| -2
|6
|minor seventh
|4L+2s
|-
|3
|1
|major second
|1L
| -3
|9v
|minor tenth
|6L+3s
|-
|4
|8
|major ninth
|6L+2s
| -4
|2v
|minor third
|1L+1s
|-
|5
|5
|major sixth
|4L+1s
| -5
|5v
|minor sixth
|3L+2s
|-
|6
|2
|major third
|2L
| -6
|8v
|minor ninth
|5L+3s
|-
|7
|9
|major tenth
|7L+2s
| -7
|1v
|minor second
|1s
|-
|8
|6^
|major seventh
|5L+1s
| -8
|4v
|diminished fifth
|2L+2s
|-
|9
|3^
|augmented fourth
|3L
| -9
|7v
|diminished octave
|4L+3s
|-
|10
|0^
|augmented unison
|1L-1s
| -10
|0v
|diminished eleventh
|6L+4s
|-
| colspan="8" style="text-align:center" |The chromatic 17-note MOS (either [[7L 10s (eleventh equivalent)|7L 10s]], [[10L 7s (eleventh equivalent)|10L 7s]], or ~[[17ed8/3]]) also has the following intervals (from some root):
|-
|11
|7^
|augmented octave
|6L+1s
| -11
|3v
|diminished fourth
|1L+2s
|-
|12
|4^
|augmented fifth
|4L
| -12
|6v
|diminished seventh
|3L+3s
|-
|13
|1^
|augmented second
|2L-1s
| -13
|9w
|diminished ninth
|5L+4s
|-
|14
|8^
|augmented ninth
|8L+1s
| -14
|2w
|diminished third
|2s
|-
|15
|5^
|augmented sixth
|5L
| -15
|5w
|diminished sixth
|2L+3s
|-
|16
|2^
|augmented third
|3L-1s
| -16
|8w
|diminished ninth
|4L+4s
|}
 
==Scale tree==
The generator range reflects two extremes: one where L = s (3\10), and another where s = 0 (2\7). Between these extremes, there is an infinite continuum of possible generator sizes. By taking freshman sums of the two edges (adding the numerators, then adding the denominators), we can fill in this continuum with compatible ~ed8/3s, increasing in number of tones as we continue filling in the in-betweens. Thus, the smallest in-between ~ed8/3 would be (3+2)\(10+7) = 5\17 – five degrees of ~[[17ed8/3]]:
{{Scale tree|Comments=3/2:L/s = 3/2;2/1:Basic Bolivar (Generators smaller than this are proper);3/1:L/s = 3/1}}The scale produced by stacks of 5\17 is the 12edo diatonic scale.


Other compatible ~ed8/3s include: ~[[37ed8/3]], ~[[27ed8/3]], ~[[44ed8/3]], ~[[41ed8/3]], ~[[24ed8/3]], ~[[31ed8/3]].
Other compatible ~ed8/3s include: ~[[37ed8/3]], ~[[27ed8/3]], ~[[44ed8/3]], ~[[41ed8/3]], ~[[24ed8/3]], ~[[31ed8/3]].
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You can also build this scale by equally dividing frequency ratio 8:3 which is not a member of an edo or stacking frequency ratio 4:3 which is not a member of an equal division of it within it.
You can also build this scale by equally dividing frequency ratio 8:3 which is not a member of an edo or stacking frequency ratio 4:3 which is not a member of an equal division of it within it.


==Rank-2 temperaments==
== Rank-2 temperaments ==
The '''Bolivar''' [[rank-2]] temperament spells its major [[tetrad]] 4:5:6:8 or 14:18:21:28<code>root-3(2g-p)-(2g-p)-(1g)</code> (p = 8/3, g = 2/1) and its minor tetrad 6:7:9:12 or 10:12:15:20 <code>root-2(p-2g)-(2g-p)-(1g)</code> (p = 8/3, g = 2/1). Basic ~17ed8/3 fits both interpretations.
The '''Bolivar''' [[rank-2]] temperament spells its major [[tetrad]] 4:5:6:8 or 14:18:21:28<code>root-3(2g-p)-(2g-p)-(1g)</code> (p = 8/3, g = 2/1) and its minor tetrad 6:7:9:12 or 10:12:15:20 <code>root-2(p-2g)-(2g-p)-(1g)</code> (p = 8/3, g = 2/1). Basic ~17ed8/3 fits both interpretations.


==='''Bolivar-Meantone'''===
=== Bolivar-Meantone ===
[[Subgroup]]: 8/3.2.5/4
[[Subgroup]]: 8/3.2.5/4


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[[Optimal ET sequence]]: ~([[17ed8/3]], [[27ed8/3]], [[44ed8/3]])
[[Optimal ET sequence]]: ~([[17ed8/3]], [[27ed8/3]], [[44ed8/3]])


==='''Bolivar-Archy'''===
=== Bolivar-Archy ===
[[Subgroup]]: 8/3.2.7/6
[[Subgroup]]: 8/3.2.7/6


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[[Optimal ET sequence]]: ~([[17ed8/3]], [[24ed8/3]], [[31ed8/3]], [[38ed8/3]])
[[Optimal ET sequence]]: ~([[17ed8/3]], [[24ed8/3]], [[31ed8/3]], [[38ed8/3]])


==7-note subsets==
== 7-note subsets ==
If you stop the chain at 7 tones, you have a [[heptatonic]] scale of the form [[3L 4s (eleventh equivalent)|3L 4s]]:
If you stop the chain at 7 tones, you have a [[heptatonic]] scale of the form [[3L 4s (eleventh equivalent)|3L 4s]]:


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The large steps here consist of L+s of the 10-tone system, and the small step is the same as L.  
The large steps here consist of L+s of the 10-tone system, and the small step is the same as L.  


==Tetrachordal structure==
== Tetrachordal structure ==
Due to the frequency of perfect fourths and fifths in this scale, it can also be analyzed as a [[tetrachord|tetrachordal scale]].
Due to the frequency of perfect fourths and fifths in this scale, it can also be analyzed as a [[tetrachord|tetrachordal scale]].

Revision as of 18:03, 3 March 2025

↖ 6L 2s⟨8/3⟩ ↑ 7L 2s⟨8/3⟩ 8L 2s⟨8/3⟩ ↗
← 6L 3s⟨8/3⟩ 7L 3s (8/3-equivalent) 8L 3s⟨8/3⟩ →
↙ 6L 4s⟨8/3⟩ ↓ 7L 4s⟨8/3⟩ 8L 4s⟨8/3⟩ ↘
Scale structure
Step pattern LLLsLLsLLs
sLLsLLsLLL
Equave 8/3 (1698.0 ¢)
Period 8/3 (1698.0 ¢)
Generator size(ed8/3)
Bright 7\10 to 5\7 (1188.6 ¢ to 1212.9 ¢)
Dark 2\7 to 3\10 (485.2 ¢ to 509.4 ¢)
Related MOS scales
Parent 3L 4s⟨8/3⟩
Sister 3L 7s⟨8/3⟩
Daughters 10L 7s⟨8/3⟩, 7L 10s⟨8/3⟩
Neutralized 4L 6s⟨8/3⟩
2-Flought 17L 3s⟨8/3⟩, 7L 13s⟨8/3⟩
Equal tunings(ed8/3)
Equalized (L:s = 1:1) 7\10 (1188.6 ¢)
Supersoft (L:s = 4:3) 26\37 (1193.2 ¢)
Soft (L:s = 3:2) 19\27 (1194.9 ¢)
Semisoft (L:s = 5:3) 31\44 (1196.3 ¢)
Basic (L:s = 2:1) 12\17 (1198.6 ¢)
Semihard (L:s = 5:2) 29\41 (1201.1 ¢)
Hard (L:s = 3:1) 17\24 (1202.8 ¢)
Superhard (L:s = 4:1) 22\31 (1205.1 ¢)
Collapsed (L:s = 1:0) 5\7 (1212.9 ¢)
View • Talk • Edit

7L 3s⟨8/3⟩ is a 8/3-equivalent (non-octave) moment of symmetry scale containing 7 large steps and 3 small steps, repeating every interval of 8/3 (1698.0 ¢). Generators that produce this scale range from 1188.6 ¢ to 1212.9 ¢, or from 485.2 ¢ to 509.4 ¢. A pathological trait these scales exhibit is that normalization to edo collapses the range for the bright generator to the octave.

Scale properties

This article uses TAMNAMS conventions for the names of this scale's intervals and scale degrees. The use of 1-indexed ordinal names is reserved for interval regions.
Template:MOS data is deprecated.

Details: Please use the following templates individually: MOS intervals, MOS genchain, and MOS mode degrees

Scale tree

Scale tree and tuning spectrum of 7L 3s⟨8/3⟩
Generator(ed8/3) Cents Step ratio Comments
Bright Dark L:s Hardness
7\10 1188.631 509.413 1:1 1.000 Equalized 7L 3s⟨8/3⟩
40\57 1191.611 506.434 6:5 1.200
33\47 1192.244 505.801 5:4 1.250
59\84 1192.674 505.371 9:7 1.286
26\37 1193.221 504.824 4:3 1.333 Supersoft 7L 3s⟨8/3⟩
71\101 1193.675 504.370 11:8 1.375
45\64 1193.938 504.107 7:5 1.400
64\91 1194.229 503.816 10:7 1.429
19\27 1194.921 503.124 3:2 1.500 Soft 7L 3s⟨8/3⟩
69\98 1195.562 502.483 11:7 1.571
50\71 1195.806 502.239 8:5 1.600
81\115 1196.014 502.031 13:8 1.625
31\44 1196.350 501.695 5:3 1.667 Semisoft 7L 3s⟨8/3⟩
74\105 1196.717 501.328 12:7 1.714
43\61 1196.983 501.062 7:4 1.750
55\78 1197.339 500.706 9:5 1.800
12\17 1198.620 499.425 2:1 2.000 Basic 7L 3s⟨8/3⟩
Scales with tunings softer than this are proper
53\75 1199.952 498.093 9:4 2.250
41\58 1200.342 497.703 7:3 2.333
70\99 1200.638 497.407 12:5 2.400
29\41 1201.056 496.989 5:2 2.500 Semihard 7L 3s⟨8/3⟩
75\106 1201.447 496.598 13:5 2.600
46\65 1201.693 496.352 8:3 2.667
63\89 1201.987 496.058 11:4 2.750
17\24 1202.782 495.263 3:1 3.000 Hard 7L 3s⟨8/3⟩
56\79 1203.677 494.368 10:3 3.333
39\55 1204.068 493.977 7:2 3.500
61\86 1204.427 493.618 11:3 3.667
22\31 1205.064 492.981 4:1 4.000 Superhard 7L 3s⟨8/3⟩
49\69 1205.858 492.187 9:2 4.500
27\38 1206.506 491.539 5:1 5.000
32\45 1207.499 490.546 6:1 6.000
5\7 1212.889 485.156 1:0 → ∞ Collapsed 7L 3s⟨8/3⟩

Other compatible ~ed8/3s include: ~37ed8/3, ~27ed8/3, ~44ed8/3, ~41ed8/3, ~24ed8/3, ~31ed8/3.

You can also build this scale by equally dividing frequency ratio 8:3 which is not a member of an edo or stacking frequency ratio 4:3 which is not a member of an equal division of it within it.

Rank-2 temperaments

The Bolivar rank-2 temperament spells its major tetrad 4:5:6:8 or 14:18:21:28root-3(2g-p)-(2g-p)-(1g) (p = 8/3, g = 2/1) and its minor tetrad 6:7:9:12 or 10:12:15:20 root-2(p-2g)-(2g-p)-(1g) (p = 8/3, g = 2/1). Basic ~17ed8/3 fits both interpretations.

Bolivar-Meantone

Subgroup: 8/3.2.5/4

Comma list: 81/80

POL2 generator: ~2/1 = 1196.3254

Mapping: [⟨1 0 -3], ⟨0 1 6]]

Optimal ET sequence: ~(17ed8/3, 27ed8/3, 44ed8/3)

Bolivar-Archy

Subgroup: 8/3.2.7/6

Comma list: 64/63

POL2 generator: ~2/1 = 1206.6167

Mapping: [⟨1 0 2], ⟨0 1 -4]]

Optimal ET sequence: ~(17ed8/3, 24ed8/3, 31ed8/3, 38ed8/3)

7-note subsets

If you stop the chain at 7 tones, you have a heptatonic scale of the form 3L 4s:

L s s L s L s

The large steps here consist of L+s of the 10-tone system, and the small step is the same as L.

Tetrachordal structure

Due to the frequency of perfect fourths and fifths in this scale, it can also be analyzed as a tetrachordal scale.

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