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

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
CompactStar (talk | contribs)
autopatrolled
5,084 edits
No edit summary
ArrowHead294 (talk | contribs)
15,213 edits
 
(10 intermediate revisions by 5 users not shown)
Line 1: Line 1:
{{Infobox MOS
{{Infobox MOS}}
| Name =
{{MOS intro}} The [[bright]] [[generator]] of these scales is close to an octave.
| Periods = 1
| nLargeSteps = 7
| nSmallSteps = 3
| Equalized = 10
| Collapsed = 7
| 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.
==Modes==
The modes contain fundamental chords with notes such that they convert a [[wikipedia:Tritone_substitution|tritone substitution]] into a diatonic chord substitution.


* LLLsLLsLLs 9|0 (Lydian ♮11)
== Scale properties ==
* LLsLLLsLLs 8|1 (Major, Ionian)
{{TAMNAMS use}}
* LLsLLsLLLs 7|2 (Mixolydian)
* 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)


==Intervals==
=== 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.
{{MOS intervals}}


2g, then, will fall between 960 cents (4\5) and 1029 cents (6\7), the range of minor sevenths.
=== Generator chain ===
{{MOS genchain}}


The "large step" will fall between 171 cents (1\7) and 240 cents (1\5), the range of major seconds.
=== Modes ===
{{MOS mode degrees}}


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.
== Scale tree ==
{| class="wikitable"
{{MOS tuning spectrum}}
!# 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]]:
{| class="wikitable center-all"
! colspan="8" rowspan="2" |Generator
!Cents
! colspan="2" |''ed17\12''
! rowspan="2" |L
! rowspan="2" |s
! rowspan="2" |L/s
! rowspan="2" |Comments
|-
!Normalized
<ref name=":0">Fractions with repeat period 2 or longer in minutes and seconds</ref>
!''Chroma-positive<ref name=":0" />''
!''Chroma-negative<ref name=":0" />''
|-
|7\10|| || || || ||
|
|
|514¢17’8”||''1190°''||''510°''||1||1||1.000||
|-
| || || || || ||40\57
|
|
|510||''1192°58’57”''||''507°1’3”''||6||5||1.200||
|-
| || || || ||33\47||
|
|
|509¢5’27”||''1193°37’1”''||''506°22'59”''||5||4||1.250||
|-
| || || || || ||59\84
|
|
|508¢28'34”||''1194°2’51”''||''505°58’9”''||9||7||1.286||
|-
| || || ||26\37|| ||
|
|
|507¢55’23”||''1194°34’3”''||''505°25’56”''||4||3||1.333||
|-
| || || || || ||71\101
|
|
|507¢2’32”||''1195°2’23”''||''504°57’37”''||11||8||1.375||
|-
| || || || ||45\64||
|
|
|506.{{Overline|6}}||''1195°18’45”''||''504°41’15”''||7||5||1.400||
|-
| || ||19\27|| || ||
|
|
|505¢15’47”||''1196°17’47”''||''503°42’13’''||3||2||1.500||L/s = 3/2
|-
| || || || ||50\71||
|
|
|504||''1197°11‘50”''||''502°48’10”''||8||5||1.600||
|-
| || || ||31\44|| ||
|
|
|503¢13’33”||''1197°43’38”''||''502°16’22”''||5||3||1.667||
|-
| || || || ||43\61||
|
|
|502¢18’8”||''1198°21’38”''||''501°39’22”''||7||4||1.750||
|-
| || || || || ||55\78
|
|
|501¢49’5”||''1198°43’4”''||''501°16’56”''||9||5||1.800||
|-
|
|
|
|
|
|
|67\95
|
|501¢29’33”
|''1198°56’51”''
|''501°3’9”''
|11
|6
|1.833
|
|-
|
|
|
|
|
|
|
|79\112
|501¢15’57”
|''1199°6’26”''
|''500°53’34”''
|13
|7
|1.857
|
|-
| ||12\17|| || || ||
|
|
|500||''1200°''||''500°''||2||1||2.000||Basic Bolivar
(Generators smaller than this are proper)
|-
|
|
|
|
|
|
|
|77\109
|498¢42’5”
|''1200°54’30”''
|''499°5’30”''
|13
|6
|2.167
|
|-
|
|
|
|
|
|
|65\92
|
|498¢21’42”
|''1201°5’13”''
|''498°54’47”''
|11
|5
|2.200
|
|-
| || || || || ||53\75
|
|
|498¢6’48”||''1201.{{Overline|3}}''||''498.{{Overline|6}}''||9||4||2.250||
|-
| || || || ||41\58||
|
|
|497¢33’39”||''1201°43’27”''||''498°16’33”''||7||3||2.333||
|-
| || || || || ||70\99
|
|
|497¢8’34”||''1202°1’13”''||''497°58’47”''||12||5||2.400||
|-
| || || ||29\41|| ||
|
|
|496¢33’6”||''1202°26’21”''||''497°33’39”''||5||2||2.500||
|-
| || || || ||46\65||
|
|
|495¢39’8”||''1203°50’46”''||''496°9’14”''||8||3||2.667||
|-
| || ||17\24|| || ||
|
|
|494¢7’4”||''1204.1{{Overline|6}}''||''495.8{{Overline|3}}''||3||1||3.000||L/s = 3/1
|-
|
|
|
|
|
|
|73\103
|
|493¢9’2”
|''1204°51’16”''
|''495°8’44”''
|13
|4
|3.250
|
|-
| || || || || ||56\79
|
|
|492¢51’26”||''1205°3’48”''||''494°56’12”''||10||3||3.333||
|-
| || || || ||39\55||
|
|
|492¢18’28”||''1205°27’16”''||''494°32’44”''||7||2||3.500||
|-
| || || || || ||61\86
|
|
|491¢48’12”||''1205°48’50”''||''494°11’10”''||11||3||3.667||
|-
| || || ||22\31|| ||
|
|
| 490¢54’33”||''1206°27’6”''||''493°32’54”''||4||1||4.000||
|-
| || || || || ||49\69
|
|
| 489¢47’45”||''1207°14’47”''||''492°45’13”''||9||2||4.500||
|-
| || || || ||27\38||
|
|
| 488.{{Overline|8}}||''1207°53’41”''||''492°6’19”''||5||1||5.000||
|-
| || || || || ||32\45
|
|
| 487.5||''1208.{{Overline|8}}''||''491.{{Overline|1}}''||6||1||6.000||
|-
|5\7|| || || || ||
|
|
| 480||''1214°17’8”''||''485°42’52’''||1||0||→ inf||
|}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]].


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==
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'''===
== 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> ({{nowrap|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> ({{nowrap|p {{=}} 8/3|g {{=}} 2/1}}). Basic ~17ed8/3 fits both interpretations.
 
=== Bolivar–Meantone ===
[[Subgroup]]: 8/3.2.5/4
[[Subgroup]]: 8/3.2.5/4


Line 445: Line 33:
[[Mapping]]: [{{val|1 0 -3}}, {{val|0 1 6}}]
[[Mapping]]: [{{val|1 0 -3}}, {{val|0 1 6}}]


[[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


Line 455: Line 44:
[[Mapping]]:  [{{val|1 0 2}}, {{val|0 1 -4}}]
[[Mapping]]:  [{{val|1 0 2}}, {{val|0 1 -4}}]


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


L s s L s L s
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.  
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]].

Latest revision as of 14:23, 5 May 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 ¢. The bright generator of these scales is close to an 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.

Intervals

Intervals of 7L 3s⟨8/3⟩
Intervals Steps
subtended 		Range in cents
Generic Specific Abbrev.
0-mosstep Perfect 0-mosstep P0ms 0 0.0 ¢
1-mosstep Minor 1-mosstep m1ms s 0.0 ¢ to 169.8 ¢
Major 1-mosstep M1ms L 169.8 ¢ to 242.6 ¢
2-mosstep Minor 2-mosstep m2ms L + s 242.6 ¢ to 339.6 ¢
Major 2-mosstep M2ms 2L 339.6 ¢ to 485.2 ¢
3-mosstep Perfect 3-mosstep P3ms 2L + s 485.2 ¢ to 509.4 ¢
Augmented 3-mosstep A3ms 3L 509.4 ¢ to 727.7 ¢
4-mosstep Minor 4-mosstep m4ms 2L + 2s 485.2 ¢ to 679.2 ¢
Major 4-mosstep M4ms 3L + s 679.2 ¢ to 727.7 ¢
5-mosstep Minor 5-mosstep m5ms 3L + 2s 727.7 ¢ to 849.0 ¢
Major 5-mosstep M5ms 4L + s 849.0 ¢ to 970.3 ¢
6-mosstep Minor 6-mosstep m6ms 4L + 2s 970.3 ¢ to 1018.8 ¢
Major 6-mosstep M6ms 5L + s 1018.8 ¢ to 1212.9 ¢
7-mosstep Diminished 7-mosstep d7ms 4L + 3s 970.3 ¢ to 1188.6 ¢
Perfect 7-mosstep P7ms 5L + 2s 1188.6 ¢ to 1212.9 ¢
8-mosstep Minor 8-mosstep m8ms 5L + 3s 1212.9 ¢ to 1358.4 ¢
Major 8-mosstep M8ms 6L + 2s 1358.4 ¢ to 1455.5 ¢
9-mosstep Minor 9-mosstep m9ms 6L + 3s 1455.5 ¢ to 1528.2 ¢
Major 9-mosstep M9ms 7L + 2s 1528.2 ¢ to 1698.0 ¢
10-mosstep Perfect 10-mosstep P10ms 7L + 3s 1698.0 ¢

Generator chain

Generator chain of 7L 3s⟨8/3⟩
Bright gens Scale degree Abbrev.
16 Augmented 2-mosdegree A2md
15 Augmented 5-mosdegree A5md
14 Augmented 8-mosdegree A8md
13 Augmented 1-mosdegree A1md
12 Augmented 4-mosdegree A4md
11 Augmented 7-mosdegree A7md
10 Augmented 0-mosdegree A0md
9 Augmented 3-mosdegree A3md
8 Major 6-mosdegree M6md
7 Major 9-mosdegree M9md
6 Major 2-mosdegree M2md
5 Major 5-mosdegree M5md
4 Major 8-mosdegree M8md
3 Major 1-mosdegree M1md
2 Major 4-mosdegree M4md
1 Perfect 7-mosdegree P7md
0 Perfect 0-mosdegree
Perfect 10-mosdegree		 P0md
P10md
−1 Perfect 3-mosdegree P3md
−2 Minor 6-mosdegree m6md
−3 Minor 9-mosdegree m9md
−4 Minor 2-mosdegree m2md
−5 Minor 5-mosdegree m5md
−6 Minor 8-mosdegree m8md
−7 Minor 1-mosdegree m1md
−8 Minor 4-mosdegree m4md
−9 Diminished 7-mosdegree d7md
−10 Diminished 10-mosdegree d10md
−11 Diminished 3-mosdegree d3md
−12 Diminished 6-mosdegree d6md
−13 Diminished 9-mosdegree d9md
−14 Diminished 2-mosdegree d2md
−15 Diminished 5-mosdegree d5md
−16 Diminished 8-mosdegree d8md

Modes

Scale degrees of the modes of 7L 3s⟨8/3⟩
UDP Cyclic
order 		Step
pattern 		Scale degree (mosdegree)
0 1 2 3 4 5 6 7 8 9 10
9|0 1 LLLsLLsLLs Perf. Maj. Maj. Aug. Maj. Maj. Maj. Perf. Maj. Maj. Perf.
8|1 8 LLsLLLsLLs Perf. Maj. Maj. Perf. Maj. Maj. Maj. Perf. Maj. Maj. Perf.
7|2 5 LLsLLsLLLs Perf. Maj. Maj. Perf. Maj. Maj. Min. Perf. Maj. Maj. Perf.
6|3 2 LLsLLsLLsL Perf. Maj. Maj. Perf. Maj. Maj. Min. Perf. Maj. Min. Perf.
5|4 9 LsLLLsLLsL Perf. Maj. Min. Perf. Maj. Maj. Min. Perf. Maj. Min. Perf.
4|5 6 LsLLsLLLsL Perf. Maj. Min. Perf. Maj. Min. Min. Perf. Maj. Min. Perf.
3|6 3 LsLLsLLsLL Perf. Maj. Min. Perf. Maj. Min. Min. Perf. Min. Min. Perf.
2|7 10 sLLLsLLsLL Perf. Min. Min. Perf. Maj. Min. Min. Perf. Min. Min. Perf.
1|8 7 sLLsLLLsLL Perf. Min. Min. Perf. Min. Min. Min. Perf. Min. Min. Perf.
0|9 4 sLLsLLsLLL Perf. Min. Min. Perf. Min. Min. Min. Dim. Min. Min. Perf.

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.

Retrieved from "https://en.xen.wiki/index.php?title=7L_3s_(8/3-equivalent)&oldid=195406"