7L 3s (8/3-equivalent): Difference between revisions
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| 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}}'''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== | ==Modes== | ||
The | 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) | * LLLsLLsLLs 9|0 (Lydian ♮11) | ||
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* sLLsLLLsLL 1|8 (Locrian) | * sLLsLLLsLL 1|8 (Locrian) | ||
* sLLsLLsLLL 0|9 (Locrian b8) | * sLLsLLsLLL 0|9 (Locrian b8) | ||
{{todo|inline=1|add definition|text=What is a diatonic chord substitution?}} | |||
==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. | 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. | 2g, then, will fall between 960 cents (4\5) and 1029 cents (6\7), the range of minor sevenths. | ||
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The "large step" will fall between 171 cents (1\7) and 240 cents (1\5), the range of major seconds. | 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. | 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" | {| class="wikitable" | ||
!# generators up | !# generators up | ||
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|4L+4s | |4L+4s | ||
|} | |} | ||
==Scale tree== | ==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]]: | 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. | {{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]]. | ||
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'''=== | ||
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[[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 | ||
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[[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== | ==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]]: | ||
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]]. | ||