Miracle 10 MODMOS: Difference between revisions
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using 3-step stacked triads, we get neutral triads on I, II, IX and X; and wolf triads on III, IV, V, VI, VI, and VIII. | using 3-step stacked triads, we get neutral triads on I, II, IX and X; and wolf triads on III, IV, V, VI, VI, and VIII. | ||
Miracle can be thought of as a cluster temperament, with 10 clusters in an octave, based around Miracle[10]. The Miracle[10] chroma is very versatile, representing 2048/2025 and 525/512 in the 5-limit; 49/48, 50/49, and 64/63 in the 7-limit; and 45/44, 55/54, and 56/55 in the 11-limit. | |||
Accordingly, any 11-limit decatonic scale of interest likely to be a MODMOS of Miracle[10]. Since there are an enormous number of possibilities, we will limit our exploration to MODMOS that bridge the gap between Miracle[10], and two rank-3 decatonic scales of interest. | |||
7-limit Miracle[10] can be described as a [[step-nested scale]] as SNS (2/1, 16/15: 225/224, 1029/1024)[10]. | 7-limit Miracle[10] can be described as a [[step-nested scale]] as SNS (2/1, 16/15: 225/224, 1029/1024)[10]. | ||
We can build similar rank-3 | We can build similar scales in rank-3 using a 16/15 generator, adding a generator of 3/2 to the 2/1 and 16/15 generators, resulting in many more 3/2s and consonant triads being made available. In such scales 1029/1024 is not tempered out, nor the ampersand comma of 5-limit Miracle (the amount by which 3/2 exceeds 6 16/15s). The two such decatonic scales are [[SNS (2/1, 3/2, 5/4: 225/224)-10|SNS (2/1, 3/2, 5/4: 225/224)[10]]], and [[SNS ((2/1, 3/2)-5, 16/15: 225/224)-10|SNS ((2/1, 3/2)[5], 16/15: 225/224)[10]]]. They can be interpreted as MODMOS of Miracle[10] (but without 1029/1024 tempered out), and the Miracle MODMOS can be interpreted as the rank-3 scales with an extra comma tempered out, without a change of structure. We explore the Miracle[10] MODMOS that map the path from Miracle[10] to these 3-SNS, and MODMOS that are similar to those. | ||
== MODMOS to SNS (2/1, 3/2, 5/4: 225/224)[10] == | == MODMOS to SNS (2/1, 3/2, 5/4: 225/224)[10] == | ||
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We will interpolate from Miracle[10] 2|7 to Miracle[10] 2|7 ♯3 ♯4 ♯5, and back a different way. | We will interpolate from Miracle[10] 2|7 to Miracle[10] 2|7 ♯3 ♯4 ♯5, and back a different way. | ||
=== Miracle[10] 2|7 ♯3 === | |||
Miracle[10] 2|7 ♯3 sLdssssLss. On the generator chain this MODMOS is #####.####.....# | |||
Miracle[10] 2|7 ♯3 sLdssssLss. On the generator chain this MODMOS is #####.####.....# | |||
16/15~15/14 7/6 11/9 21/16 7/5 3/2 8/5 7/4 15/8 2/1 | 16/15~15/14 7/6 11/9 21/16 7/5 3/2 8/5 7/4 15/8 2/1 | ||
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using the 3-step stacked triads, we get major on X; neutral on I, II, VII, and IX; minor on III; wolf triads on IV and V; and augmented triads on VI and VII. | using the 3-step stacked triads, we get major on X; neutral on I, II, VII, and IX; minor on III; wolf triads on IV and V; and augmented triads on VI and VII. | ||
=== Miracle[10] 2|7 ♯3 ♯4 === | |||
Let's try Miracle[10] 2|7 ♯3 ♯4 sLsdsssLss. On the generator chain this MODMOS is ####..####....## | Let's try Miracle[10] 2|7 ♯3 ♯4 sLsdsssLss. On the generator chain this MODMOS is ####..####....## | ||
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using the 3-step stacked triads, we get major on I, and X; neutral on II and IX; minor on III, and IV; wolf triads on V and VI, and augmented triads on VII and VIII. | using the 3-step stacked triads, we get major on I, and X; neutral on II and IX; minor on III, and IV; wolf triads on V and VI, and augmented triads on VII and VIII. | ||
=== Miracle[10] 2|7 ♯3 ♯4 ♯5 === | |||
Now go to Miracle[10] 2|7 ♯3 ♯4 ♯5 sLssdssLss, which is also the 3-SN scale mLmmsmmLmm, but with 243/242 tempered out to make 11/9 and 27/22 equivalent. | |||
Now go to Miracle[10] 2|7 ♯3 ♯4 ♯5 sLssdssLss, which is also the 3-SN scale mLmmsmmLmm, but with 243/242 tempered out to make 11/9 and 27/22 equivalent. | |||
On the generator chain this MODMOS is ###...####...### | On the generator chain this MODMOS is ###...####...### | ||
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using the 3-step stacked triads, we get major on I, II, and X; minor on III, IV, and V; wolf triads on VI, and IX, and augmented triads on VII and VIII. | using the 3-step stacked triads, we get major on I, II, and X; minor on III, IV, and V; wolf triads on VI, and IX, and augmented triads on VII and VIII. | ||
And back a different way: | And back a different way: | ||
=== Miracle[10] 2|7 ♯4 ♯5 === | |||
Miracle[10] 2|7 ♯4 ♯5 ssLsdssLss. On the generator chain this MODMOS is ###..#####...## | Miracle[10] 2|7 ♯4 ♯5 ssLsdssLss. On the generator chain this MODMOS is ###..#####...## | ||
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using the 3-step stacked triads, we get major on I, II; minor on IV, and V; wolf triads on III, VI, VII, and IX, an augmented triad on VIII; and a neutral triad on X. | using the 3-step stacked triads, we get major on I, II; minor on IV, and V; wolf triads on III, VI, VII, and IX, an augmented triad on VIII; and a neutral triad on X. | ||
=== Miracle[10] 2|7 ♯5 === | |||
Miracle[10] 2|7 ♯4 ♯5 sssLdssLss. On the generator chain this MODMOS is ###.######...# | |||
Miracle[10] 2|7 ♯5 sssLdssLss. On the generator chain this MODMOS is ###.######...# | |||
16/15~15/14 8/7 11/9 4/3 7/5 3/2 8/5 7/4 15/8 2/1 | 16/15~15/14 8/7 11/9 4/3 7/5 3/2 8/5 7/4 15/8 2/1 | ||
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using the 3-step stacked triads, we get neutral triads on I and X, major on II; minor on V; and wolf triads on III, IV, VI, VII, VIII, and IX. | using the 3-step stacked triads, we get neutral triads on I and X, major on II; minor on V; and wolf triads on III, IV, VI, VII, VIII, and IX. | ||
=== Miracle[10] 2|7 ♯4 === | |||
And we'll also try Miracle[10] 2|7 ♯4 ssLdsssLss. On the generator chain this MODMOS is ####.#####....# | And we'll also try Miracle[10] 2|7 ♯4 ssLdsssLss. On the generator chain this MODMOS is ####.#####....# | ||
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using the 3-step stacked triads, we get major on I; neutral triads on II, IX, and X; minor on IV; wolf triads on III, V, VI, and VII; and an augmented triad on VIII. | using the 3-step stacked triads, we get major on I; neutral triads on II, IX, and X; minor on IV; wolf triads on III, V, VI, and VII; and an augmented triad on VIII. | ||
=== Summary === | |||
If we want to minimize the number of wolf triads, out of these scales, ♯3, ♯3 ♯4, and ♯3 ♯4 ♯5 are best. | |||
If we want to minimize the number of wolf triads, out of these scales, ♯3, ♯3 ♯4, and ♯3 ♯4 ♯5 are best. | |||
If we want to maximize the number of major and minor triads, ♯3 ♯4 ♯5 is best. | If we want to maximize the number of major and minor triads, ♯3 ♯4 ♯5 is best. | ||
<nowiki>#</nowiki>3 #4 is cool because you get 2 major, 2 minor, 2 neutral, 2 augmented, and 2 wolf triads. | <nowiki>#</nowiki>3 #4 is cool because you get 2 major, 2 minor, 2 neutral, 2 augmented, and 2 wolf triads. | ||
No major or minor triads can be found in MODMOS of Miracle[10] with a span of less than 14 generators, since 5/4 is found at 13 generators. | No major or minor triads can be found in MODMOS of Miracle[10] with a span of less than 14 generators, since 5/4 is found at 13 generators. | ||
We get to 11/8 with MODMOS that span 16 generators, which include Miracle[10] 2|7 ♯3, ♯3 ♯4, and ♯3 ♯4 ♯5, which have an 11/8 above III. | We get to 11/8 with MODMOS that span 16 generators, which include Miracle[10] 2|7 ♯3, ♯3 ♯4, and ♯3 ♯4 ♯5, which have an 11/8 above III. | ||
♯3 ♯4 ♯5 is symmetric. | ♯3 ♯4 ♯5 is symmetric. | ||
♯3, ♯3 ♯4, ♯4, ♯5, and ♯4 ♯5 all have inverses that are different MODMOS. | ♯3, ♯3 ♯4, ♯4, ♯5, and ♯4 ♯5 all have inverses that are different MODMOS. | ||
Of these, the inverse of ♯3 ♯4 is of particular interest. ##....####..#### on the generator chain: | Of these, the inverse of ♯3 ♯4 is of particular interest. ##....####..#### on the generator chain: | ||
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Same triads as it's inverse, just ordered differently. | Same triads as it's inverse, just ordered differently. | ||
The generator chain sequences ###...####...###, ####..####....##, and ##....####..#### give the best scales in the opinion of the author. | |||
Let's try ##..######..## | === Miracle[10] 2|7 ♯5 ♯6 === | ||
Let's also try ##..######..## | |||
Miracle[10] 2|7 ♯5 ♯6 sssLsdsLss. | Miracle[10] 2|7 ♯5 ♯6 sssLsdsLss. | ||
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using the 3-step stacked triads, we get neutral triads on I and VI, major on II; minor on V; and wolf triads on III, IV, VII, VIII, IX, and X. | using the 3-step stacked triads, we get neutral triads on I and VI, major on II; minor on V; and wolf triads on III, IV, VII, VIII, IX, and X. | ||
== Tonal modes: == | == Tonal modes: == | ||
A tonal mode is a mode with major or minor triads on 1/1 and 4/3 and/or 3/2, (and with at least 2 major triads and 2 minor triads) | A tonal mode is a mode with major or minor triads on 1/1 and 4/3 and/or 3/2, (and with at least 2 major triads and 2 minor triads) | ||