Miracle 10 MODMOS

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Miracle[10]

First, Miracle[10]:

In the mode 2|7, sssssssLss it approximates

16/15~15/14 8/7 11/9 21/16 7/5 3/2 8/5 7/4 15/8 2/1

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].

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], and 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] (and similar)

SNS (2/1, 3/2, 5/4: 225/224)[10] in mode 1 has step pattern mLmmsmmLmm, and step signature and mapping 2L 7m 1s = (35/32~49/45, 16/15~15/14, 135/128~21/20). m is the small step of Miracle[10], 16/15~15/14; L is the large step of Miracle[10], 35/23 (12/11 in the 11-limit); and s is the diminished step of Miracle[10], 21/20.

Accordingly, mLmmsmmLmm can be written as sLssdssLss.

It approximates the JI intervals 16/15~15/14 7/6 5/4 4/3 7/5 3/2 8/5 7/4 15/8 2/1

We can write it as Miracle[10] 2|7 ♯3 ♯4 ♯5 or Miracle[10] 8|1 ♭6 ♭7 ♭8.

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 #####.####.....#

16/15~15/14 7/6 11/9 21/16 7/5 3/2 8/5 7/4 15/8 2/1

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 ####..####....##

16/15~15/14 7/6 5/4 21/16 7/5 3/2 8/5 7/4 15/8 2/1

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.

On the generator chain this MODMOS is ###...####...###

16/15~15/14 7/6 5/4 4/3 7/5 3/2 8/5 7/4 15/8 2/1

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:

Miracle[10] 2|7 ♯4 ♯5

Miracle[10] 2|7 ♯4 ♯5 ssLsdssLss. On the generator chain this MODMOS is ###..#####...##

16/15~15/14 8/7 5/4 4/3 7/5 3/2 8/5 7/4 15/8 2/1

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 ###.######...#

16/15~15/14 8/7 11/9 4/3 7/5 3/2 8/5 7/4 15/8 2/1

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 ####.#####....#

16/15~15/14 8/7 5/4 21/16 7/5 3/2 8/5 7/4 15/8 2/1

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 maximize the number of major and minor triads, ♯3 ♯4 ♯5 is best.

#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.

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, ♯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:

From Miracle[10] 7|2 ssLsssssss: 16/15~15/14 8/7 5/4 4/3 10/7 32/21 18/11 7/4 15/8 2/1

It is Miracle[10] 7|2 ♭8 ♭9 ssLsssdsLs: 16/15~15/14 8/7 5/4 4/3 10/7 32/21 8/5 12/7 15/8 2/1

Let's put it in a minor mode:

Miracle[10] 1|8 ♭4 ♭5 ssdsLsssLs: 16/15~15/14 8/7 6/5 9/7 7/5 3/2 8/5 12/7 15/8 2/1.

using the 3-step stacked triads, we get minor on I, II; neutral on III and X; augmented on IV, and V; wolf triads on VI and VII, and major on VIII, and IX.

Same triads as it's inverse, just ordered differently.

The generator chain sequences ###...####...###, ####..####....##, and ##....####..#### give the best scales in the opinion of the author.

Miracle[10] 2|7 ♯5 ♯6

Let's also try ##..######..##

Miracle[10] 2|7 ♯5 ♯6 sssLsdsLss.

16/15~15/14 8/7 11/9 4/3 10/7 3/2 8/5 7/4 15/8 2/1.

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.

Going further than Miracle[10] 2|7 ♯3 ♯4 ♯5

Miracle[10] 2|7 ♯2 ♯3 ♯4 ♯5

Now try going further, to Miracle[10] 2|7 ♯2 ♯3 ♯4 ♯5 LsssdssLss. On the generator chain this MODMOS is ###....###...####.

12/11 7/6 5/4 4/3 7/5 3/2 8/5 7/4 15/8 2/1

using the 3-step stacked triads, we get major on I and X; major-diminished triad (11/9, 6/5) on II; minor on III, IV, and V; wolf triads IX, and augmented triads on VI, VII and VIII.

Miracle[10] 1|8 ♯3 ♯4 ♯5 / Miracle[10] 2|7 ♯2 ♯3 ♯4

It's inverse is sLssdsssLs, Miracle[10] 1|8 ♯3 ♯4 ♯5. On the generator chain this MODMOS is ####...###....###

16/15 7/6 5/4 4/3 7/5 3/2 8/5 12/7 15/8 2/1

using the 3-step stacked thirds, we get major on I, II, and X; minor-diminished triad (6/5,11/9) on III; minor on IV, and V, wolf triad on VI, and augmented triads on VII, VIII, and IX.

Miracle[10] 1|8 ♯3 ♯4

sLsdssssLs. #####..###.....##

16/15 7/6 5/4 21/16 7/5 3/2 8/5 12/7 15/8 2/1

using the 3-step stacked thirds, we get major on I and X; a neutral triad on II; minor-diminished triad on III; minor on IV; wolf triads on IV, VI, and IX; and augmented triads on VII and VIII

Miracle[10] 2|7 ♯2 ♯3 ♯4 ♯5 ♯6

LssssdsLss. ##.....###..#####

12/11 7/6 5/4 4/3 10/7 3/2 8/5 7/4 15/8 2/1

using the 3-step stacked triads, we get major on I; a major-diminished triad on II; neutral triad on III; minor on IV and V; wolf triads on VI, IX, and X, and augmented triads on VII and VIII.

Miracle[10] 1|8 ♯2 ♯3 ♯4 ♯5

LsssdsssLs. ####....##....####

12/11 7/6 5/4 4/3 7/5 3/2 8/5 12/7 15/8 2/1

using the 3-step stacked thirds, we get major on I and X; minor on IV, and V; major-diminished triad on II; minor-diminished triad on III; wolf triad IX; and augmented triads on VI, VII, and VIII.

Miracle[10] 1|8 ♯2 ♯3 ♯4

LssdssssLs. #####...##.....###

12/11 7/6 5/4 21/16 7/5 3/2 8/5 12/7 15/8 2/1

using the 3-step stacked thirds, we get major on I and X; minor-diminished triads on II and III; minor on IV; wolf triads on V, VI, and IX; and augmented triads on VII, and VIII.

Miracle[10] 1|8 ♯2 ♯3 ♯4 ♯5 ♯6

LssssdssLs. ###.....##...#####

12/11 7/6 5/4 4/3 10/7 3/2 8/5 12/7 15/8 2/1

using the 3-step stacked thirds, we get major on I; major-diminished triads on II and III; minor on IV and V; wolf triads on VI, IX, and X; and augmented triads on VII, and VIII.

Tonal modes:

A tonal mode, here, is a mode with major or minor triads on 1/1 and 4/3 and/or 3/2, and with at least two major triads, and two minor triads.

Of all above scales, only the following modes are tonal.

Major with minor subdominant

###..#####...## (2 major, 2 minor, 1 augmented, 1 neutral, 4 wolves)

2|7 ♯4 ♯5 ssLsdssLss 16/15 8/7 5/4 4/3 7/5 3/2 8/5 7/4 15/8 2/1

##..#####...### (2 major, 2 minor, 1 augmented, 1 neutral, 4 wolves)

7|2 ♭7 ♭8 ssLssdsLss 16/15 8/7 5/4 4/3 10/7 3/2 8/5 7/4 15/8 2/1

####..####....## (2 major, 2 minor, 2 augmented, 2 neutral, 2 wolves)

1|8 ♯4 ♯5 ssLsdsssLs 16/15 8/7 5/4 4/3 7/5 3/2 8/5 12/7 15/8 2/1

##....####..#### (2 major, 2 minor, 2 augmented, 2 neutral, 2 wolves)

8|1 ♭7 ♭8 sLsssdsLss 16/15 7/6 5/4 4/3 10/7 3/2 8/5 7/4 15/8 2/1

###...####...### (3 major, 3 minor, 2 augmented, 2 wolves)

2|7 ♯3 ♯4 ♯5 or 8|1 ♭6 ♭7 ♭8 sLssdssLss 16/15 7/6 5/4 4/3 7/5 3/2 8/5 7/4 15/8 2/1

1|8 ♯4 ♯5 ♯6 or 7|2 ♭7 ♭8 ♭9 and ssLssdssLs 16/15 8/7 5/4 4/3 10/7 3/2 8/5 12/7 15/8 2/1

###....###...#### (2 major, 3 minor, 3 augmented, 1 major-diminished, 1 wolf)

8|1 ♭7 ♭8 ♭9 sLsssdssLs 16/15 7/6 5/4 4/3 10/7 3/2 8/5 12/7 15/8 2/1

9|0 ♭6 ♭7 ♭8 LsssdssLss 12/11 7/6 5/4 4/3 7/5 3/2 8/5 7/4 15/8 2/1

####...###....### (3 major, 2 minor, 3 augmented, 1 minor-diminished, 1 wolf)

1|8 ♯3 ♯4 ♯5 sLssdsssLs 16/15 7/6 5/4 4/3 7/5 3/2 8/5 12/7 15/8 2/1

0|9 ♯4 ♯5 ♯6 ssLssdsssL 16/15 8/7 5/4 4/3 10/7 3/2 8/5 12/7 11/6 2/1

####....##....#### (2 major, 2 minor, 3 augmented, 1 major-diminished, 1 minor-diminished, 1 wolf)

1|8 ♯2 ♯3 ♯4 ♯5 or 9|0 ♭6 ♭7 ♭8 ♭9 LsssdsssLs 12/11 7/6 5/4 4/3 7/5 3/2 8/5 12/7 15/8 2/1

0|9 ♯3 ♯4 ♯5 ♯6 or 8|1 ♭7 ♭8 ♭9 ♭10 sLsssdsssL 16/15 7/6 5/4 4/3 10/7 3/2 8/5 12/7 11/6 2/1

Minor with major dominant (modes starting on V of the above modes)

###..#####...## (2 major, 2 minor, 1 augmented, 1 neutral, 4 wolves)

1|8 ♭2 ♭3 ♭4 dssLssssLs 21/20 9/8 6/5 21/16 7/5 3/2 8/5 12/7 15/8 2/1 (4:5:6:7 on dominant)

##..#####...### (2 major, 2 minor, 1 augmented, 1 neutral, 4 wolves)

1|8 ♭3 ♭4 sdsLssssLs 16/15 9/8 6/5 21/16 7/5 3/2 8/5 12/7 15/8 2/1 (4:5:6:7 on dominant)

####..####....## (2 major, 2 minor, 2 augmented, 2 neutral, 2 wolves)

1|8 ♭2 ♭3 ♭4 ♭5 dsssLsssLs 21/20 9/8 6/5 9/7 7/5 3/2 8/5 12/7 15/8 2/1

##....####..#### (2 major, 2 minor, 2 augmented, 2 neutral, 2 wolves)

2|7 ♭3 ♭4 sdsLsssLss: 16/15 9/8 6/5 21/16 7/5 3/2 8/5 7/4 15/8 2/1 (4:5:6:7 on dominant)

###...####...### (3 major, 3 minor, 2 augmented, 2 wolves)

2|7 ♭2 ♭3 ♭4 dssLsssLss 21/20 9/8 6/5 21/16 7/5 3/2 8/5 7/4 15/8 2/1 (4:5:6:7 on dominant)

1|8 ♭3 ♭4 ♭5 sdssLsssLs 16/15 9/8 6/5 9/7 7/5 3/2 8/5 12/7 15/8 2/1

###....###...#### (2 major, 3 minor, 3 augmented, 1 major-diminished, 1 wolf)

2|7 ♭3 ♭4 ♭5 sdssLssLss 16/15 9/8 6/5 9/7 7/5 3/2 8/5 7/4 15/8 2/1

3|6 ♭2 ♭3 ♭4 dssLssLsss 21/20 9/8 6/5 21/16 7/5 3/2 18/11 7/4 15/8 2/1 (4:5:6:7 on dominant)

####...###....### (3 major, 2 minor, 3 augmented, 1 minor-diminished, 1 wolf)

2|7 ♭2 ♭3 ♭4 ♭5 dsssLssLss 21/20 9/8 6/5 9/7 7/5 3/2 8/5 7/4 15/8 2/1

1|8 ♭3 ♭4 ♭5 ♭6 sdsssLssLs 16/15 9/8 6/5 9/7 11/8 3/2 8/5 12/7 15/8 2/1

####....##....#### (2 major, 2 minor, 3 augmented, 1 major-diminished, 1 minor-diminished, 1 wolf)

3|6 ♭2 ♭3 ♭4 ♭5 dsssLsLsss 21/20 9/8 6/5 9/7 7/5 3/2 18/11 7/4 15/8 2/1

2|7 ♭3 ♭4 ♭5 ♭6 sdsssLsLss 16/15 9/8 6/5 9/7 11/8 3/2 8/5 7/4 15/8 2/1

MODMOS to SNS ((2/1, 3/2)[5], 16/15: 225/224)[10]

SNS ((2/1, 3/2)[5], 16/15: 225/224)[10] in mode 1 has step pattern smLmsmLmsm, and step signature and mapping 2L 5m 3s = (10/9, 16/15~15/14, 135/128~21/20). m is the small step of Miracle[10], 16/15~15/14; s is the diminished step of Miracle[10], 21/20; and L is the augmented step of Miracle[10], 10/9.

Accordingly, smLmsmLmsm can be written as dsAsdsAsds. This is clearly a much more complex MODMOS than those above.

It approximates the JI intervals 21/20 9/8 5/4 4/3 7/5 3/2 5/3 16/9 15/8 2/1

We can write it as Miracle[10] 7|2 ♭2 ♭3 ♭6 ♭7 ♯8 ♯9, or as Miracle[10] 6|3 ♭2 ♭3 ♯4 ♯5 ♯8 ♯9.

using the 3-step stacked triads, we get major on I, V, and VII; augmented on II (14/11, 5/4) and III (5/4, 14/11), minor on IV, VIII, and X, large major (14/11, 33/28) on VI, small minor (33/28, 14/11) on IX.

So here we get 3 major, 3 minor, 2 augmented, and instead of 2 wolves, we get a large major and a small minor!

Now let's consider a path from Miracle[10] to this MODMOS. (tbc)

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