User:Unque/Dietic Minor

The Dietic Minor scale is a family of closely related 10-note ternary scales that act as aberrismic extensions of Smitonic. The scale can be generalized under the pattern 4L 3M 3s; the most prototypical arrangements of the scale are LsmLLsmLsm and LsmLsLmsLm, though other permutations exist (see below).

Theory and History

Conception

The Aeolian mode of the diatonic scale (scale pattern LsLLsLL), while harmonically rich, has a minor problem in the form of the flat seventh degree; this interval provides a minor triad over the fifth degree rather than the more desirable major triad, as well as leaving no leading tone into the tonic note. As such, many musicians opt to use an altered "melodic minor" scale, which has a raised seventh degree (scale pattern LsLLsAs).

This presents a problem of its own - the raised seventh is an augmented tone above the minor sixth, which creates a disjointed sound in melodies that walk up this scale. Many musicians opt to intonate the augmented step as a normal whole tone (scale pattern LsLLsLs), but this scale now falls short of the octave by one Apotome. In common-practice theory, the most common solution for this is to extend the preceding limma between the fifth and sixth into a whole tone (scale pattern LsLLLLs), which allows the scale to reach the octave; however, this approach largely abandons the minor tonality of the scale, with all but the third degree now being that of the Ionian scale.

29 equal divisions of the octave provides an elegant solution that retains this minor tonality, reaches the octave, and avoids disjunct augmented tones: by noticing that the Apotome is divided evenly into three steps, and there exist three limmas in the scale, we can disperse this damage across all three semitones rather than placing it all onto one of them. This creates a form of the Smitonic scale (scale pattern LsLLsLs), where the large step is 9/8 and the small step 2187/2048. This pattern can be thought of as a 3-limit essentially tempered scale that tempers out the Mystery comma. While this scale does meaningfully retain the minor tonality, the harmonies are quite distorted compared to the diatonic counterparts. The minor third and perfect fifth are both stretched a diesis wider than in diatonic, while the sixth and seventh degrees are stretched by an entire limma.

So perhaps instead of stretching the semitones, the dieses could be employed as step sizes - these smaller steps would act as commatic adjustments that preserve the diatonic harmony in the context of a Smitonic scale. There are a number of arrangements that can be employed in this pattern, though they can be generalized under the form LmLLmLm with three small steps inserted at various points among the scale degrees. The name "Dietic Minor" was given to this scale, as it is a modification of the harmonic minor scale that adds dietic steps as a practical consideration.

Notation

Because the L and m steps are anchored in place with the s steps being inserted around them, a given permutation of the Dietic Minor scale can be described based on where the s steps are inserted. Here, I will use Roman numeral notation to indicate the degree (1-indexed) after which an s step is slotted into the pattern; for instance, the pattern LsmLLsmLsm will be indicated as "s(II-V-VII)," and the pattern LsmLsLmsLm will be indicated as "s(II-IV-VI)."

Aberrismic Theory

As it turns out, this proposed solution is quite similar to the principles that spawned Groundfault's aberrismic theory, a concept based on the insertion of small steps called "aberrismas" into a MOS scale to solve structural issues. The small steps of the Dietic Minor scale can be conceived as aberrismas, especially in tunings outside of 29edo where these aberrismas are no longer the single step "diesis."

It should be noted, however, that Ground's approach to aberrismic scales involves certain other features that are foregone in Dietic Minor, such as formation by MOS substitution and the satisfaction of Monotone-MOS conditions; the structure of Dietic Minor does not generalize well to either of these properties, and while closely related to Ground's approach, the two might not be considered entirely equivalent.

Tuning Ranges

While the scale was designed for use in 29edo, a number of other tuning systems of similar sizes support the scale pattern. Below are listed the EDOs (up to 41) that support the Dietic Minor scale:

Tuning Ranges
EDO	Step Sizes
21edo	3:2:1
25edo	4:2:1
28edo	4:3:1
29edo	5:2:1
31edo	4:3:2
32edo	5:3:1
33edo	6:2:1
35edo	5:3:2
35edo	5:4:1
36edo	6:3:1
37edo	7:2:1
38edo	5:4:2
39edo	6:4:1
39edo	6:3:2
40edo	7:3:1
41edo	5:4:3
41edo	8:2:1

Pattern Arrangements

s(II-V-VII)

s(II-V-VII), or LsmLLsmLsm, is the most obvious arrangement of the scale - an aberrisma is inserted before each semitone.

Modes of s(II-V-VII)
Rotational Order	Pattern	Notes
1	LsmLLsmLsm
2	smLLsmLsmL
3	mLLsmLsmLs	Contains Perfect Fourth
4	LLsmLsmLsm
5	LsmLsmLsmL	It's a palindrome in spirit!
6	smLsmLsmLL	Contains Perfect Fifth
7	mLsmLsmLLs
8	LsmLsmLLsm
9	smLsmLLsmL
10	mLsmLLsmLs

s(II-IV-VI)

s(II-IV-VI), or LsmLsLmsLm, is another intuitive arrangement of the scale. The aberrismas are arranged such that the scale has a maximum variety of three, which makes the scale significantly more harmonically coherent while sacrificing some of the melodic qualities of the s(II-V-VII) arrangement.

Modes of s(II-V-VII)
Rotational Order	Pattern	Notes
1	LsmLsLmsLm
2	smLsLmsLmL	Contains Perfect Fifth
3	mLsLmsLmLs
4	LsLmsLmLsm
5	sLmsLmLsmL
6	LmsLmLsmLs
7	msLmLsmLsL
8	sLmLsmLsLm
9	LmLsmLsLms	Contains Perfect Fourth
10	mLsmLsLmsL

s(I-V-VII)

s(I-V-VII), or sLmLLsmLsm, is an arrangement that might seem unintuitive at first - after all, the aberrismas appear in a weird cluster in the upper end of the scale rather than being dispersed as one would expect. However, this placement allows for the Perfect Fourth and Fifth to occur over two degrees instead of one, and additionally to be available over the same degree, which makes the harmonies of this permutation significantly smoother.

Modes of s(V-VI-VII)
Rotational Order	Pattern	Notes
1	sLmLLsmLsm
2	LmLLsmLsms	Contains Perfect Fourth and Fifth
3	mLLsmLsmsL
4	LLsmLsmsLm
5	LsmLsmsLmL	Contains Perfect Fifth
6	smLsmsLmLL	Contains Perfect Fourth
7	mLsmsLmLLs
8	LsmsLmLLsm
9	smsLmLLsmL
10	msLmLLsmLs