Modal systematization of soid-family scales

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This is my (Joseph Ruhf's) proposed notation for scales which repeat at an arbitrary second-octave interval. Scales for which this notation works include:

I refer to this notation as Long Common Practice (LCP) and the Reformed Church Modes (RCM).


New uses for classic names

These scales are classified into modal families based on which interval is taken as the formal chroma equivalence (a [near] 2:1, if one exists in the scale, will always be perceived as substantially chroma-equivalent even if it falls between two notes which are required to have different names). There is no particular comma these scales are defined as tempering out (although Middletown used to be specifically a distorted meantone which tempered out 64/63).

Designating a particular pitch as the formal chroma equivalence enables the modal center to be named relative to it. These names, which are independent of the notation used for the actual notes*, are as follows:

Quantity (±comma) Mode
1\1edo-22\21edo Perfect Minor Neapolitan Phrygian Soft
22\21-16\15~15\14 Thirdtone
16\15~15\14-11\10 Semitone
11\10-10\9 Intense
10\9-9\8 Neapolitan/Natural-Harmonic Intense Phrygian-Soft Aeolian
9\8-8\7 Natural-Harmonic Aeolian Soft
8\7-15\13 Flattone
15\13-7\6 Meantone
7\6-13\11 Intense Superpyth
13\11-6\5 Ultrapyth
6\5-11\9 Natural-Harmonic/Melodic Intense Aeolian-Subpental Dorian
11\9-5\4 Melodic Dorian Subpental
5\4-14\11 Pental
14\11-9\7 Superpental
9\7-22\17 Perfect Neutral Dorian-Mixolydian Mohajira
22\17-13\10 Beatles
13\10-21\16 Perfect Major Melodic Mixolydian Subpental
21\16-4\3 Pental
4\3-39\29 Superpental Soft
39\29-40\29 Intense
40\29-7\5 Melodic/Natural-Harmonic Superpental Mixolydian-Soft Ionian
7\5-17\12 Natural-Harmonic Ionian Soft
17\12-10\7 Intense
10\7-22\15 Ambiguous Natural-Harmonic/Acoustic Major/False Melodic Minor/Diminished Intense Ionian-Lydian/Pseudodorian/Locrian
22\15-23\15 Acoustic Major/False Melodic Minor/Diminished Lydian/Pseudodorian/Locrian
23\15-11\7 Acoustic Major/False Melodic Minor/Diminished/Neapolitan Minor Lydian/Pseudodorian/Locrian-Soft Phrygian
11\7-19\12 Pluperfect/Abundant Minor Neapolitan Phrygian Soft
19\12-8\5 Intense
8\5-47\29 Neapolitan/Natural-Harmonic Intense Phrygian-Subpental Aeolian
47\29-48\29 Natural-Harmonic Aeolian Subpental Intense
48\29-5\3 Soft
5/3-27\16 Pental
27\16-17\10 Superpental
17\10-29\17 Natural-Harmonic/Melodic Aeolian-Dorian Beatles
29\17-12\7 Mohajira
12\7-19\11 Melodic Dorian Subpental
19\11-7\4 Pental
7\4-16\9 Superpental
16\9-9\5 Pluperfect/Abundant "Neutral" Superpental Dorian-Soft Mixolydian
9\5-20\11 Pluperfect/Abundant Major Melodic Mixolydian Soft Ultrapyth
20\11-11\6 Superpyth
11\6-24\13 Meantone
24\13-13\7 Flattone
13\7-15\8 Intense
15\8-17\9 Melodic/Natural-Harmonic Intense Mixolydian-Soft Ionian
17\9-19\10 Natural-Harmonic Ionian Soft
19\10-27\14~29\15 Semitone
27\14~29\15-41\21 Thirdtone
41\21-2\1 Intense

The names "Perfect" and "Pluperfect/Abundant" refer to the common limitation of a vocal melody to within an eleventh for the sake of overall perceptual coherence. The "Perfect" modes also match where LCP may consider just a triad (or tetrad) to be reasonably complete and therefore the basic chordal harmony.

Two noted potential bugs of the RCM are that only the tenths, in reference to their value as the compound form of the third which is the modal degree of the diatonic scale, are in reformed modes which match their qualities and Lydian and Locrian are technically two names of the same reformed mode. On the other hand, one noted feature of the RCM (unlike the common-practice church modes) is that they, by definition, do not refer to a specific gamut (or subgamut of a larger whole gamut) of notes to which a composition is presumed to be confined.

As a result, the requirement of diatonicity, if retained, is under-specified and a whole gamut of 20, 22, 26, 32 or even more notes falls within the same mode as long as it has a formal chroma equivalence which falls within the same general region of the spectrum. There is therefore not necessarily "attribute clash" between the seventh degree and Lydian leading tone of a diatonic scale. This opens unusual possibilities, such as compositions with a common-practice modality with a minor 7th which is not disrupted by the Lydian leading tone of the scale.

The notation-independent functional name of the (near) 2:1, if one exists in the scale, is "Viridiant", and is a reference to green (viridis in Latin) being perfectly equally opposite, according to color theory, to red (ruber) and blue (caesius) by way of the (near) 2:1 being called a perfect octave.


Extending common practice diatonic scales to repeat beyond the octave

In syntonic temperaments, the seven notes of the diatonic scale are considered the basic components of linear melody and relatively easy to stabilize over most chords of the key.

But this reform leaves the requirement of diatonicity, if retaining it, under-specified, and it would be nice to have some form of full specificity to apply anywhere in the spectrum (Bohlen-Pierce is fine, but it leaves one out of luck where the tritave is not to be chroma-equivalent).

This is where LCP comes into the picture. It provides these names for the extensions of the common practice diatonic scales to repeat beyond the octave, that is the Reformed Authentic modes:

Minor Ninths (G-A-B-C-Q-D-E-F) - Phrygian Mode:

Major Ninths (F-G-A-B-H-C-D-E) - Aeolian Mode

New Neapolitan Scale:

2L 6s and 6L 2s - Macroshrutis

I II III IV V VI VII VIII
Symmetric Major Phrygian so le ti/si du bo/ko re mi fa
Aeolian fo se li ta/sa bu/ku do ra mi
Tetrachordal Major Phrygian do ra mi fu bo/ko se li ta/sa
Aeolian da ro ma fi bu/ku so le ti/si

4L 4s - Macro-diminished

3L 5s and 5L 3s - Grandfather

Minor Tenths (G[-J]-A-B-C-Q-D[-S]-E-F) - Dorian Mode

Major Tenths (F-G[-J]-A-B-H-C-D[-S]-E) - Mixolydian Mode

Middletown

3L 6s and 6L 3s - Macro-augmented[9]

I II III IV V VI VII VIII IX
Symmetric Major Dorian so le

lu

ti/si

ve

du

ti/si

bo

du

re

ko

mo

re

ki

mi

fa
Mixolydian fo se li

bu

ta/sa

li

bu

ta/sa

do

ku

ra

do

ku

ra

mi
Tetrachordal Major Dorian do ra

ru

mi

ve

fu

mi

bo

fu

se

ko

lo

se

ki

li

ta/sa
Mixolydian da ro ma

bu

fi

ma

bu

fi

so

ku

le

so

ku

le

ti/si

4L 5s and 5L 4s - Montrose

2L 7s and 7L 2s - Terra Rubra

Perfect Elevenths - Ionian Mode

Augmented Elevenths - Lydian Mode

Galveston Bay Temperament Area

2L 8s and 8L 2s, 5L 5s - Galveston Symmetric, Pentachordal Major, Macro-Blackwood

4L 6s and 6L 4s - Baytown

3L 7s and 7L 3s - Bolivar

Diminshed Twelfths - Locrian Mode

Perfect Twelfths - Phrygian Mode

Sigmatic

5L 6s and 6L 5s - Sesquimachine

4L 7s and 7L 4s, 3L 8s and 8L 3s - (Un)Fair Sigma and Mu

2L 9s and 9L 2s - Arcturus[11]

Minor Thirteenths - Aeolian Mode

Major Thirteenths - Dorian Mode (aka Kiriage Magan)

Bijou deck of scales

2L 10s and 10L 2s, 3L 9s and 9L 3s, 4L 8s and 8L 4s - Macro-augmented[12], Macro-diminished[12], (Bifold, Trifold, Quadrifold) Symmetric; Hexachordal, Pentachordal, Tetrachordal Major

6L 6s - Macro-Hexe

5L 7s and 7L 5s - Chromatic Major

Minor Fourteenths - Mixolydian Mode

Major Fourteenths - Ionian Mode (aka Nagashi)

(Tetrad and Pentatonic - Mangan Temperament

Hexa- and Heptatonic - Haneman Temperament

Enneatonic plus or minus one - Baiman Temperament

Hen- and dodecatonic - Sanbaiman Temperament)

Triskaidekatonic - Yakuman Temperament List

(1L 12s and 12L 1s - Kazoe Yakuman)

7L 6s and 6L 7s - Daichīsei and Daisharin

9L 4s and 4L 9s - Shōsūshī and Daisūshī

10L 3s and 3L 10s - Shōsangen and Daisangen

5L 8s and 8L 5s - Ryūīsō

2L 11s and 11L 2s - Kokushimusō


Chord progressions

Due to the fact that the fifth of a common practice diatonic scale can work normally in the extensions beyond the basic ninth, transliterations of chord progressions from 12edo into these LCP scales are fairly trivial, although using any but an eleventh practically assumes that commas (particularly the septimal quarter tone of 36/35) tempered out by 12edo are to be observed in order to have a more stable minor seventh degree. Also, the transliterations are by definition modally ambiguous because they assume extra notes in the harmony that 12edo does not use in those contexts as a rule.

However, the transliteration is not so immediately trivial when the scale is the basic ninth because the fifth of a common practice diatonic scale must work abnormally, being the midpoint of the nine-tone scale. Nevertheless, transliterations of chord progressions from 12edo into the LCP scales of this family will come straight across relatively clearly modally and even into the 12edo-based modes (although Phrygian is difficult to use well because it can generally cut so close to the octave), at least as long as augmented sixth chords are not to be transliterated in a pre-Romantic context (12edo tempers out the augmented comma, transliterating these into [incomplete] dominant 8th chords, which are technically unstable but also technically misleading by enharmonic equivalence). As a result of the fifth that must work abnormally, root position triads actually have a stronger tonality than they do in common practice, being composed of a set of intervals in which there are two that are qualitatively and quantitatively different from each other, which also obtaining for tetrads when an eleventh is equivalent or pentads when a thirteenth is equivalent although the fifth returns to being able to work normally then. The names for these root position triads are:

m6 n6 M6
m3 Minor - 5:6:8 Husayni (Persian) - 11:13:18 Dorian - 6:7:10
n3 Bayati/Turkish Minor - 9:11:14 Neutral - 13:16:21 (18:22:29, 19:23:31, 22:27:36, 25:31:41) Rast - 9:11:15
M3 Hindu~Chahargah (Persian) “fourth”ward - 7:9:11 Chahargah (Persian) - 8:10:13 (11:14:18, 14:18:23) Major - 7:9:12

There are also "full" and "defective" ways of transliterating chord progressions into LCP modes which are ninths, elevenths and thirteenths due to the scale having a degree which is exactly at its midpoint. However, the ninths offer all the extra possibilities with no extra necessities unless you care about having great diversity of "defective" ways of transliterating chord progressions into the mode. Also, transliterations into ninths work as follows:

d5-v5 P5 ^5-A5
m3 Dorian - 6:7:10~Husayni (Persian) - 11:13:18 Minor - 5:6:8 (Major #3~Major fourthward - 3:4:5)
n3 Bayati/Turkish Minor - 9:11:14 Neutral - 13:16:21 (18:22:29, 19:23:31, 22:27:36, 25:31:41) Rast - 9:11:15
M3 (Minor b3 - 9:10:14) Major - 7:9:12 Chahargah (Persian) - 8:10:13 (11:14:18, 14:18:23), “fourth”ward - 7:9:11 (Hindu)~(Italian - 4:5:7)

The twelve reformed minor keys are as follows

I II III IV V VI VII VIII
Phrygian MOS G Major A Minor B Minor C Hindu Q Minor D Major E Minor F Minor
#7 A Dorian Q Hindu E# Double Diminished
Aeolian MOS,*b4 Mode 5 ♮8 F Major G Major A Dorian B Minor H Major C Major D Dorian E Minor
b4 G Dorian Bb Major D Minor
*MOS b7 G Hindu B Minor b3 H Dorian Db Italian
*b4 b7 G Minor Bb Dorian Db Major
*MOS Mode 2 ♮8 G Major A Dorian B Dorian H Major C Major D Major E Minor F# Minor
*MOS Mode 3 ♮8 A Dorian B Minor H Italian C Major D Dorian E Major F Major G# Minor
*MOS Mode 4 ♮8 B Minor H Major C Italian D Dorian E Minor F Major G Major A# Minor
*b4 Mode 2 ♮8 G Dorian A Dorian Bb Italian H Major C Major D Hindu E Minor F# Minor
*b4 Mode 3 ♮8 A Dorian Bb Major H Italian C Major D Minor E Hindu F Major G# Minor b3
*b4 Mode 6 ♮8 C Major D Minor E Dorian F Major G Dorian A Hindu Bb Major H# Minor
*b4 Mode 7 ♮8 D Minor E Minor F Italian G Dorian A Dorian Bb Major #3 H Major C# Minor
*b4 Mode 8 ♮8 E Minor F Major G Dorian #6 A Dorian Bb Major H Major #3 C Major D# Double Diminished