User:Moremajorthanmajor/Hierarchy of soid-family modes

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:

• edIX (Neapolitan temperament)
• edX (Middletown Valley temperaments)
• edt (Bohlen-Pierce among others)

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

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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 (±13¢)	Mode
1\1edo-11\10edo	Perfect Phrygian
11\10-10\9	Perfect Intense Phrygian
10\9-9\8	Perfect Intense Phrygian-Soft Aeolian
9\8-8\7	Perfect Soft Aeolian
8\7-6\5	Perfect Aeolian
6\5-11\9	Perfect Aeolian-Dorian
11\9-5\4	Perfect Subpental Dorian
5\4-14\11	Perfect Pental Dorian
14\11-9\7	Perfect Superpental Dorian
9\7-13\10	Perfect Dorian-Mixolydian
13\10-21\16	Perfect Subpental Mixolydian
21\16-4\3	Perfect Pental Mixolydian
4\3-39\29	Perfect Soft Superpental Mixolydian
39\29-40\29	Perfect Intense Superpental Mixolydian
40\29-7\5	Perfect Mixolydian-Ionian
7\5-10\7	Perfect Ionian
10\7-22\15	Ionian-Lydian/Locrian
22\15-23\15	Lydian/Locrian
23\15-11\7	Lydian/Locrian-Phrygian
11\7-8\5	Pluperfect Phrygian
8\5-47\29	Pluperfect Phrygian-Aeolian
47\29-48\29	Pluperfect Intense Subpental Aeolian
48\29-5\3	Pluperfect Soft Subpental Aeolian
5/3-27\16	Pluperfect Pental Aeolian
27\16-17\10	Pluperfect Subpental Aeolian
17\10-12\7	Pluperfect Aeolian-Dorian
12\7-19\11	Pluperfect Subpental Dorian
19\11-7\4	Pluperfect Pental Dorian
7\4-16\9	Pluperfect Superpental Dorian
16\9-9\5	Pluperfect Dorian-Mixolydian
9\5-13\7	Pluperfect Mixolydian
13\7-15\8	Pluperfect Intense Mixolydian
15\8-17\9	Pluperfect Intense Mixolydian-Soft Ionian
17\9-19\10	Pluperfect Soft Ionian
19\10-2\1	Pluperfect Ionian

The names "Perfect" and "Pluperfect" refer to the common limitation of a vocal melody to within a tenth 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 Mixolydian modality 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.

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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 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:

Minor Ninths (F-G-A-B-H-C-D-E; da/fo-ro/se-ma/li-fi/ta/sa-bu/ku-so/da-le/ra-ti/si/mi) - Phrygian Mode New Neapolitan Scale:

Major Ninths (F-G-A-B-H-C-D-E; da/fo-ro/se-ma/li-fi/ta/sa-bu/ku-so/da-le/ra-ti/si/mi) - Aeolian Mode New Neapolitan Scale:

2L 6s and 6L 2s - Symmetric, Tetrachordal Major

4L 4s - Macro-diminished

3L 5s and 5L 3s - Grandfather

Minor Tenths - Dorian Mode Middletown (F-G[-J]-A-B-H-C-D[-S]-E; da/fo-ro/se[-bu]-ma/li-fi/ta/sa-bu/ku-so/da-le/ra[-ku]-ti/si/mi)

Major Tenths - Mixolydian Mode Middletown (F-G[-J]-A-B-H-C-D[-S]-E; da/fo-ro/se[-bu]-ma/li-fi/ta/sa-bu/ku-so/da-le/ra[-ku]-ti/si/mi)

3L 6s and 6L 3s - Symmetric, Tetrachordal Major, Macro-augmented[9]

4L 5s and 5L 4s - Montrose

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

Perfect Elevenths - Ionian Mode Galveston Bay Temperament Area

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 Sigmatic

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 Bijou deck of scales

Major Thirteenths - Dorian Mode 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 Yakuman Temperament List

Major Fourteenths - Ionian Mode Yakuman Temperament List

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

5L 8s and 8L 5s - Ryūīsō

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

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

2L 11s and 11L 2s - Kokushimusō

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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, at least as long as augmented sixth chords are not to be transliterated into Aeolian mode in a pre-Romantic context (12edo tempers out the augmented comma, transliterating these into 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. The names for these root position triads are: