User:Overthink/22edo decatonic theory

Todo: Citation to Paul Erlich's paper, Add diagram

In classical music, we often use the diatonic scale, which is generated by meantone temperament. This gives us relatively simple access to 5-limit harmony, while avoiding wolf intervals. There is a system many consider to be analogous to diatonic in the 7-limit; the decatonic scale of pajara temperament. Strictly speaking, there are two decatonic scales: the symmetrical scale of the 2L 8s mos pattern, with large and small steps in the pattern ssLssssLss; and the pentachordal near-mos scale, with steps in the pattern ssLsssLsss. Here we will mainly be focusing on the pentachordal scale.

Interval classification

We will classify intervals based on the 2L 8s mos, with names "unison", "second", etc. However, since we have 10 note classes, the interval of equivalence is not an "octave" as in diatonic, but a "hendecave". In this system, 1sns, 2nds, 5ths, 6ths, 7ths, 10ths, and 11ves are perfect, while 3rds, 4ths, 8ths, and 9ths are imperfect. Pajara has a half-octave (600 ¢) period representing both 7/5 and 10/7, and a ~3/2 generator (~709 ¢ in 22edo). The half-octave is a perfect 6th, and the 3/2 is a perfect 7th. (From now on, interval names are decatonic unless specified.) Here is a table of intervals classified by the 2L 8s scale.

Decatonic intervals

Name	Cents (22edo)	Steps of 22edo	JI ratios
Perfect 1sn	0.0	0	1/1
Augmented 1sn	54.5	1	25/24, 28/27, 49/48
Diminished 2nd	54.5		36/35, 81/80
Perfect 2nd	109.1	2	16/15, 15/14, 21/20
Augmented 2nd	163.6	3	10/9
Minor 3rd	218.2	4	8/7, 9/8
Major 3rd	272.7	5	7/6
Minor 4th	327.3	6	6/5
Major 4th	381.8	7	5/4
Diminished 5th	436.4	8	9/7, 32/25
Perfect 5th	490.9	9	4/3, 21/16
Augmented 5th	545.5	10	11/8
Diminished 6th	545.5		27/20, 15/11
Perfect 6th	600.0	11	7/5, 10/7
Augmented 6th	654.5	12	40/27, 22/15
Diminished 7th	654.5		16/11
Perfect 7th	709.1	13	3/2, 32/21
Augmented 7th	763.6	14	14/9, 25/16
Minor 8th	818.2	15	8/5
Major 8th	872.7	16	5/3
Minor 9th	927.3	17	12/7
Major 9th	981.8	18	7/4, 16/9
Diminished 10th	1036.4	19	9/5
Perfect 10th	1090.9	20	15/8, 28/15, 40/21
Augmented 10th	1145.5	21	27/14, 48/25, 96/49
Diminished 11ve	1145.5		35/18, 160/81
Perfect 11ve	1200.0	22	2/1

The white keys

In diatonic, the white keys are in a mode such that, on C, there is a major triad on the tonic, dominant, and subdominant. Here, the main chords are tetrads, with the major tetrad being P1−M4−P7−M9, approximating 4:5:6:7, and the minor tetrad being P1−m4−P7−m9, approximating 1/(7:8:10:12). More about this will be discussed later. Unfortunately, there is no mode of the pentachordal scale which places the major tetrad on the tonic, dominant, and subdominant of the same key, but the ssLsssLsss mode gets as close as possible. Hence, we will place the white keys on this mode above C.

The white keys

Note	Interval above C	Cents	Steps
C	P1	0.0	0
R	P2	109.1	2
D	m3	218.2	4
E	M4	381.8	7
F	P5	490.9	9
T	P6	600.0	11
G	P7	709.1	13
A	M8	872.7	16
S	M9	981.8	18
B	P10	1090.9	20
C	P11	1200.0	22

The notes are named so that C, D, E, F, G, A, and B represent the Zarlino diatonic scale, with R, S, and T filling the gaps.

Modes

Complete the table

Name	Brightness	Step pattern	Intervals	Note names on C	Appears in white keys on
Scandium	+5	LsssLsssss	P1, A2, M3, M4, P5, A6, A7, M8, M9, P10, P11	C, R#, D#, E, F, T#, G#, A, S, B, C	D
Titanium	+3	LsssssLsss	P1, A2, M3, M4, P5, P6, P7, M8, M9, P10, P11	C, R#, D#, E, F, T, G, A, S, B, C	G
Vanadium	+3	sLsssLssss	P1, P2, M3, M4, P5, P6, A7, M8, M9, P10, P11	C, R, D#, E, F, T, G#, A, S, B, C	R
Chromium	+1	sLsssssLss	P1, P2, M3, M4, P5, P6, A7, M8, M9, P10, P11	C, R, D#, E, F, T, G, Ab, S, B, C	T
Manganese (Major)	+1	ssLsssLsss	P1, P2, M3, M4, P5, P6, A7, M8, M9, P10, P11	C, R, D, E, F, T, G, A, S, B, C	C

Chords

First explain what chords are there

Using the Root−4th−7th−9th construction, we find the tetrads on each note.

Tetrads on white keys

Note	Notes of chord	Intervals of chord	Name
C	C−E−G−S	P1−M4−P7−M9	Major
R	R−F−A−B	P1−M4−A7−M9	Aug.
D	D−T−S−C	P1−M4−A7−M9	Aug.
E	E−G−B−R	P1−m4−P7−m9	Minor
F	F−A−C−D	P1−M4−P7−m9	Major−minor
T	T−S−R−E	P1−M4−P7−M9	Major
G	G−B−D−F	P1−M4−P7−M9	Major
A	A−C−E−T	P1−m4−P7−m9	Minor
S	S−R−F−G	P1−m4−P7−m9	Minor
B	B−D−T−A	P1−m4−P7−M9	Minor−major