22edo/Vector's approach

This page documents Vector's info on 22EDO, for the Tuning of the Year project.

Notes and intervals

Intro

22edo contains 22 notes per octave, and thus 22 distinct intervals within the octave to use in your music.

Zarlino (intense diatonic) is the basic diatonic for 22edo, as with 15edo, so that the same porcupine-based notation may be used for both. The scale has two different types of whole tones, and was chosen so that the basic major and minor triads would represent 4:5:6 and its minor counterpart.

However, the main scale that will be used is jaric (pajara[10]), which has 10 interval names taken from Leriendil's decatonic system mixed with standard interval names.

Both scales have the same chroma size (and moreso, thirds, fourths, fifths, and sixths are very similar between the systems, with an allowance for the special notation of wolf intervals in diatonic), making interval logic pretty easy to apply to both.

Name	Type	Degree	Cents	Latitude	Approximate Ratios	Note (Diatonic)	Note (Pajara)
perfect unison	Perfect consonance	0	0	(unison)	1/1	C	0
augmented unison	Primary dissonance	1	54.5	-	25/24, 81/80, 36/35, 33/32	C#	0#
minor second	Primary dissonance	2	109.1	-	16/15, 15/14	Db	1
major second	Primary dissonance	3	163.6	-	12/11, 11/10, 10/9	D	1#
minor unilatus	Secondary consonance	4	218.2	extraslendric (35°)	9/8, 8/7	D#	2
major unilatus	Secondary consonance	5	272.7	subminor (21°)	7/6	Ebb	2#
minor third	Imperfect consonance	6	327.3	supraminor (7°)	6/5, 11/9	Eb	3b
major third	Imperfect consonance	7	381.8	submajor (7°)	5/4	E	3
diminished fourth	Secondary dissonance	8	436.4	supermajor (21°)	9/7, 14/11	E#	4b
perfect fourth	Perfect consonance	9	490.9	extraslendric (35°)	4/3	F	4
augmented fourth / diminished median	Secondary dissonance	10	545.5	-	15/11, 11/8	F#	4#
perfect median	(See note)	11	600.0	-	10/7, 7/5	Gbb	5
diminished fifth / augmented median	Secondary dissonance	12	654.5	-	16/11, 22/15	Gb	6b
perfect fifth	Perfect consonance	13	709.1	(fifth)	3/2	G	6
augmented fifth	Secondary dissonance	14	763.6		11/7, 14/9	G#	7b
minor sixth	Imperfect consonance	15	818.2		8/5	Ab	7
major sixth	Imperfect consonance	16	872.7		18/11, 5/3	A	7#
minor antilatus	Secondary consonance	17	927.3		12/7	A#	8b
major antilatus	Secondary consonance	18	981.8		7/4, 16/9	Bbb	8
minor seventh	Secondary dissonance	19	1036.4		9/5, 20/11, 11/6	Bb	8#
major seventh	Secondary dissonance	20	1090.9		28/15, 15/8	B	9
diminished octave / augmented seventh	Primary dissonance	21	1145.5		48/25, 64/33, 35/18	B#	9#
perfect octave	Perfect consonance	22	1200.0		2/1	C	0

While the perfect median is, by itself, a discordant interval (as it is the same interval as the 12edo tritone), its key structural properties in 22edo, as well as being both 7/5 and 10/7 and thus found in the harmonic 4:5:6:7 chord, elevate it to being a form of consonance in certain contexts, especially in harmonic tetrads.

Just intonation

Microtonal theorists and composers like to use just intonation as a basis for their theory. Meaning that they think of harmony in terms of either just intonation or some abstraction away from it. Just intonation is the tuning where all intervals are pure integer ratios, which is easy to tune, and provides a distinct harmonious sound. The most important ratios of just intonation are 2/1, 3/2, and 4/3, along with to a lesser extent 5/3 and 5/4. These are the ratios that are most likely to be picked up by historical musical cultures, and explains the convergence on pentatonic and heptatonic scale forms.

However, just intonation has caught modern microtonal theorists' attention for a different reason: the idea of stacking (multiplying or dividing) "prime" intervals to obtain more complex intervals. Stacking is multiplying, but it looks like adding on paper because pitch is logarithmic.

The decatonic scale

The 10-form (represented as pajara[10] here) is arguably a more intuitive categorization scheme for intervals than the 7-form we're used to. While most microtonal systems simply add new accidentals and interval qualities, it's best to think of 22edo as adding three distinct interval ordinal categories "in between the gaps" of the standard 7.

Unilatus (1lt)

From the Latin for "carried once", this refers to the separation of roughly one 5edo-step, or two decatonic steps, and sits between the standard seconds and thirds. The minor unilatus in 22edo is the 4-step interval representing 8/7 and 9/8. Meanwhile, the major unilatus represents 7/6. Two unilati make a fourth, just as two thirds make a fifth.

Antilatus (4lt)

The antilatus is the separation of roughly four 5edo-steps, or eight decatonic steps, and sits roughly between the standard sixths and sevenths. The major antilatus represents 7/4, and the minor antilatus represents 12/7. The antilatus essentially gives the simplest 7-limit intervals their own interval category, much as the simplest 5-limit intervals do, properly cementing 22edo as a 7-limit system.

Median (Med)

The perfect median is the interval of exactly 600 cents - that is, the perfect semioctave. The median splits the octave in two, reflecting the tritone as an intuitive interval category even in heptatonic systems, where the fourth and the fifth intersect to create it. But here, it is its own degree. Antilati are separated from thirds by the perfect median, and same with sixths from unilati. The median also solves the pesky 14/11 problem in heptatonic schemes: either 11/8 is a median, in which case the separation between it and a 7/4 antilatus (that is, 14/11) is a third, or it is a fourth, in which case 14/11 is also a fourth - however, the diminished fourth is the basic decatonic "imperfect" fourth, rather than the augmented fourth, so it is less of a problem.

Interval arithmetic

Interval arithmetic can be applied to decatonic intervals. Here is an isomorphic interval table to aid with it.

Degree	Quality
	-2	-1	0	1	2
1sn	-	P1sn	A1sn	AA1sn	3A1sn
2nd	d2nd	m2nd	M2nd	A2nd	AA2nd
1lt	d1lt	m1lt	M1lt	A1lt	AA1lt
3rd	d3rd	m3rd	M3rd	A3rd	AA3rd
4th	dd4th	d4th	P4th	A4th	AA4th
Med	ddMed	dMed	PMed	AMed	AAMed
5th	dd5th	d5th	P5th	A5th	AA5th
6th	dd6th	d6th	m6th	M6th	A6th
4lt	dd4lt	d4lt	m4lt	M4lt	A4lt
7th	dd7th	d7th	m7th	M7th	A7th
8ve	3d8ve	dd8ve	d8ve	P8ve	A8ve

Some key rules to keep track of:

• 4th + 1lt of opposite quality (perfect 4th + minor 1lt, diminished 4th + major 1lt) = perfect 5th
• 3rd + 3rd of opposite quality = perfect 5th
• Perfect Med + 2nd = 5th (minor 2nd -> perfect 5th, major 2nd -> augmented 5th)
• Perfect Med + 1lt = 6th of same quality
• Perfect Med + 3rd = 4lt of same quality (e.g. major 3rd -> major 4lt)
• Perfect Med + 4th = 7th (diminished 4th -> minor 7th, perfect 4th -> major 7th)
• Perfect Med + Med of opposite quality = perfect 8ve

Note that as a rule, the 3-letter abbreviation for an interval is always used. Otherwise, for example, "4" would be ambiguous between 4lt (antilatus) and 4th (fourth).

Triads and tetrads

Triads bounded by P5th

Tertian triads

There are two types of tertian triads where the bounding interval is a perfect fifth, and the third is found in a pajara scale.

Name	1	2	Bounding interval	Edostep	Chart
Major triad	Major 3rd	Minor 3rd	Perfect 5th	[0 7 13]	█▒▒▒▒▒▒▓▒▒▒▒▒▓▒▒▒▒▒▒▒▒█
Minor triad	Minor 3rd	Major 3rd	Perfect 5th	[0 6 13]	█▒▒▒▒▒▓▒▒▒▒▒▒▓▒▒▒▒▒▒▒▒█

Unilatal triads

There are four types of triads where the bounding interval is a perfect 5th, and the middle interval is either a unilatus or a fourth found in a pajara scale. These are suspended triads, and notably, the diatonic "supermajor" and "subminor" thirds are categorized under this group.

Name	1	2	Bounding interval	Edostep	Chart
Sus P4th triad	Perfect 4th	Minor 1lt	Perfect 5th	[0 9 13]	█▒▒▒▒▒▒▒▒▓▒▒▒▓▒▒▒▒▒▒▒▒█
Sus d4th triad	Diminished 4th	Major 1lt	Perfect 5th	[0 8 13]	█▒▒▒▒▒▒▒▓▒▒▒▒▓▒▒▒▒▒▒▒▒█
Sus M1lt triad	Major 1lt	Diminished 4th	Perfect 5th	[0 5 13]	█▒▒▒▒▓▒▒▒▒▒▒▒▓▒▒▒▒▒▒▒▒█
Sus m1lt triad	Minor 1lt	Perfect 4th	Perfect 5th	[0 4 13]	█▒▒▒▓▒▒▒▒▒▒▒▒▓▒▒▒▒▒▒▒▒█

Tetrads with P5th

Harmonic tetrads

These represent the conventional structure of a 4:5:6:7 harmonic tetrad, but generalized such that any middle interval (and thus any top interval) can be used. They take the form of a fifth-bounded triad, with an additional note a perfect median above the middle note. This is the characteristic kind of chord found in pajara temperament, and can also be thought of the stacking of a fifth-bounded triad and a fourth-bounded triad. Non-tertian versions are not shown.

Name	Type	1	2	3	4	Bounding interval 1	Bounding interval 2	Bounding interval 3	Edostep	Chart
Major harmonic tetrad	Tertian	Major 3rd	Minor 3rd	Major 1lt	Minor 1lt	Perfect 5th	Major 4lt	Perfect 8ve	[0 7 13 18]	█▒▒▒▒▒▒▓▒▒▒▒▒▓▒▒▒▒▓▒▒▒█
Minor harmonic tetrad	Tertian	Minor 3rd	Major 3rd	Minor 1lt	Major 1lt	Perfect 5th	Minor 4lt	Perfect 8ve	[0 6 13 17]	█▒▒▒▒▒▓▒▒▒▒▒▒▓▒▒▒▓▒▒▒▒█

Diatonic tetrads

These represent the conventional structure of an 8:10:12:15 diatonic major tetrad, but generalized such that any middle interval (and thus any top interval) can be used. They take the form of a fifth-bounded triad, with an additional note a perfect fifth above the middle note. The sus4 diatonic tetrad is the same as the triad version up to inversion.

Name	Type	1	2	3	Bounding interval 1	Bounding interval 2	Edostep	Chart
Sus d4th diatonic tetrad	Unilatal	Diminished 4th	Major 1lt	Diminished 4th	Perfect 5th	Diminished 8ve	[0 8 13 21]	█▒▒▒▒▒▒▒▓▒▒▒▒▓▒▒▒▒▒▒▒▓█
Major diatonic tetrad	Tertian	Major 3rd	Minor 3rd	Major 3rd	Perfect 5th	Major 7th	[0 7 13 20]	█▒▒▒▒▒▒▓▒▒▒▒▒▓▒▒▒▒▒▒▓▒█
Minor diatonic tetrad	Tertian	Minor 3rd	Major 3rd	Minor 3rd	Perfect 5th	Minor 7th	[0 6 13 19]	█▒▒▒▒▒▓▒▒▒▒▒▒▓▒▒▒▒▒▓▒▒█
Sus M1lt diatonic tetrad	Unilatal	Major 1lt	Diminished 4th	Major 1lt	Perfect 5th	Major 4lt	[0 5 13 18]	█▒▒▒▒▓▒▒▒▒▒▒▒▓▒▒▒▒▓▒▒▒█
Sus m1lt diatonic tetrad	Unilatal	Minor 1lt	Perfect 4th	Minor 1lt	Perfect 5th	Minor 4lt	[0 4 13 17]	█▒▒▒▓▒▒▒▒▒▒▒▒▓▒▒▒▓▒▒▒▒█

Other triads

Some of these have names based on temperaments, provided by Stalefleas.

Name	Edostep	Chart
-	[0 4 8]	█▒▒▒▓▒▒▒▓▒▒▒▒▒▒▒▒▒▒▒▒▒█
-	[0 4 9]	█▒▒▒▓▒▒▒▒▓▒▒▒▒▒▒▒▒▒▒▒▒█
-	[0 4 10]	█▒▒▒▓▒▒▒▒▒▓▒▒▒▒▒▒▒▒▒▒▒█
-	[0 4 11]	█▒▒▒▓▒▒▒▒▒▒▓▒▒▒▒▒▒▒▒▒▒█
-	[0 4 12]	█▒▒▒▓▒▒▒▒▒▒▒▓▒▒▒▒▒▒▒▒▒█
-	[0 5 9]	█▒▒▒▒▓▒▒▒▓▒▒▒▒▒▒▒▒▒▒▒▒█
Orwell triad	[0 5 10]	█▒▒▒▒▓▒▒▒▒▓▒▒▒▒▒▒▒▒▒▒▒█
Utonal diminished triad	[0 5 11]	█▒▒▒▒▓▒▒▒▒▒▓▒▒▒▒▒▒▒▒▒▒█
Orwell minor triad	[0 5 12]	█▒▒▒▒▓▒▒▒▒▒▒▓▒▒▒▒▒▒▒▒▒█
-	[0 5 14]	█▒▒▒▒▓▒▒▒▒▒▒▒▒▓▒▒▒▒▒▒▒█
-	[0 6 10]	█▒▒▒▒▒▓▒▒▒▓▒▒▒▒▒▒▒▒▒▒▒█
Otonal diminished triad	[0 6 11]	█▒▒▒▒▒▓▒▒▒▒▓▒▒▒▒▒▒▒▒▒▒█
Keemic triad / diminished triad	[0 6 12]	█▒▒▒▒▒▓▒▒▒▒▒▓▒▒▒▒▒▒▒▒▒█
Sensaminor triad	[0 6 14]	█▒▒▒▒▒▓▒▒▒▒▒▒▒▓▒▒▒▒▒▒▒█
-	[0 6 15]	█▒▒▒▒▒▓▒▒▒▒▒▒▒▒▓▒▒▒▒▒▒█
-	[0 7 11]	█▒▒▒▒▒▒▓▒▒▒▓▒▒▒▒▒▒▒▒▒▒█
Orwell major triad	[0 7 12]	█▒▒▒▒▒▒▓▒▒▒▒▓▒▒▒▒▒▒▒▒▒█
Magic triad / augmented triad	[0 7 14]	█▒▒▒▒▒▒▓▒▒▒▒▒▒▓▒▒▒▒▒▒▒█
Magic minor triad	[0 7 15]	█▒▒▒▒▒▒▓▒▒▒▒▒▒▒▓▒▒▒▒▒▒█
-	[0 7 16]	█▒▒▒▒▒▒▓▒▒▒▒▒▒▒▒▓▒▒▒▒▒█
-	[0 8 12]	█▒▒▒▒▒▒▒▓▒▒▒▓▒▒▒▒▒▒▒▒▒█
Sensamajor triad	[0 8 14]	█▒▒▒▒▒▒▒▓▒▒▒▒▒▓▒▒▒▒▒▒▒█
Magic major triad	[0 8 15]	█▒▒▒▒▒▒▒▓▒▒▒▒▒▒▓▒▒▒▒▒▒█
Sensamagic triad	[0 8 16]	█▒▒▒▒▒▒▒▓▒▒▒▒▒▒▒▓▒▒▒▒▒█
-	[0 8 17]	█▒▒▒▒▒▒▒▓▒▒▒▒▒▒▒▒▓▒▒▒▒█
-	[0 9 14]	█▒▒▒▒▒▒▒▒▓▒▒▒▒▓▒▒▒▒▒▒▒█
-	[0 9 15]	█▒▒▒▒▒▒▒▒▓▒▒▒▒▒▓▒▒▒▒▒▒█
-	[0 9 16]	█▒▒▒▒▒▒▒▒▓▒▒▒▒▒▒▓▒▒▒▒▒█
-	[0 9 17]	█▒▒▒▒▒▒▒▒▓▒▒▒▒▒▒▒▓▒▒▒▒█
Archytas triad	[0 9 18]	█▒▒▒▒▒▒▒▒▓▒▒▒▒▒▒▒▒▓▒▒▒█