22edo interval names

[Dave Keenan]

Here is a way of having our cake and eating it too. By which I mean that, for different purposes we can name intervals
(a) according to the notation of the pitches (in a chain-of-best-fifths notation), or
(b) according to the sound of the interval or dyad (in isolation),
and the two names do not conflict.

Mnemonic for "pai" is "Pental comma hIGH". Mnemonic for "pao" is "Pental comma dOWn".

Pitch				22-edo
up				intervals
from
C		Notation-based name		Sound-based name
--------------------------------------------------------------------
C		1	unison			P1	perfect unison
D		2	flat second		sm2	sub-minor second
D		2	pai-flat second		m2	minor second
D		2	pao second		M2	major second
D		2	second			SM2	super-major second
E		3	flat third		sm3	sub-minor third
E		3	pai-flat third		m3	minor third
E		3	pao third		M3	major third
E		3	third			SM3	super-major third
F		4	fourth			P4	perfect fourth
F		4	pai fourth		S4	super fourth
F		4	pao-sharp fourth	Aug4	augmented fourth
G		5	pao fifth		s5	sub fifth
G		5	fifth			P5	perfect fifth
A		6	flat sixth		sm6	sub-minor sixth
A		6	pai-flat sixth		m6	minor sixth
A		6	pao sixth		M6	major sixth
A		6	sixth			SM6	super-major sixth
B		7	flat seventh		sm7	sub-minor seventh
B		7	pai-flat seventh	m7	minor seventh
B		7	pao seventh		M7	major seventh
B		7	seventh			SM7	super-major seventh
C		8	octave			P8	perfect octave

If required for clarification, notation-based interval names with no accidentals can be preceded by the word "natural".

The notation-based names are derived directly from the pitch names up from C. The derivation of the sound-based names is described here: http://dkeenan.com/Music/EdoIntervalNames.pdf

Here are some alternative names corresponding to alternative pitch spellings.

Pitch				22-edo
up				intervals
from
C		Notation-based name		Sound-based name
--------------------------------------------------------------------
C		1	unison			P1	perfect unison
C		1	pai unison		S1	super unison
D		2	pai-flat second		m2	minor second
D		2	pao second		M2	major second
E		3	pao-flat third		dim3	diminished third
E		3	flat third		sm3	sub-minor third
E		3	pai-flat third		m3	minor third
E		3	pao third		M3	major third
F		4	pao fourth		s4	sub fourth
F		4	fourth			P4	perfect fourth
F		4	pai fourth		S4	super fourth
G		5	pao-flat fifth		dim5	diminished fifth
G		5	pao fifth		s5	sub fifth
G		5	fifth			P5	perfect fifth
G		5	pai fifth		S5	super fifth
A		6	pai-flat sixth		m6	minor sixth
A		6	pao sixth		M6	major sixth
A		6	sixth			SM6	super-major sixth
A		6	pao-sharp sixth		Aug6	augmented sixth
B		7	pai-flat seventh	m7	minor seventh
B		7	pao seventh		M7	major seventh
C		8	pao octave		s8	sub octave
C		8	octave			P8	perfect octave

[Dave Keenan]

For some other ideas on microtonal interval naming, see Kite Giedraitis' colour notation here:
http://xenharmonic.wikispaces.com/Kite's+color+notation
and in part II of his book, Alternative Tunings: Theory, Notation and Practice, including the alt-tuner manual, here:
http://www.tallkite.com/AlternativeTunings.html

[William Lynch]

Ok so cam's approach is interesting and I like it. Is this the interval name for 22 EDO now? In this, I do not think it's needed to specify major and minor with pai and pao right?

pao2
min2
pai2
pao2
maj2
min3
pai3
pao3
maj3
perfect? fourth
vai fourth / waoaug fourth
? fourth is it augmented?
vao fifth/ waidim fifth
perfect? fifth
min6
pai6
pao6
maj6
min7
pai7
pao7
maj7
octave

[cam.taylor]

Not quite.

If we're only using the names for the 22 Sagittal set (i.e. the only accidentals in our mixed Sagittal set being , , ), then we might have:

1
minor 2
pai(minor) 2
pao(major) 2
major 2
minor 3
pai(minor) 3
pao(major) 3
major 3/pao 4
4
pai 4/dim 5
paoaug4/paidim 5
aug 4/pao 5
5
pai 5/minor 6
pai(minor) 6
pao(major) 6
major 6
minor 7
pai(minor) 7
pao(major) 7
major 7
8

[William Lynch]

Ok makes sense but can it be understood in a universal sense still? The difference between Vai fourth and Pai fourth?

[cam.taylor]

We would use vai and vao if you wanted the 11-limit implications, but the pai/pao notation only really translates 5-limit information. If you wanted to mix and match you could, as the two commas are of course equal here, while tai/tao disappears. However, while using both vai/vao and pai/pao pairs you could easily translate information across tunings, they mean the same thing in 22edo, so you only need one pair. The pair recommended on the Sagittal article is pai/pao, which will link 22 with other common EDOs like 34, 41, 53, 72.

If you want to be able to translate your intervals and chords to ANY tuning, I'd recommend notating in JI, so that a 4:5:6:7:9:11 on C would be notated C-E-G-B-D-F, however the effective temperaments in play will change how some intervals and chords relate to each other.

[William Lynch]

But I imagine as well that somebody who knew both 24 and 22 in sagittal could tell that pai4 and vao4 can both be interpreted 11/8 but written differently, right? I mean, I don't really mind writing everything in closest matching JI but I'm wondering how odd this makes something to read. OR does JI notating make it easier to interpret and read?

[cam.taylor]

As are tempered out in 24EDO, it's all written using , i.e. showcasing 2.3.11 rather than 2.3.5. Btw 11:8 would be vai4, not vao4.