22edo interval names

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Dave Keenan
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22edo interval names

Postby Dave Keenan » Fri Sep 18, 2015 12:18 am

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:b: :b:2 flat second sm2 sub-minor second
D:b::/|: :/|::b:2 pai-flat second m2 minor second
D:\!: :\!:2 pao second M2 major second
D 2 second SM2 super-major second
E:b: :b:3 flat third sm3 sub-minor third
E:b::/|: :/|::b: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:b: :b:6 flat sixth sm6 sub-minor sixth
A:b::/|: :/|::b:6 pai-flat sixth m6 minor sixth
A:\!: :\!:6 pao sixth M6 major sixth
A 6 sixth SM6 super-major sixth
B:b: :b:7 flat seventh sm7 sub-minor seventh
B:b::/|: :/|::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:b::/|: :/|::b:2 pai-flat second m2 minor second
D:\!: :\!:2 pao second M2 major second
E:b::\!: :\!::b:3 pao-flat third dim3 diminished third
E:b: :b:3 flat third sm3 sub-minor third
E:b::/|: :/|::b: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:b::/|: :/|::b: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:b::/|: :/|::b: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:b::/|: :/|::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

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Dave Keenan
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Re: 22edo interval names

Postby Dave Keenan » Fri Sep 18, 2015 4:50 pm

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
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Joined: Mon Sep 21, 2015 9:27 pm

Re: 22edo interval names

Postby William Lynch » Mon Sep 21, 2015 10:20 pm

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

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cam.taylor
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Re: 22edo interval names

Postby cam.taylor » Tue Sep 22, 2015 9:13 am

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
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Joined: Mon Sep 21, 2015 9:27 pm

Re: 22edo interval names

Postby William Lynch » Tue Sep 22, 2015 11:05 am

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

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cam.taylor
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Re: 22edo interval names

Postby cam.taylor » Tue Sep 22, 2015 4:26 pm

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:!)::b:-D-F:/|\:, however the effective temperaments in play will change how some intervals and chords relate to each other.

William Lynch
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Joined: Mon Sep 21, 2015 9:27 pm

Re: 22edo interval names

Postby William Lynch » Tue Sep 22, 2015 9:31 pm

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?

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cam.taylor
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Re: 22edo interval names

Postby cam.taylor » Wed Sep 23, 2015 8:58 am

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.


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