Intervals and chords for 15 edo?

William Lynch
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Intervals and chords for 15 edo?

Postby William Lynch » Wed Apr 20, 2016 3:38 am

How do I name intervals and chords in 15? Im using nominals as 5 edo on CDEGA with B and C equated and E and F equated.

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Dave Keenan
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Re: Intervals and chords for 15 edo?

Postby Dave Keenan » Thu Apr 21, 2016 12:26 pm

I've just copied and pasted from the Facebook thread, so it doesn't get lost.


Dave Keenan As you can read here:
http://dkeenan.com/Music/EdoIntervalNames.pdf
I would name the intervals of 15edo
P1 m2 M2 A2/d3 m3 M3 P4 A4 d5 P5 m6 M6 A6/d7 m7 M7
but that has nothing to do with how the pitches are notated.
I'll let Cam tell you how he'd use Sagispeak to name the intervals.

I note that I have responded to some of your posts on
viewtopic.php?p=204#p204
I also note that you won't have to keep checking the Sagittal forum if you go to your user control panel there and turn on notifications.
Like · Reply · Yesterday at 09:34
William Lynch
William Lynch Ok right I want to use sagispeak intervals as its consistent with the notation.
Like · Reply · 1 · Yesterday at 09:37 · Edited
Dave Keenan
Dave Keenan Cam, you might want to make use of the enharmonics F = E and B = C in generating Sagispeak names for the 15edo intervals.
Like · Reply · 1 · Yesterday at 09:38
Dave Keenan
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Cam Taylor
Cam Taylor I think it rather depends on what you're notating. I wouldn't limit myself to 5 nominals, but use all 7, even knowing that F#=G, B=C and E=F, A=Bb, etc, one can still notate a (rather large) major second and a (very small) minor third in different ways, even though they might sound the same, much like the major second and diminished third in 12edo, or augmented second and diminished third in 19.

The sagittal pdf says use 5edo nominals pls pai/pao, so that'd be the easiest way. Unless of course you want to spell the rather wide fifth as something other than a perfect fifth, where you might use 60edo notation.

So for me, the gamut of useful 15edo intervals might look like this:

*D perfect unison*, Eb minor second (0)
D/| pai unison, *E/|b pai minor second* (80)
D\!# pao augmented unison, *E\! pao major second* (160)
*E major second*, F minor third (240)
*F/| pai minor third* (320)
*F\!# pao major third* (400)
F# major third, *G perfect fourth*, Ab diminished fifth (480)
*G/| pai fourth*, A/|b pai diminished fifth (560)
G\!# pao augmented fourth, *A\! pao fifth* (640)
G# augmented fourth, *A perfect fifth*, Bb minor sixth (720)
*B/|b pai minor sixth* (800)
*B\! pao major sixth* (880)
B major sixth, *C minor seventh (960)*
*C/| pai minor seventh*, D/|b pai diminished unison (1040)
*C\!# pao major seventh*, D\! pao octave (1120)
C# major seventh, *D perfect octave* (0)

However the most commonly used intervals would probably be the ones marked with asterisks. Some on the list would rarely be used "on purpose", but would arise during regular harmonic and melodic progressions in a chromatic environment.
Like · Reply · 2 · 16 hours ago
Cam Taylor
Cam Taylor This gives you triad names like:
0-240-720 minor, sus (1st inversion)
0-320-720 pai (minor)
0-400-720 pao (major)
0-480-720 major, sus
0-240-480 diminished *****, sus2sus4
0-320-560 pai diminished, "harmonic diminished"
0-320-640 phai diminished
0-480-960 augmented *****, sus (2nd inversion)
0-400-880 pao augmented, major six (omit 5)
0-400-800 phao augmented

Because this is right on one edge of the diatonic spectrum, the diminished and augmented chords built from chains of 15edo fifths are not really much like regular dim and aug chords, but that's how they can be constructed

Notice 15 is such a small edo with such tight wraps that many chords and intervals become tempered together. Fun. 15 shares it's phao augmented chord [16:20:25], or stacked 5:4s, with 3N EDOs: 3, 6, 9, 12, 18, 21, 24, 27, 30, 33, 36, 39, 42.
Like · Reply · 1 · 17 hrs
William Lynch
William Lynch Cam cool though I don't get the phai and phao, why do we use these for more extreme categories?
Like · Reply · 23 hrs
Cam Taylor
Cam Taylor You don't have to, that's just the name ofthe JI [16:20:25] chord, as it would be notated D F\!# A\\!#, or
"D, F pao sharp, A phao sharp". "phao" is \\!, or a lowering by two syntonic commas, which is needed for identities involving 25.
25:16 is a phao augmented fifth (e.g. D A\\!#)
25:24 is a phao augmented unison (e.g. D D\\!#)
25:18 is a phao augmented fourth (e.g. D G\\!#)

and their inverses

32:25 is a phai diminished fourth (e.g. D G//|b)
48:25 is a phai diminished octave (e.g. D D//|b)
36:25 is a phai diminished fifth (e.g. D A//|b)
Like · Reply · 1 · Yesterday at 10:43

William Lynch
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Re: Intervals and chords for 15 edo?

Postby William Lynch » Thu Apr 21, 2016 2:10 pm

Thanks Dave. We need tables like this for all the EDO honestly but it's a good start. I'm not sure about chord names though. What does phao and phai mean here? But is this actually correct? Augmented and diminished still mean what they mean even in a 5n EDO where it means altering a chain of fifths by a sharp or flat making it enharmonically the same as a note from 5 EDO.

C augmented chord is C E G# which is enharmonic to C E A
C diminished chord is C Eb Gb which is enharmonic to C D E.

It's a little weird but then so are 5n edo anyway haha.

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Dave Keenan
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Re: Intervals and chords for 15 edo?

Postby Dave Keenan » Thu Apr 21, 2016 2:29 pm

William Lynch wrote:... What does phao and phai mean here? But is this actually correct? ... It's a little weird but then so are 5n edo anyway haha.

Indeed they are weird, so their notation and nomenclature is bound to be weird too.

I think Cam may have made an unwarranted assumption about phai :/ /|: and phao :\ \!: here. They aren't in any of the 5n pitch notations I gave. And just because pai :/|: and pao :\!: are a certain number of degrees of some edo, doesn't necessarily mean that phai :/ /|: and phao :\ \!: are twice that. And if they are not, we shouldn't use them. Pai :/|: and pao :\!: are inconsistent for the 5n's beyond 15edo anyway. They notate the best approximations to 5-limit minor thirds (5:6) correctly, but not 5-limit major thirds (4:5).


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