## Yer: a Sagittal study

Dave Keenan
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### Re: Yer: a Sagittal study

I think it's whenever the closest 12edo pitch differs from the one notated, when you ignore all but the conventional accidentals.

That's certainly an option. But it seems a little silly to do that for D#-49 when you could just write 51. I'd probaby let the offset go to 59 or 60 before doing that. And even then, maybe it'd be better to respell the note to be an alteration of the nearest 12edo note, to avoid the problem.

That "D#-40" kind of thing would be a better way to deal with cents on Revo, than either using cents greater than 60, or expecting recognition of the number of shafts as giving sharps and flats. But I still think we should recommend that Evo be used when cents are given, so as to minimise such occurrences.

Where I said cents "above the staff" in the education material should be changed to "above or below the staff".

cmloegcmluin
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### Re: Yer: a Sagittal study

It occurred to me this morning, now that I've been studying regular temperament theory for many months, to look into what a temperament that tempers out the commas that figure prominently in Yer would look like.

If you temper out only the Blumeyer comma, you get this 2.11.13.17.19 subgroup mapping:

[ ⟨	1	0	0	0	-7	]
⟨	0	1	0	0	1	]
⟨	0	0	1	0	1	]
⟨	0	0	0	1	1	] ⟩

Expressed as a join of ETs, that's 13&113&137&194:
http://x31eq.com/cgi-bin/rt.cgi?ets=194 ... 1_13_17_19
Which helps to confirm that my choice of 137-ET as an ET to notate the Yer EFG with was reasonable.
So one ~19 is up one each of the ~11, ~13, and ~17 here.
So that's "Blumeyer temperament"!

Now if I temper out the Blumeyer comma and the yama comma
(and therefore also the blume comma; in fact, the canonical form of the comma-basis appears to be blume & yama, with the Blumeyer comma being a linear combination of them), then you get a rank-3 temperament, with mapping:

[ ⟨	1	0	0	11	4	]
⟨	0	1	0	-2	-1	]
⟨	0	0	1	0	1	] ⟩

Again that's still in the 2.11.13.17.19 subgroup. And this one is the join of 13&24&33:
http://x31eq.com/cgi-bin/rt.cgi?ets=33_ ... 1_13_17_19
With generators 1200.4457, 4149.8305, 4442.6991 (that 1st generator modulo the period is 548.4934 and the 2nd generator is 841.362). So one ~17 is down two ~11, and one ~19 is down one ~11 and up a ~13. Cool! This I guess you'd call "Yer temperament" then.

I realize it doesn't make too much sense to post this stuff here without addressing the issue of notation. Reviewing the material on the wiki at present about Sagittal notations of regular temperaments, I'm not sure there's much advice I can really leverage here. These temperaments lack fifth-like generators. So I wonder if this suggests that something more like a prime factor notation would be appropriate for temperaments like this, but with the symbols standing for the approximated primes.

cmloegcmluin
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### Re: Yer: a Sagittal study

It looks like one good way to go with this Yer temperament would be a 13-note MV3 scale, a 7L 4M 2s, with the pattern LMLLMLsLMLMLs, where L ≈ 103.4¢, M ≈ 85.9¢, and s ≈ 66.2¢. That gives you a lattice like this:

 - (-2,2) 11·13·17·19 13·19/11 585.4¢ (-1,2) 13·19   1133.7¢ (0,2) 11·13·19   482.0¢ (1,2) 11·13²   1030.3¢ (-3,1) 17·19   396.1¢ (-2,1) 13·17 11·17·19 944.4¢ (-1,1) 11·13·17 19 292.7¢ (0,1) 13 11·19 841.0¢ (1,1) 11·13   189.3¢ (-3,0) 17·19/13 17/11 755.1¢ (-2,0) 17   103.4¢ (-1,0) 11·17   651.7¢ (0,0) 1 11²·17 0¢ -

I found this by first simply transferring the original Yer EFG into this temperament. That gave me a really similar looking lattice, with a rows of lengths 4, 5, and 4 as well, with the only difference being that the empty cells were in the opposite corners as you see above. The problem with that lattice, though, is that the resulting scale had not three, not four, but five different step sizes: 37.2¢, 66.2¢, 85.9¢, 103.4, and 189.3¢. After a bit of fidgeting (I don't yet understand the underlying theory of MV3 scales well enough to make deeply intentional decisions) I managed to find the above scale. I just deleted the cells for 11 and 13.17.19 that came from the EFG, and replaced them with the cells for 11.13² and 17/11.

My gut tells me that it is distributionally even, because the s's separate two symmetrical chunks of L's and M's with the greatest weights of the L's positioned opposite the s's, to balance things out. (Edit: yes, it appears it qualifies. Found a rotation of it here as "LLMLsLMLMLsLM".) (Edit: no wait, it's been pointed out to me that it doesn't qualify as MV3, because that pattern is found in the "Conditional (MM=Ls)" section, and while it's quite close, L + s = 103.4 + 66.2 = 169.6 but 169.6/2 = 84.8, not 85.9 which is my M.)

So the scale is:

 ! yerMV3.scl
!
yer temperament 13-note scale
13
!
103.4
189.3
292.7
396.1
482.0
585.4
651.7
755.1
841.0
944.4
1030.3
1133.7
(2/1)

I guess if you really wanted, you could treat that 11th harmonic like a fifth. Perhaps I could re-center the coordinates so that the center cell was the (0,0).