5-limit misty (12p&51p) notation

herman.miller
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5-limit misty (12p&51p) notation

Misty [<3 5 6] <0 -1 4]> is one of the few 5-limit rank 2 temperaments that would benefit from using schisma accents. I worked out a chart of 5-limit notation for misty[99] and I was able to notate most pitches with 5-limit sagittals with two exceptions. For those two pitches (a quarter tone below G and a quarter tone above A), I'm using to represent 250/243, which as a tempered interval is 48.5 cents in TOP misty. I think this is one case where the schisma accents may be better than the alternative of finding new Sagittal accidentals for all the gaps in between the unaccented Sagittals. This could also be a potential notation for 99-EDO if we don't already have one, and it would make a good keyboard layout for 99-EDO as well.

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Dave Keenan
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Re: 5-limit misty (12p&51p) notation

I see that Misty is not in Middle Path. And the mapping given in the Xen Wiki
https://en.xen.wiki/w/Misty_family
is the far more complex:
[⟨3 0 26],
⟨0 1 -4]⟩

Comma: 67108864/66430125 = [26 -12 -3⟩
POTE generator: ~3/2 = 703.111
Map: [<3 0 26|, <0 1 -4|]
EDOs: 12, 51, 63, 75, 87, 99, 285, 384

So then I found that Graham has the mapping you give.
http://x31eq.com/cgi-bin/rt.cgi?ets=12+51&limit=5
[ ⟨ 3 5 6 ]
⟨ 0 -1 4 ] ⟩

TE Generator Tunings (cents)
⟨399.8596, 96.8546]

Does anyone have the secret decoder ring that shows why "Misty" is an appropriate name for one of these mappings and not the other. Or can you convince us that they are somehow equivalent? What is the relationship between the 97 cent and 703 cent generators that makes them so?

They both agree that the comma [26, -12, -3⟩ is being tempered out.

Dave Keenan
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Re: 5-limit misty (12p&51p) notation

OK. I see that the Wiki entry is simply wrong. The given mapping does not match the given generator. Both mappings agree that the period is approx 1/3-octave (approx 400 cents). If prime 3 is 1 gen and 0 periods then the generator must be 1903 cents, not 703 cents. Or if the gen is 703 cents, the mapping of 3 must be 1 gen minus 3 periods. And prime 5 must be 14 periods minus 4 gens.

It seems best to me, to fix it by changing the mapping from
Map: [<3 0 26|, <0 1 -4|]
to
Map: [<3 -3 14|, <0 1 -4|]

All the other members of the family are wrong in the same way. These errors were present in the original article from Gene. Am I missing something?

herman.miller
Posts: 33
Joined: Sun Sep 06, 2015 8:27 am

Re: 5-limit misty (12p&51p) notation

[⟨3 0 26], ⟨0 1 -4]⟩ and [ ⟨ 3 5 6 ] ⟨ 0 -1 4 ] ⟩ are equivalent -- the wedge product for both is <<3 -12 -26]] -- but [<3 -3 14|, <0 1 -4|] has a wedge product of <<3 -12 -2]] so there must be some mistake. I'd leave the mapping and correct the generator size to 1903 cents.

7-limit misty (which has the same TOP tuning) was on Gene's "114 7-limit temperaments" list from 2004 with the mapping that agrees with the 5-limit version I'm using. The earliest mention of 5-limit misty I found is from 2002, without any specific mapping.
Number 38 {3136/3125, 5120/5103} Misty

[3, -12, -30, -26, -56, -36] [[3, 5, 6, 6], [0, -1, 4, 10]]
TOP tuning [1199.661465, 1902.491566, 2787.099767, 3368.765021]
TOP generators [399.8871550, 96.94420930]
bad: 53.622498 comp: 12.585536 err: .338535

Dave Keenan
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Re: 5-limit misty (12p&51p) notation

herman.miller wrote: Tue Mar 23, 2021 1:23 pm [⟨3 0 26], ⟨0 1 -4]⟩ and [ ⟨ 3 5 6 ] ⟨ 0 -1 4 ] ⟩ are equivalent -- the wedge product for both is <<3 -12 -26]] -- but [<3 -3 14|, <0 1 -4|] has a wedge product of <<3 -12 -2]] so there must be some mistake. I'd leave the mapping and correct the generator size to 1903 cents.
You've just shown the usefulness of the wedge product — its independence of the choice of generator.

I messed up. For the 703 cent generator it should be:
Map: [<3 3 14|, <0 1 -4|]

Wedge product: ⟨⟨3×1 - 3×0, 3×-4 - 14×0, 3×-4 - 1×14]] = ⟨⟨3 -12 -26]]

Dave Keenan
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Re: 5-limit misty (12p&51p) notation

But for an octave-repeating notation, we only care about the number of periods modulo the number per octave, so the mapping we care about is just the octave-equivalent mapping:

3 5
<0 -1] 400c
<1 -4] 703c

So we have 3 chains of fifths. We can notate one chain of up to 35 notes in the conventional manner, and we only need one up/down pair of accidentals to switch to the other 2 chains. Isn't that just and so that C:E is a tempered 5/4?

herman.miller
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Joined: Sun Sep 06, 2015 8:27 am

Re: 5-limit misty (12p&51p) notation

You could use and up to a point. If you want to notate the full 99-note MOS, you'll need triple sharps and triple flats for some of the notes, since E is 4 generators above C, which is 4 generators above E. Here's what that would look like with single and double sharps. You could always use the double-comma symbols ( and ) to fill in the gaps.

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Dave Keenan
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Re: 5-limit misty (12p&51p) notation

Ah yes. Good point. And good idea to use the 25-comma symbols and  . We almost get the 87 note MOS without them.

But I see that gives a notation that is highly non-monotonic in nominals (a bad thing), unlike your original notation.

Here's the 99edo notation George and I agreed on back in 2006.
See viewtopic.php?p=2740#p2740

deg99 symbol semantics
1 245C, i.e. 49C 5C (secondary comma)
2 5C
3 49C
4 25C, i.e. 5C + 5C
5 5:49M, i.e. 49C + 5C

herman.miller
Posts: 33
Joined: Sun Sep 06, 2015 8:27 am

Re: 5-limit misty (12p&51p) notation

That works reasonably well for 5-limit misty, although 7-limit misty [<3 5 6 6] <0 -1 4 10]> would map as (-5, +21), but that differs from (+3, -12) by only 0.06 cents which is negligible. Now if we take as the 23-comma 736/729, we can use a 2.3.5.23-limit temperament with 23 mapped to (+16, -10), and use 46/45 for (+3, -12).

I still like the mnemonic simplicity of for 5-limit misty, but something like or also seems reasonable. The one thing that doesn't fit with 99-EDO is the (-4, +17) that I was notating as , which is the same 99-EDO pitch as (+4, -16) .

Dave Keenan
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Re: 5-limit misty (12p&51p) notation

Yes. Having gone through the exercise, I agree that your original Misty notation is a good one. But I think your latest suggestion of is better, because it is just as memorable and avoids the extra width of the accents, and avoids having a wider symbol be a smaller alteration due to the fact that the accent points in the opposite direction to the arrow.