Notation for George Secor's High-Tolerance Temperament

[cmloegcmluin]

I've been working on updating the configuration files for Sagittal notation which Huygen-Fokker's Scala microtonal software uses, and there's one file which is giving me a particular bit of trouble: sag_htt.par, reproduced below:

Code: Select all

! Sagittal high-tolerance temperament notation. George Secor. 2003-Aug-28, updated 2020-Jun-14
!
max_abs_slope	7.5
c_exp_range	-6	10
nc_exp_range	-1	5
fifth_size	703.5787
oct_size	2/1
!
!Lower bound	Symbol	Value		Short
-70.385		(!)	704/729		w
-62.5		\!/	32/33		v
-55.0		(!)	704/729		w
-43.3		\!	80/81		\
-35.0		!)	63/64		t
-32.0		!/	54/55		k
-29.0		\!	80/81		\
-26.0		!)	63/64		t
-15.0
15.0		|)	64/63		f
26.0		/|	81/80		/
29.0		|\	55/54		y
32.0		|)	64/63		f
35.0		/|	81/80		/
43.3		(|)	729/704		m
55.0		/|\	33/32		^
62.5		(|)	729/704		m
70.385

There's something a bit fishy about it, I think: the repetition of the symbols and for discontinuous capture zones. I don't know what to make of it.

@Dave Keenan has suggested I plug it into Scala along with the scale it is meant to notate, which I believe I have correctly identified in the Scala scale archives and reproduced below:

Code: Select all

! secor29htt.scl
!
George Secor's 29-tone 13-limit high-tolerance temperament (5/4 & 7/4 exact)
 29
!
58.08980
97.04984
140.19633
179.15637
207.15739
265.24719
296.73557
347.35372
5/4
414.31478
472.40458
496.42131
554.51111
593.47114
633.37025
679.56197
703.57869
761.66849
800.62853
843.77502
882.73506
910.73608
7/4
992.84261
1050.93241
1089.89245
1117.89347
1175.98327
2/1

Dave points out that there is more than one size of fifth in there, and that it appears to be a well temperament.

The problem is that I don't feel super confident about the advanced harmonic theory that Dave and George grappled with to develop Sagittal. Maybe I'll get there one day! I just don't have enough of the mathematical and musical practice backgrounds to confidently notate this thing.

I was hoping that someone else might enjoy the puzzle of reverse-engineering this tuning in the form of its Sagittal notation, so that I could then confirm that Scala generates the correct notation for it using the configuration file we also maintain for that exact purpose. I know we have some puzzle lovers on this forum... come out, come out, wherever you are...

I have gone ahead and scooped up some useful resources for the endeavor:

https://yahootuninggroupsultimatebackup ... 53712.html
https://yahootuninggroupsultimatebackup ... .html#6889
https://yahootuninggroupsultimatebackup ... .html#7574

There's also this quote, from an unknown author, I found in a text file (not a post) from the same backup:

a fabulous
near-just, non-equal well temperament included in
the Scala freeware archive under the filename
secor29htt.scl, best represented in Sagittal High
Tolerance Temperament notation (SAHTT). Among many
virtues, this temperament allows otonal ogdoads to
odd limit 15 in six key signatures. It has three
chains of tempered "fifth" (near 3/2) intervals
plus one inserted note.

And this, which I lost track of where I found (sorry):

a "near-just" temperament which, as a 29-note set, uses
multiple chains of the mildly tempered fifths to offer a great variety
of pure or near-pure ratios.

I've attached the .mp3 file George mentions on one of these posts, which for whatever reason my code has failed to link up (vindicating Dave's insistence that not all possible files were processed properly). Quite a special little performance it is.

[volleo6144]

There's something a bit fishy about it, I think: the repetition of the symbols and for discontinuous capture zones. I don't know what to make of it.

And also , , , and

I don't either. Now we have capture zones with gaps in them. Great.

[cmloegcmluin]

Indeed...

One last thing we should say that could be helpful:

BTW, George agreed when I pointed out recently, that "high tolerance" is a misnomer, but too late to change the name. It is really a "low tolerance temperament" or equivalently a "high precision temperament".

[volleo6144]

A fourth message defining the fifth itself and noting a number of commas of the temperament appears here, which is linked to from the sagittal.org homepage.

In order for this temperament's commas to have a 12edo keyspan of zero, 8:11 has to be an augmented—not perfect—fourth ( = P1; = A1) and 8:13 has to be a major—not minor—sixth ( = P1; = A1). As such, and are considered to be P1s and and to be A1s in the discussion below. (And 4:5 and 4:7 have to be M3 and m7, but that's already unambiguous.)

The fifth (703.6¢, the closest non-b edo being 29) is defined as one-ninth of 13:504, which is a minor 38th. The difference between this and a Pythagorean augmented 37th is 28431:28672 (7:13C, a diminished second), which is therefore tempered out.

Three of the fifths are intended to represent 13:44 (which is a major 13th), which means that 351:352 (11:13k, which incidentally already has an Olympian-level exact representation: ) is tempered out as well.

As 5/4 and 7/4 are independent generators, no more commas appear to be intended, and we get a rank-4, 13-limit temperament, the TM basis of which is 11:13k (351:352) and 91:121k (363:364).

8:11 is tempered to a major third below 4:7, or 648:891.
8:13 is tempered to an apotome below 4:7, or 2187:3584.

The chain of fifths on 1/1 goes fourthward to the m7 (16/9) and fifthward to the M7 (243/128). (8 notes)
The chain of fifths on 5/4 goes fourthward to the M2 (10/9) and fifthward to the A5 (405/256). (7 notes)
The chain of fifths on 7/4 goes fourthward to the d6 (28672/19683 and 13/9) and fifthward to the P5 (189/128). (13 notes. This particular chain is particularly extensive, because the fourthward region of this chain is also where the 11- and 13-limit otonalities appear.)

There's an added note (296.73557¢, about half of the ninth-comma out from 19/16) that doesn't appear to have a 13-limit untempered version.

[Dave Keenan]

That's a masterful analysis. Thanks @volleo6144.

@cmloegcmluin, can you easily post the notation that Scala gives to it, alongside the cents/ratios?

-- Dave

[cmloegcmluin]

In Revo flavor, as George would have liked:

0		C
58.08980	C	D
97.04984	C	D
140.19633	D	C
179.15637	D	C
207.15739	D
265.24719	D	E
296.73557	E	D
347.35372	E	D
5/4		E	D
414.31478	E	F
472.40458	F	E
496.42131	F	E
554.51111	F	G
593.47114	F	G
633.37025	F	G
679.56197	G	F
703.57869	G
761.66849	G	A
800.62853	G	A
843.77502	A	G
882.73506	A	G
910.73608	A
7/4		B	A
992.84261	B	A
1050.93241	B	A
1089.89245	B	A
1117.89347	B	C
1175.98327	C	B
2/1		C

At a glance, that looks reasonable enough to me, in terms of the cents w/r/t the symbols. I only had a few minutes so I optimized for getting y'all the requested information. I'll read volleo6144's analysis for breakfast!

And this is what Scala gives in the Notation Selector, which is at least as strange to me as the config file:

Code: Select all

0: C
  1: C|)    D\!!!/
  2: C(|)   D!!!)
  3: C#(!)  D\!!/
  4: C||)   D!!)
  5: D!!/   C/||\
  6: D\!/   C|||)
  7: D!)    C(|||)
  8: D
  9: D|)    E(!!!)
 10: D/|    E\!!!
 11: D(|)   E\!!/
 12: E\!!/  D||)
 13: E!!/   D/||\
 14: E(!)   D|||)
 15: E\!    D/|||
 16: E      D(|||)
 17: E      Fb(|)
 18: E/|    F(!)
 19: F!)    E/|\
 20: F      E||\
 21: F/|    G(!!!)
 22: F(|)   G\!!!
 23: F#(!)  G\!!/
 24: F||)   G!!)
 25: F/||\  Gb(|)
 26: G(!)   F/|||
 27: G\!    F(|||)
 28: G
 29: G|)    A\!!!/
 30: G(|)   A\!!!
 31: G#(!)  A\!!/
 32: G||\   A\!!/
 33: A!!/   G/||\
 34: A\!/   G|||)
 35: A\!    G(|||)
 36: A
 37: A      B(!!!)
 38: A/|    B\!!!
 39: A(|)   B!!!)
 40: B\!!/  A||\
 41: B!!)   A/||\
 42: B(!)   A/||\
 43: B\!    A/|||
 44: B!)    A(|||)
 45: B      Cb(|)
 46: B|)    C(!)
 47: C!)    B/|\
 48: C

[Dave Keenan]

So it's a very tricky notation file that George devised. The tuning has chains of two different sizes of fifth [actually more than 2 sizes, corrected below], but the .par file can only specify one size. But by using disjoint ranges for the same symbol, he was able to get it to work correctly for both fifth sizes. So no need for any changes other than the new short-ASCII.

Thanks for that, guys.

BTW, the "unknown author" was almost certainly George. No one else understood it well enough to describe it like that.

[cmloegcmluin]

Okay, the issue is resolved then w/r/t the Scala .par updates.

Much may remain to be said about such a scale, however, and doubtlessly much remains to be said writing music with it.

[Dave Keenan]

Ha! You just knew I'd have to hit the quote button to see what the codes were for those emoji! So you knew I'd find this:

[size=703.57869]:brain:[/size][size=679.56197]:boom:[/size]

And it's just as well, because you've made me realise I was wrong. I take it those "size=" numbers were intended to represent the two (perfect) fifth sizes that I was claiming, in cents. But when I saw 679.56197 I thought, is that a typo? That's not a perfect fifth I'd expect to find in something intended to be a high-precision temperament.

So I did what I should have done long ago, and knocked up a spreadsheet to calculate every interval in the scale. I found that there are in fact 4 different perfect fifth sizes (and 679.6 isn't one of them). They are:

  2 of 696.1 ¢
  1 of 699.6 ¢
24 of 703.6 ¢
  1 of 706.8 ¢
  1 of 715.5 ¢ (pretty much a wolf)

As Gene described in the article linked from the Sagittal home page, and as @volleo6144 noted, the majority 703.6 cent fifths come in 3 chains, separated by a 4:5 and a 4:7.

So my current hypothesis is that the disjoint bins for the same symbol are not for different fifth sizes but for different chains of the same fifth size. The chains on the 5/4 and 7/4 can also be said to be offset from the chain on the 1/1 by a tempered 5-comma and a tempered 7-comma. The same-symbol bins are about 14 cents apart. There are at most two bins for each symbol, so either the 5-comma and 7-comma end up close enough to each other in size, or one of them ends up close enough to zero, to get way with 2 bins.

Much may remain to be said about such a scale, however, and doubtlessly much remains to be said writing music with it.

Indeed. It needs some kind of lattice diagram.

[cmloegcmluin]

I’m glad you found that. Especially so since it serendipitously led to deeper (or more correct) insight about the scale.

I feel like at one point I experienced a flicker of a thought that the different fifths were internal intervals, not external. But clearly I didn’t act on it. Thanks for looking into that.

I believe this pushes that .par file over the line from merely "tricky" to what we in the biz would call "hacky".

I’d love to see such a lattice! Not something I'm going to get to anytime soon, but I feel like it's only a matter of time before someone finds this topic and can't resist