Notation for George Secor's High-Tolerance Temperament

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Dave Keenan
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Re: Notation for George Secor's High-Tolerance Temperament

Post by Dave Keenan »

I'm just posting this here as a backup in case anything should happen to Robert Walker's tuning-math archive, since we have so little information from George in relation to his 29-HTT tuning.
Gene Ward Smith wrote: On Yahoo! Group: tuning-math
Message: 11113
Date: Sat, 26 Jun 2004 00:57:10
Subject: HTT temperament
From: Gene Ward Smith

On Yahoo! Groups : MakeMicroMusic Messages * [with cont.] (Wayb.)
George Secor wrote: Gene, for your enlightenment: 29-HTT consists (except for one filler
tone) of 3 chains of fifths of ~703.5787c, or exactly (504/13)^
(1/9). The 3 chains of fifths contain tones 1/1, 5/4, and 7/4,
respectively, and the tones in each chain are taken to as many places
as are required to result in otonal ogdoads on roots Bb, F, C, G, D,
and A. This also gives very-near-just diatonic (5-limit) scales in 5
different keys.
Since the fifth is (504/13)^(1/9), we immediately have that

(504/13)/(3/2)^9 = 28672/28431

is a comma of the temperament, which must go up to the 13 limit at
least. It seems clear also that three of these slightly sharp fifths
are intended to represent 44/13, which means

(44/13)/(3/2)^3 = 352/351

is another comma of the system. It does not appear any more commas are
intended, since the 5/4 and 7/4 are introduced as independent
generators. This means that the HTT temperament is a two-comma
temperament in the 13-limit; the TM basis for which turns out to be
352/351 and 364/363. This is a spacial temperament, meaning one with
four generators, counting octaves. In this case we can take the
generators to be the approximation to 2,3,5,and 7, and the commas then
give us

11 ~ 896/81
13 ~ 28672/2187

Five is not a factor of the commas, so we can make this into a
no-fives system very easily. The reduction to the 11-limit is
896/891-spacial in the 11-limit, again of course a no-fives comma.
Aside from George's tuning, we have all the usual rms, minimax, TOP
etc. tunings if we want them. An interesting question is what 7-limit
JI scales would be good ones to temper using HTT; the question of high
dimensional temperaments has of course not been much explored.
Possibly looking at how near the TOP tunings are for various commas
would be useful in discovering pairings which make sense.

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Re: Notation for George Secor's High-Tolerance Temperament

Post by Dave Keenan »

Here's a lattice for the 29-HTT tuning, showing the 3 chains of 703.58 cent fifths and the fill-in note (yellow). Any fifth whose size isn't given is 703.58 cents.

stretched just enough to dewolf B:F:#:			   split the difference between G:#::\!: and B:b:
		  |																			 |
		  |			    700	D:\!: ----	A:\!: ----	E:\!: ----	B:\!: ----	F:#::\!: ---	C:#::\!: ---	G:#::\!: 696 D:#::\!:E:b: 696
715	F:#:G:(!):-707-D:(!): ---	A:(!): ---/	E:(!):\---/	B:(!):\---/	F:/|\:\---/	C:/|\:\---/G:/|\:A:b::!):--/D:/|\:E:b::!):--/	B:b::!):\---- F:!): --- C:!): -- G:!):G:\!: 700
				    696	B:b: ----	F  ----	C  ----	G  ----	D  ----	A  ----	E  ----	B  715      


Here's the shape of a 15-odd-limit ogdoad (octad) on this lattice. You obtain the 6 ogdoads by shifting it left and right.

		  									5		15
	  				13		•		•		11		•	  	•  		•		7
										1		3		9

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Re: Notation for George Secor's High-Tolerance Temperament

Post by Dave Keenan »

Next puzzle: What 16 note subset of 29-HTT did George use for Coming on Clouds? :)


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Re: Notation for George Secor's High-Tolerance Temperament

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