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.)Since the fifth is (504/13)^(1/9), we immediately have thatGeorge 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.

(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.

## Notation for George Secor's High-Tolerance Temperament

- Dave Keenan
- Site Admin
**Posts:**2095**Joined:**Tue Sep 01, 2015 2:59 pm**Location:**Brisbane, Queensland, Australia-
**Contact:**

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

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.

- Dave Keenan
- Site Admin
**Posts:**2095**Joined:**Tue Sep 01, 2015 2:59 pm**Location:**Brisbane, Queensland, Australia-
**Contact:**

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

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.

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.

stretched just enough to dewolf B:F split the difference between G and B | | | 700 D ---- A ---- E ---- B ---- F --- C --- G 696 DE 696 715 FG-707-D --- A ---/ E\---/ B\---/ F\---/ C\---/GA--/DE--/ B\---- F --- C -- GG 700 696 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

- Dave Keenan
- Site Admin
**Posts:**2095**Joined:**Tue Sep 01, 2015 2:59 pm**Location:**Brisbane, Queensland, Australia-
**Contact:**

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

Next puzzle: What 16 note subset of 29-HTT did George use for

*Coming on Clouds*?- cmloegcmluin
- Site Admin
**Posts:**1689**Joined:**Tue Feb 11, 2020 3:10 pm**Location:**San Francisco, California, USA**Real Name:**Douglas Blumeyer (he/him/his)-
**Contact:**

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

No progress made on the puzzle, but I do have a bunch of links to the Yahoo tuning list archive with relevant information about 29-HTT:

https://yahootuninggroupsultimatebackup ... 7512#17512

https://yahootuninggroupsultimatebackup ... _6820#6964

https://yahootuninggroupsultimatebackup ... 0549#10549

https://yahootuninggroupsultimatebackup ... 9612#39689

https://yahootuninggroupsultimatebackup ... html#53712

https://yahootuninggroupsultimatebackup ... 7512#17512

https://yahootuninggroupsultimatebackup ... _6820#6964

https://yahootuninggroupsultimatebackup ... 0549#10549

https://yahootuninggroupsultimatebackup ... 9612#39689

https://yahootuninggroupsultimatebackup ... html#53712