HTT temperament

On /makemicromusic/topicId_6820.html#6964
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.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> On /makemicromusic/topicId_6820.html#6964
> 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.

Yes. The essence of high-tolerance temperament (or HTT) is that the
ratios of 7, 11, and 13 are all in a single chain of fifths, but the
size of the fifth has been set so that 8:9 has the same error as
8:13, making 9:13 exact. This gives 8:11 and 7:11 almost the same
error, which in turn makes 11:13 almost exact.

> 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.

Yes. Margo does this by omitting the chain of fifths containing 5/4,
which reduces the total number of tones to a number that she can put
on two conventional keyboards, while maintaining the conventional
octave distance.

> 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;

I don't know where you might go with that idea, since HTT was
intended to bridge 7 with 11 and 11 with 13.

--George

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:

> Yes. The essence of high-tolerance temperament (or HTT) is that the
> ratios of 7, 11, and 13 are all in a single chain of fifths, but the
> size of the fifth has been set so that 8:9 has the same error as
> 8:13, making 9:13 exact. This gives 8:11 and 7:11 almost the same
> error, which in turn makes 11:13 almost exact.

If 7 is an independent generator it isn't in the same chain of fifths.
One way to put it in is via hemififths, where the no-fives commas
have a basis 144/143, 243/242 and 364/363. Adding 196/195 to the mix
gives you fives, or using it as a generator gives you a planar
temperament. This, however, will not give you something exactly like
HTT as you have described it, which seems to need 7 *not* to be in the
chain of fifths.

> Yes. Margo does this by omitting the chain of fifths containing 5/4,
> which reduces the total number of tones to a number that she can put
> on two conventional keyboards, while maintaining the conventional
> octave distance.

Hemififths has a 24-note MOS, which she could put in her two
conventional keyboards very easily. Five would then be too complex to
play much of a role, though there are plenty of 7/5s to play with.
TOP tuning for it makes the error of five, over log 5, the same as the
error in 13, over log 13. If we ignore 5 the TOP tuning now wants to
make the error of 11, over log 11, the same as the error in 13, over
log 13, and so 13/11 becomes nearly exact, which corresponds to your
remark about the tuning of this interval in HTT. Making the octaves
and 13/11 both pure gives a fifth of (44/13)^(1/3), and so a
hemififths generator of (44/13)^(1/6). This is actually a little
sharper even than the 58-et fifth, and this whole approach may not
give you as much accuracy as you require. However, the 24 note MOS of
58-et with a 17 step generator is interesting; it has step sizes of 3
and 1: 3,3,3,1,3,3,1,3,3,3,1,3,3,1,3,3,3,1,3,3,1,3,3,1 if anyone feels
inspired to try it.