72

I sorted the generators using goodness of the white+black keys in the
11-limit, and as might be expected the miracle system of 41+31 came
out on top. However, no one generator will encompass all of the
possibilities of an et, and 46+26, for one, gives
it worthy competition.

41+31

There may not seem much left to say about the miracle system, so I'll
use the occasion to point out that one possible "middle way" is to
employ a temperament of a temperament, or even a temperament of a
temperament of a temperament. The 10-note scale mentioned in
connection with 3/31 is 7777777779 in this tuning, and we also have
Blackjack, Canasta and whatever the 41-note temperament is being
called. I would count Canasta, the 31-from-72 system, as a
temperament of 31 notes, and not a scale; considered from that point
of view one might ask what happens if we use it, rather than 31-et,
for meantone versions of diatonic, 12-notes, etc. In fact what
happens is that we get a lot of different versions of these things as
we modulate from one key to another. For instance, three of the 31
keys give us a diatonic scale which is simply the 72-version of JI
diatonic, and others give a variety of other 7-note scales.

46+26

From 26/46 = 13/23 ~ 4/7 we obtain 7/23 < 11/36 < 4/13 as our
generator, with (7+11)/(23+36) = 18/59 = 36/118 belonging to the same
family. For anyone particularly interested in pure triads the 36/118
temperament is highly recommendable. Generator steps are -6, -11, 2,
3, -2, -6 for the odd primes to 17, but of course this is modulo the
half-octave, so it doesn't beat miracle. Scale possibilities include
(11 11 14)x2, (11 11 11 3)x2, and (8383833)x2; in addition
(2333233323333)x2 is an alternative to Blackjack or Canasta.

50+22

From 50/22 = 25/11 ~ 9/4 we get 9/25 < 13/36 < 4/11; we also have
(13+4)/(36+11) = 17/47 = 34/94 as a closely-related generator.
Generator steps are 6, -1, 10, -3, -10 up to 13; and we have a
10-note scale with pattern (3 10 3 10 10)x2, as well as an
alternative to Blackjack with pattern (33343334334)x2, and 22 notes.

53+19

Judging by its origins from 53 and 19, we would expect this to be a
triad machine, and in fact it is--the 72-version of kleismic. The
first four odd primes are mapped to 6, 5, 22, and -21, and
5/19 < 19/72 < 14/53 locates the generator. It has scales of form
4 15 4 15 4 15 4 15 15, 4 4 11 4 4 11 4 4 11 4 11 and 444744474447447.

45+27

The 27-et is a good scale division, but 45 seems to have little to
recommend it, so one might wonder what it is doing on this list. In
fact, appearances are deceptive, and this is very interesting indeed.

We have 45/27 = 5/3 ~ 2, and so we locate our generator by 1/3 < 3/8
< 2/5. This gives us -2, -3, -2, 3, -2 for our generator steps to odd
primes; the factor of 9 gives us 18, 27, 27, 36, 63 for complexity up
to 11, which is enough to show that this isn't what I find
interesting. Between 1/3 and 3/8 we have (1+3)/(3+8) = 4/11 = 36/99,
and between 4/11 and 3/8 we have (4+3)/(11+8) = 7/19 = 63/171,
suggesting a relationship between this generator and the
171-division, and the possibility of getting an excellent temperament
for the 7-limit. Looking in the vicinity of 4/11 at denominators less
than 75, we find

4/11 << 11/30 < 18/49 < 25/68 < 7/19 << 3/8,

and multiplying numerators and denominators by 9 gives us

36/99 << 99/270 < 162/441 < 225/612 < 63/171 << 27/72;

this seems to be the miraculous set of generators I thought I had
found before. All of these have -2, -3, -2 for the generators of 3,
5, and 7, 441 and 270 share 14 as a generator for 11, and 171, 612,
441 and 270 all have 9 as a generator for 13. While the full power of
this therefore can't be used by a 72-temperament, since 72/9 = 8, it
does show us that we can temper the 72-et so as to get very accurate
3, 5, and 7s using this generator--and that we could also temper 99
and 171, so as to get some excellent 13 and 11 harmonies as well.

In-Reply-To: <9qjmvi+11beq@eGroups.com>
Gene wrote:

> I sorted the generators using goodness of the white+black keys in the
> 11-limit, and as might be expected the miracle system of 41+31 came
> out on top. However, no one generator will encompass all of the
> possibilities of an et, and 46+26, for one, gives
> it worthy competition.

How are you calculating these? An exhaustive search of all generators for
"good" ETs would be a good complement to my search of pairs of ETs.

> 46+26
>
> >From 26/46 = 13/23 ~ 4/7 we obtain 7/23 < 11/36 < 4/13 as our
> generator, with (7+11)/(23+36) = 18/59 = 36/118 belonging to the same
> family. For anyone particularly interested in pure triads the 36/118
> temperament is highly recommendable. Generator steps are -6, -11, 2,
> 3, -2, -6 for the odd primes to 17, but of course this is modulo the
> half-octave, so it doesn't beat miracle. Scale possibilities include
> (11 11 14)x2, (11 11 11 3)x2, and (8383833)x2; in addition
> (2333233323333)x2 is an alternative to Blackjack or Canasta.

By my calculations it does beat miracle in the 17-limit. Complexity
measure of 40, minimax error of 6.7 cents provided you sacrifice the
equalness of the temperament. Compared to a complexity of 49 and error of
4.9 cents for miracle.

> 50+22
>
> >From 50/22 = 25/11 ~ 9/4 we get 9/25 < 13/36 < 4/11; we also have
> (13+4)/(36+11) = 17/47 = 34/94 as a closely-related generator.
> Generator steps are 6, -1, 10, -3, -10 up to 13; and we have a
> 10-note scale with pattern (3 10 3 10 10)x2, as well as an
> alternative to Blackjack with pattern (33343334334)x2, and 22 notes.

Or generator steps of 6, -1, 10, -3, -10, 6 up to 17. Complexity of 44,
minimax error of 5.8 cents, comes between the two above.

There's also 29+43. Mapping -6, 19, 26, 21, 10, 30. 17-limit complexity
of 42, so you get an otonality in the 43 note MOS, and accurate to within
6.7 cents. You also get 5 15-limit otonalities.

And 15+57. Mapping -6, -5, 2, -3, -14. 15-limit complexity of 48,
accurate to 2.8 cents.

And 58+14. Mapping -6, -17, -10, -15, -26. 6 15-limit otonalities in the
58 note MOS, to within 3.3 cents.

> 45+27
>
> The 27-et is a good scale division, but 45 seems to have little to
> recommend it, so one might wonder what it is doing on this list. In
> fact, appearances are deceptive, and this is very interesting indeed.
>
> We have 45/27 = 5/3 ~ 2, and so we locate our generator by 1/3 < 3/8
> < 2/5. This gives us -2, -3, -2, 3, -2 for our generator steps to odd
> primes; the factor of 9 gives us 18, 27, 27, 36, 63 for complexity up
> to 11, which is enough to show that this isn't what I find
> interesting. Between 1/3 and 3/8 we have (1+3)/(3+8) = 4/11 = 36/99,
> and between 4/11 and 3/8 we have (4+3)/(11+8) = 7/19 = 63/171,
> suggesting a relationship between this generator and the
> 171-division, and the possibility of getting an excellent temperament
> for the 7-limit. Looking in the vicinity of 4/11 at denominators less
> than 75, we find

Yep, it's top of my list of 7-limit microtemperaments. Complexity of 27
(means you miss out on any otonalities with the 27-note MOS) and worst
error of 0.2 cents. So are you planning to get a 45 note instrument and
tune it up?

Graham

--- In tuning@y..., graham@m... wrote:

> How are you calculating these? An exhaustive search of all
generators for
> "good" ETs would be a good complement to my search of pairs of ETs.

I sorted them by pairs, and looked at the good ones. You are right
that the safest method would be an exhaustive search, but as I
pointed out, generator steps, given for two vals hn and hm, can be
calculated from mhn - nhm in terms of the errors in relative cents of
hn and hm.

> There's also 29+43. Mapping -6, 19, 26, 21, 10, 30. 17-limit
complexity
> of 42, so you get an otonality in the 43 note MOS, and accurate to
within
> 6.7 cents. You also get 5 15-limit otonalities.

> And 15+57. Mapping -6, -5, 2, -3, -14. 15-limit complexity of 48,
> accurate to 2.8 cents.

> And 58+14. Mapping -6, -17, -10, -15, -26. 6 15-limit otonalities
in the
> 58 note MOS, to within 3.3 cents.

These are possibilities I was going to include, but it got too late
and I got too lazy. The 43+29 has (5)x15 7 and (5)x14 2 as scales;
the 57+15 gives us (5 5 14)x3, (5559)x3, and (55554)x3 as scales, and
has (141414144)x3 in the Blackjack-Canasta range; 58+14 has
(5555 16)x2, (55555 11)x2 and (5555556)x2 as some scale possibilities.

> Yep, it's top of my list of 7-limit microtemperaments. Complexity
of 27
> (means you miss out on any otonalities with the 27-note MOS) and
worst
> error of 0.2 cents. So are you planning to get a 45 note
instrument and
> tune it up?

That's a possibility, but I was thinking more of using it to tune 72.
What's a tempered 72 Canasta have going for it, and is this a middle
way, or a muddle way? Having some intervals effectively pure seems to
me an interesting alternative.

Me:
> > Yep, it's top of my list of 7-limit microtemperaments. Complexity
> of 27
> > (means you miss out on any otonalities with the 27-note MOS) and
> worst
> > error of 0.2 cents. So are you planning to get a 45 note
> instrument and
> > tune it up?

Gene:
> That's a possibility, but I was thinking more of using it to tune 72.
> What's a tempered 72 Canasta have going for it, and is this a middle
> way, or a muddle way? Having some intervals effectively pure seems to
> me an interesting alternative.

The main advantage of Canasta over 45 notes of that microtemperament is
that you only need 31 notes, and get as many 7-limit chords. 72-equal's
near enough pure anyway. But I do prefer the melody for a tuning
somewhere between 41 and 72, so that's the advantage you get by not tuning
to 72-equal. If that's what you meant.

The right instrument would be easily retunable, so you could try all these
ideas.

Graham

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., graham@m... wrote:

> > Yep, it's top of my list of 7-limit microtemperaments.
Complexity
> of 27
> > (means you miss out on any otonalities with the 27-note MOS) and
> worst
> > error of 0.2 cents. So are you planning to get a 45 note
> instrument and
> > tune it up?
>
> That's a possibility, but I was thinking more of using it to tune
72.
> What's a tempered 72 Canasta have going for it,

Gene, if you try to calculate the optimal MIRACLE generator in
various different ways, you'll see that most or all of the results
are within a few hundredths of a cent of 72-tET. This won't be true
for the other systems, such as the one above.

> and is this a middle
> way, or a muddle way?

If you only have 21 or 31 notes on your instrument, it won't much
matter whether you're using an exact 72-tET generator or a slight
deviation from it, so the question of whether it's "middle" is a
bit "muddled".

> Having some intervals effectively pure seems to
> me an interesting alternative.

In 72-tET, 7:6 is "effectively pure", I'd say . . .

--- In tuning@y..., genewardsmith@j... wrote:
> I sorted the generators using goodness of the white+black keys in
the
> 11-limit, and as might be expected the miracle system of 41+31 came
> out on top. However, no one generator will encompass all of the
> possibilities of an et, and 46+26, for one, gives
> it worthy competition.
>
> 41+31
>
> There may not seem much left to say about the miracle system, so
I'll
> use the occasion to point out that one possible "middle way" is to
> employ a temperament of a temperament, or even a temperament of a
> temperament of a temperament.

Yes indeed! I find that Meantones of Miracles are particularly
fertile. Thanks to Rami Vitale (although he didn't approve) we know
that a max-even 19 of Canasta, e.g. 19 of 31 of 72, includes many
Byzantine scales. Lots of interesting tetrachords. This may also have
advantages for guitarists that want to keep standard open tuning, but
without having zillions of fretlets or bent frets.

--- In tuning@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning@y..., genewardsmith@j... wrote:
> > I sorted the generators using goodness of the white+black keys in
> the
> > 11-limit, and as might be expected the miracle system of 41+31
came
> > out on top. However, no one generator will encompass all of the
> > possibilities of an et, and 46+26, for one, gives
> > it worthy competition.
> >
> > 41+31
> >
> > There may not seem much left to say about the miracle system, so
> I'll
> > use the occasion to point out that one possible "middle way" is
to
> > employ a temperament of a temperament, or even a temperament of a
> > temperament of a temperament.

Where was this posted? Is it in the archives? How did I miss it? Help!
>
> Yes indeed! I find that Meantones of Miracles are particularly
> fertile.
> Thanks to Rami Vitale (although he didn't approve) we know
> that a max-even 19 of Canasta, e.g. 19 of 31 of 72, includes many
> Byzantine scales. Lots of interesting tetrachords. This may also
have
> advantages for guitarists that want to keep standard open tuning,
but
> without having zillions of fretlets or bent frets.

You're referring, I think, to the use of 7-limit periodicity blocks
where 224:225 is tempered out, 81:80 is not tempered out, and the
third unison vector, for example 245:243 is also not tempered
out . . . yes? What if 245:243 _is_ tempered out?

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
> Where was this posted? Is it in the archives? How did I miss it?
Help!

/tuning/topicId_29240.html#29240
You should have been able to get to it by click the "Up thread"
button/link from my reply.

> > Yes indeed! I find that Meantones of Miracles are particularly
> > fertile.
> > Thanks to Rami Vitale (although he didn't approve) we know
> > that a max-even 19 of Canasta, e.g. 19 of 31 of 72, includes many
> > Byzantine scales. Lots of interesting tetrachords. This may also
> have
> > advantages for guitarists that want to keep standard open tuning,
> but
> > without having zillions of fretlets or bent frets.
>
> You're referring, I think, to the use of 7-limit periodicity blocks
> where 224:225 is tempered out, 81:80 is not tempered out, and the
> third unison vector, for example 245:243 is also not tempered
> out . . . yes?

I have no idea. Take its tonic to be 0 on a chain of secors. The scale
consists of these notes (as secor numbers) -8,-7,-6,-5, -2,-1,0,1,
5,6,7, 11,12,13,14, 17,18,19,20.

> What if 245:243 _is_ tempered out?

Beats me.

--- In tuning@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
> > Where was this posted? Is it in the archives? How did I miss it?
> Help!
>
> /tuning/topicId_29240.html#29240
> You should have been able to get to it by click the "Up thread"
> button/link from my reply.

Well that's 2000 messages ago! We were already talking
about "goodness" then?

We shouldn't forget the Boston-area composer Ezra Sims who also uses 72-tet.

--John