Canonical homomorphisms revisted

The problem with my first definition is that it works beautifully when
it works, but it doesn't always work (1/2 time in the 5-limit, 1/4 of
the time in the 7-limit, and so forth.) Taking it as a source of
inspiration, here is a definition which works in general.
Unfortunately it no longer is as slick as goose grease.

If T is a temperament, call q a "subgroup comma" for T if q>1 is not a
power of anything else, and if only three primes are involved in its
factorization (these commas are easily found from the wedgie.) Call a
prime "reducible" if it appears by itself in either numerator or
denominator for the factorization of a subgroup comma. If T is
p-limit, let P be the set of primes <= p, and let R be the set of
reducible primes. Then N = P\R is the set of non-reducible primes. If
card(N) = g, where g is the number of generators (2 for a linear
temperament, 3 for a planar temperament, and so forth) then set G = N.
If card(N)<g, then fill it out by adding the smallest of the remaining
primes, and set the result to G; if card(N)>g, then reduce it by
removing the largest of the primes in N, and call that G. We end with
a set G of primes such that card(G)=g, and we may use G as a set of
generators for the temperament T.

Strange as it may seem, this definition actually corresponds to my
previous one in those cases where the previous one gives a
homomorphism. It also seems to give us reasonable results, or at least
so it seems to me, YMMV. Here is what we get for meantone and miracle;
I give the mapping of the generators (2 and 3/2 in the case of
meantone, 2 and 16/15 in the case of miracle), and the mapping applied
to primes:

5-limit meantone 81/80
Generators [2, 5^(1/4)]
Prime mapping [2, 2*5^(1/4), 5]

7-limit meantone [1, 4, 10, 4, 13, 12]
Generators [2, 2^(3/10)*7^(1/10)]
Prime mapping [2, 2*2^(3/10)*7^(1/10), 2*2^(1/5)*7^(2/5), 7]

11-limit meantone [1, 4, 10, 18, 4, 13, 25, 12, 28, 16]
Generators [2, 2^(3/10)*7^(1/10)]
Prime mapping [2, 2*2^(3/10)*7^(1/10), 2*2^(1/5)*7^(2/5), 7,
7/4*2^(2/5)*7^(4/5)]

11-limit meanpop [1, 4, 10, -13, 4, 13, -24, 12, -44, -71]
Generators [7^(13/71)*11^(10/71), 7^(11/71)*11^(3/71)]
Prime mapping [7^(13/71)*11^(10/71), 7^(24/71)*11^(13/71),
7^(44/71)*11^(12/71), 7, 11]

5-limit miracle 34171875/33554432
Generators [3^(7/25)*5^(6/25), 1/5*3^(3/25)*5^(24/25)]
Prime mapping [3^(7/25)*5^(6/25), 3, 5]

7-limit miracle [6, -7, -2, -25, -20, 15]
Generators [3^(7/25)*5^(6/25), 1/5*3^(3/25)*5^(24/25)]
Prime mapping [3^(7/25)*5^(6/25), 3, 5, 3^(3/5)*5^(4/5)]

11-limit miracle [6, -7, -2, 15, -25, -20, 3, 15, 59, 49]
Generators [5^(15/59)*11^(7/59), 1/5*5^(57/59)*11^(3/59)]
Prime mapping [5^(15/59)*11^(7/59), 5^(3/59)*11^(25/59), 5,
5^(49/59)*11^(15/59), 11]

what do you make of monz's use of the term "internal homomorphism" on
this page:

http://sonic-arts.org/td/gill/duodene.htm

?

anyway, so miracle doesn't want to have 2 as a generator? then what
does the scale "really" look like?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> The problem with my first definition is that it works beautifully
when
> it works, but it doesn't always work (1/2 time in the 5-limit, 1/4
of
> the time in the 7-limit, and so forth.) Taking it as a source of
> inspiration, here is a definition which works in general.
> Unfortunately it no longer is as slick as goose grease.
>
> If T is a temperament, call q a "subgroup comma" for T if q>1 is
not a
> power of anything else, and if only three primes are involved in its
> factorization (these commas are easily found from the wedgie.) Call
a
> prime "reducible" if it appears by itself in either numerator or
> denominator for the factorization of a subgroup comma. If T is
> p-limit, let P be the set of primes <= p, and let R be the set of
> reducible primes. Then N = P\R is the set of non-reducible primes.
If
> card(N) = g, where g is the number of generators (2 for a linear
> temperament, 3 for a planar temperament, and so forth) then set G =
N.
> If card(N)<g, then fill it out by adding the smallest of the
remaining
> primes, and set the result to G; if card(N)>g, then reduce it by
> removing the largest of the primes in N, and call that G. We end
with
> a set G of primes such that card(G)=g, and we may use G as a set of
> generators for the temperament T.
>
> Strange as it may seem, this definition actually corresponds to my
> previous one in those cases where the previous one gives a
> homomorphism. It also seems to give us reasonable results, or at
least
> so it seems to me, YMMV. Here is what we get for meantone and
miracle;
> I give the mapping of the generators (2 and 3/2 in the case of
> meantone, 2 and 16/15 in the case of miracle), and the mapping
applied
> to primes:
>
> 5-limit meantone 81/80
> Generators [2, 5^(1/4)]
> Prime mapping [2, 2*5^(1/4), 5]
>
> 7-limit meantone [1, 4, 10, 4, 13, 12]
> Generators [2, 2^(3/10)*7^(1/10)]
> Prime mapping [2, 2*2^(3/10)*7^(1/10), 2*2^(1/5)*7^(2/5), 7]
>
> 11-limit meantone [1, 4, 10, 18, 4, 13, 25, 12, 28, 16]
> Generators [2, 2^(3/10)*7^(1/10)]
> Prime mapping [2, 2*2^(3/10)*7^(1/10), 2*2^(1/5)*7^(2/5), 7,
> 7/4*2^(2/5)*7^(4/5)]
>
> 11-limit meanpop [1, 4, 10, -13, 4, 13, -24, 12, -44, -71]
> Generators [7^(13/71)*11^(10/71), 7^(11/71)*11^(3/71)]
> Prime mapping [7^(13/71)*11^(10/71), 7^(24/71)*11^(13/71),
> 7^(44/71)*11^(12/71), 7, 11]
>
> 5-limit miracle 34171875/33554432
> Generators [3^(7/25)*5^(6/25), 1/5*3^(3/25)*5^(24/25)]
> Prime mapping [3^(7/25)*5^(6/25), 3, 5]
>
> 7-limit miracle [6, -7, -2, -25, -20, 15]
> Generators [3^(7/25)*5^(6/25), 1/5*3^(3/25)*5^(24/25)]
> Prime mapping [3^(7/25)*5^(6/25), 3, 5, 3^(3/5)*5^(4/5)]
>
> 11-limit miracle [6, -7, -2, 15, -25, -20, 3, 15, 59, 49]
> Generators [5^(15/59)*11^(7/59), 1/5*5^(57/59)*11^(3/59)]
> Prime mapping [5^(15/59)*11^(7/59), 5^(3/59)*11^(25/59), 5,
> 5^(49/59)*11^(15/59), 11]

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> what do you make of monz's use of the term "internal homomorphism" on
> this page:
>
> http://sonic-arts.org/td/gill/duodene.htm

It makes no sense to me. Oh well. :)

> anyway, so miracle doesn't want to have 2 as a generator? then what
> does the scale "really" look like?

72-et with slightly flattened octaves, perhaps.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
>
> > what do you make of monz's use of the term "internal
homomorphism" on
> > this page:
> >
> > http://sonic-arts.org/td/gill/duodene.htm
>
> It makes no sense to me. Oh well. :)
>
> > anyway, so miracle doesn't want to have 2 as a generator? then
what
> > does the scale "really" look like?
>
> 72-et with slightly flattened octaves, perhaps.

no, i meant something with about as many notes per octave as
blackjack or canasta, but using the generators you posted instead of
2 . . . are those just the secor and a flattened octave? i didn't
bother to check . . .

I have nothing to say, but I wish I did. Does anybody
else have anything to say? If so, you may say it now.

-Carl

>I have nothing to say,

Other than, if I understand this, it's what I've been pining
for all along -- defining temperaments with maps. Unforch,
I haven't tracked the reasoning behind what why maps found
with this technique should be canonical. Would anybody
besides Gene consider them natural mappings for their fav.
temperament(s)?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >I have nothing to say,
>
> Other than, if I understand this, it's what I've been pining
> for all along -- defining temperaments with maps. Unforch,
> I haven't tracked the reasoning behind what why maps found
> with this technique should be canonical.

Maybe I shouldn't call them that anymore.

>> Other than, if I understand this, it's what I've been pining
>> for all along -- defining temperaments with maps. Unforch,
>> I haven't tracked the reasoning behind what why maps found
>> with this technique should be canonical.
>
>Maybe I shouldn't call them that anymore.

Why?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> Other than, if I understand this, it's what I've been pining
> >> for all along -- defining temperaments with maps. Unforch,
> >> I haven't tracked the reasoning behind what why maps found
> >> with this technique should be canonical.
> >
> >Maybe I shouldn't call them that anymore.

My nice, clean, overwhelmingly nifty definition only worked part of
the time, and its replacement does the same job, but the definition is
hardly nifty anymore.

In practical terms the darn thing works, however.

Gene,

Any news on this front? Might it be useful for the notation
effort?

-Carl

>From: "Gene Ward Smith" <gwsmith@svpal.org>
>Date: Sun, 13 Apr 2003
>Subject: [tuning-math] Canonical homomorphisms
>
>The problem with my first definition is that it works beautifully when
>it works, but it doesn't always work (1/2 time in the 5-limit, 1/4 of
>the time in the 7-limit, and so forth.) Taking it as a source of
>inspiration, here is a definition which works in general.
>Unfortunately it no longer is as slick as goose grease.
>
>If T is a temperament, call q a "subgroup comma" for T if q>1 is not a
>power of anything else, and if only three primes are involved in its
>factorization (these commas are easily found from the wedgie.) Call a
>prime "reducible" if it appears by itself in either numerator or
>denominator for the factorization of a subgroup comma. If T is
>p-limit, let P be the set of primes <= p, and let R be the set of
>reducible primes. Then N = P\R is the set of non-reducible primes. If
>card(N) = g, where g is the number of generators (2 for a linear
>temperament, 3 for a planar temperament, and so forth) then set G = N.
>If card(N)<g, then fill it out by adding the smallest of the remaining
>primes, and set the result to G; if card(N)>g, then reduce it by
>removing the largest of the primes in N, and call that G. We end with
>a set G of primes such that card(G)=g, and we may use G as a set of
>generators for the temperament T.
>
>Strange as it may seem, this definition actually corresponds to my
>previous one in those cases where the previous one gives a
>homomorphism. It also seems to give us reasonable results, or at least
>so it seems to me, YMMV. Here is what we get for meantone and miracle;
>I give the mapping of the generators (2 and 3/2 in the case of
>meantone, 2 and 16/15 in the case of miracle), and the mapping applied
>to primes:
>
>5-limit meantone 81/80
>Generators [2, 5^(1/4)]
>Prime mapping [2, 2*5^(1/4), 5]
>
>7-limit meantone [1, 4, 10, 4, 13, 12]
>Generators [2, 2^(3/10)*7^(1/10)]
>Prime mapping [2, 2*2^(3/10)*7^(1/10), 2*2^(1/5)*7^(2/5), 7]
>
>11-limit meantone [1, 4, 10, 18, 4, 13, 25, 12, 28, 16]
>Generators [2, 2^(3/10)*7^(1/10)]
>Prime mapping [2, 2*2^(3/10)*7^(1/10), 2*2^(1/5)*7^(2/5), 7,
>7/4*2^(2/5)*7^(4/5)]
>
>11-limit meanpop [1, 4, 10, -13, 4, 13, -24, 12, -44, -71]
>Generators [7^(13/71)*11^(10/71), 7^(11/71)*11^(3/71)]
>Prime mapping [7^(13/71)*11^(10/71), 7^(24/71)*11^(13/71),
>7^(44/71)*11^(12/71), 7, 11]
>
>5-limit miracle 34171875/33554432
>Generators [3^(7/25)*5^(6/25), 1/5*3^(3/25)*5^(24/25)]
>Prime mapping [3^(7/25)*5^(6/25), 3, 5]
>
>7-limit miracle [6, -7, -2, -25, -20, 15]
>Generators [3^(7/25)*5^(6/25), 1/5*3^(3/25)*5^(24/25)]
>Prime mapping [3^(7/25)*5^(6/25), 3, 5, 3^(3/5)*5^(4/5)]
>
>11-limit miracle [6, -7, -2, 15, -25, -20, 3, 15, 59, 49]
>Generators [5^(15/59)*11^(7/59), 1/5*5^(57/59)*11^(3/59)]
>Prime mapping [5^(15/59)*11^(7/59), 5^(3/59)*11^(25/59), 5,
>5^(49/59)*11^(15/59), 11]

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> Gene,
>
> Any news on this front? Might it be useful for the notation
> effort?

I was happy just to finally get it to work. You could base notation
systems on it, I suppose, but there doesn't seem any clear connection.