A chart of syntonic comma temperaments

http://www.io.com/~hmiller/png/syntonic.png

This is a chart of 7-limit temperaments that temper out the syntonic comma 81;80. The horizontal axis is deviation from 3:1 and the vertical axis is the deviation from 7:1. This time I limited the list to 7-limit consistent ET's.

hi Herman,

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...>
wrote:

> http://www.io.com/~hmiller/png/syntonic.png
>
> This is a chart of 7-limit temperaments that temper out
> the syntonic comma 81;80. The horizontal axis is deviation
> from 3:1 and the vertical axis is the deviation from 7:1.
> This time I limited the list to 7-limit consistent ET's.

so according to the criteria in this chart, the temperament
with the lowest error for both 3 and 7 is 36-ET gawel?

i've been missing a lot on the tuning lists until lately,
so i don't even know about the gawel family.

-monz

oops ... of course, i see that 36-ET is also catler and
mothra, as well as gawel. i don't know about either of
those two, either.

-monz

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:

> hi Herman,
>
> --- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...>
> wrote:
>
> > http://www.io.com/~hmiller/png/syntonic.png
> >
> > This is a chart of 7-limit temperaments that temper out
> > the syntonic comma 81;80. The horizontal axis is deviation
> > from 3:1 and the vertical axis is the deviation from 7:1.
> > This time I limited the list to 7-limit consistent ET's.
>
>
>
> so according to the criteria in this chart, the temperament
> with the lowest error for both 3 and 7 is 36-ET gawel?
>
> i've been missing a lot on the tuning lists until lately,
> so i don't even know about the gawel family.
>
>
>
> -monz

hi Herman,

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...>
wrote:

> http://www.io.com/~hmiller/png/syntonic.png
>
> This is a chart of 7-limit temperaments that temper out
> the syntonic comma 81;80. The horizontal axis is deviation
> from 3:1 and the vertical axis is the deviation from 7:1.
> This time I limited the list to 7-limit consistent ET's.

are the numbers associated with each temperament family
on this chart wedgies? if not, then what are they?

on the meantone one, <<1, 4, 10, 4, 13, 12|| ,
the "1, 4, 10" part at least looks familiar as the
generator mapping for primes 3, 5, and 7. am i on
the right track?

please explain one, using the meantone one as an example.
if your triangular arrangement is appropriate, please
show that as well. thanks.

-monz

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:

> are the numbers associated with each temperament family
> on this chart wedgies? if not, then what are they?

Wedgies, yes.

> on the meantone one, <<1, 4, 10, 4, 13, 12|| ,
> the "1, 4, 10" part at least looks familiar as the
> generator mapping for primes 3, 5, and 7. am i on
> the right track?
>
> please explain one, using the meantone one as an example.

1 4 10

This is how many generator steps, assuming octave periods, it takes to
get to 3, 5, and 7

4 13

This is how many generator steps, assuming tritave (3) periods, it
takes to get to 5 and 7

12

3 is the number of generator steps, using a period of 5^(1/4) (1/4
meantone fifth) to get to 7; 12 is 3*4

Mostly we are just interested in 1, 4 and 10, but leaving off the rest
makes things hard to compute. It is a lot like leaving the 2 off a
comma, and just giving the octave equivalent comma; and in fact the
two proceedures are related. Anything of the form

<<1 4 10 a b 4b - 10a||

is a valid wedgie, but not much good unless it is meantone. For
instance, <<1 4 10 4 14 16|| has TM basis {81/80, 252/125} and
<<1 4 10 5 15 10|| has TM basis {160/81, 125/63}. Both of these are
octave equivalent to {81/80, 126/125}. If we take <<1 4 10 0 0 0||
we get a TM basis consisting of odd ratios, {81/5, 225/7}. This may be
regarded as the octave equivalent form of meantone, and its easy to
get meantone from it by reducing the commas to the half-octave.

You can take any generator mapping you like at will, fill it out with
zeros to get an octave-equivalent wedgie, reduce the commas to the
half-octave, take the wedgie from this, and you have a temperament for
that generator mapping. Mostly, the results aren't interesting.

monz wrote:
> are the numbers associated with each temperament family
> on this chart wedgies? if not, then what are they?
> > on the meantone one, <<1, 4, 10, 4, 13, 12|| ,
> the "1, 4, 10" part at least looks familiar as the > generator mapping for primes 3, 5, and 7. am i on
> the right track?
> > please explain one, using the meantone one as an example.
> if your triangular arrangement is appropriate, please
> show that as well. thanks.

Yes, you could use the triangular arrangement:

. 1 4 10 (note that <<1, 4, 4|| is the wedgie for 81;80)
. 4 13
. 12

Gene explained a couple of days ago about how the first row of the wedgie in this form is related to the mapping (depending on how the generators are defined, you might need to negate them, as in the case of meantone with a fourth as the generator, and divide by the number of periods in an octave).

Looking at an 11-limit version of meantone, <<1, 4, 10, -13, 4, 13, -24, 12, -44, -71||, you can see the 7-limit meantone wedgie in the triangular arrangement:

. 1 4 10 -13
. 4 13 -24
. 12 -44
. -71

Since meantone on the chart is shown as a line between 12 and 19, you can get any other information you need from Graham Breed's temperament finder (http://x31eq.com/temper/twoet.html): put in "12", "19", and "7" into the boxes, and this is what you get:

13/31, 503.4 cent generator

basis:
(1.0, 0.419517976278)

mapping by period and generator:
[(1, 0), (2, -1), (4, -4), (7, -10)]

mapping by steps:
[(19, 12), (30, 19), (44, 28), (53, 34)]

highest interval width: 10
complexity measure: 10 (12 for smallest MOS)
highest error: 0.004480 (5.377 cents)
unique

Similarly, "mothra" <<3, 12, -1, 12, -10, -36|| is shown as a line from 5 to 26. Follow this link for the stats:

http://x31eq.com/cgi-bin/temperament.cgi?et1=5&et2=26&limit=7

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...>
wrote:
> http://www.io.com/~hmiller/png/syntonic.png
>
> This is a chart of 7-limit temperaments that temper out the
syntonic
> comma 81;80. The horizontal axis is deviation from 3:1 and the
vertical
> axis is the deviation from 7:1. This time I limited the list to 7-
limit
> consistent ET's.

What you call Mothra, I call Cynder, since it's basically the same as
the Slendric or Wonder temperament, but with 5 thrown into the primal
mix.

What you call Hemifourths, I call Semaphore.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

> What you call Mothra, I call Cynder, since it's basically the same as
> the Slendric or Wonder temperament, but with 5 thrown into the primal
> mix.

> What you call Hemifourths, I call Semaphore.

Actually, I had proposed Godzilla for that, but unless we use the rest
of the 8/7 generator lineup that doesn't suggest as much as Semaphore.

There's something attractive about giving similar names to similar
generators, though, since we have more than one comma, but just the
one generator. In the case of Mothra, I hardly think that a
temperament with a comma of 81/80 is "basically the same" as Wonder,
so I don't think Cynder as a name has a lot to recommend it on that
basis. Were you thinking of Rodan=Supersupermajor?

Godzilla has (4/3)/(8/7)^2 = 49/48 going for it, as well as
2^(-20) 7^8 5^(-1).

Mothra has (3/2)/(8/7)^3 = 1029/1024, but we can hardly ignore the 81/80.

Rodan has 1029/1024 also, as well as 2^(-50) 5 7^17, and so it would
seem to be a lot more like something which could be called Wonder with
5 tossed in. Gamera is a microtemperament, with 4375/4374 its most
familiar comma.