Re: [tuning-math] handy text breakdown of monz's chart

re:
http://www.ixpres.com/interval/dict/eqtemp.htm

> From: Carl Lumma <carl@lumma.org>
> To: <tuning-math@yahoogroups.com>
> Sent: Friday, February 22, 2002 1:39 AM
> Subject: [tuning-math] handy text breakdown of monz's chart
>
>

excellent! thanks for that, Carl!

i've corrected your table, inserted the missing ratios (easy
to calculate from the vector), and added all the other vanishing
commas for which Gene has supplied that data.

i've also added this table to the webpage, along with a graph which
plots the vanishing commas on a 5-limit lattice.

altho i haven't really looked into it yet, i believe that the
angles and lengths of the vectors of the vanishing commas on
this "Small 5-limit intervals" lattice have something to do
with the patterns of shading and coloring on the big gallery
of lattices i just added recently to the bottom part of the page.

a suggestion to all: look up the vectors of the vanishing commas
for all families of temperaments to which one particular EDO belongs,
then compare that with the shaded lattice for that EDO. then
post your comments to the list; most likely, they'll find their
way into the webpage, which i envision as being broken up into
a whole library of webpages on individual EDOs.

from the webpage (unfortunately i think the line length is going
to cause this table to be mangled slightly) :

Below is a table listing each of the vanishing commas depicted in
the diagram above, and their associated temperaments. Where the
name has "--" the temperament family has not yet been assigned a name,
and where it has "x" I also did not draw the red line on the diagram
because of lack of space.

name ratio ~cents [2 3 5] vector EDOs

x 244140625:229582512 106.4395534 [ -4 -15 12] 48, 63,
78, 15
-- 6561:6250 84.0714376 [ -1 8 -5] 23, 35,
47, 12
-- 536870912:512578125 80.16399602 [ 29 -8 -7] 59, 71,
83, 95, 12
x 30517578125:29386561536 65.38069499 [-11 -15 15] 45, 60,
75, 90, 15
-- 648:625 62.565148 [ 3 4 -4] 28, 40,
52, 64, 12
-- 262144:253125 60.61142721 [ 18 -4 -5] 25, 37,
49, 61, 73, 12
x 1953125:1889568 57.27341613 [ -5 -10 9] 26, 41,
97, 56, 71, 15
-- 16875:16384 51.11985806 [-14 3 4] 29, 48,
19, 47, 28
porcupine* 250:243 49.16613727 [ 1 -5 3] 29, 22,
59, 37, 15
diesic 128:125 41.05885841 [ 7 0 -3] 15, 42,
27, 39, 12
-- 48828125:47775744 37.72084732 [-16 -6 11] 19, 94,
75, 56, 37
x 1594323:1562500 34.90530033 [ -2 13 -8] 19, 96,
77, 58, 39
-- 1990656:1953125 32.95157954 [ 13 5 -9] 47, 31,
77, 46, 61, 15
ampersand 34171875:33554432 31.56728925 [-25 7 6] 41, 72, 31
magic 3125:3072 29.61356846 [-10 -1 5] 22, 63,
41, 60, 79, 19, 35
x 2.54E+12:2.5E+12 28.75174226 [-11 26 -13] 26, 91, 65
[minimal diesis] 20000:19683 27.65984767 [ 5 -9 4] 48, 41,
75, 34, 95, 61, 27
Pyth. Comma 531441:524288 23.46001038 [-19 12 0] 12, 96,
84, 72, 60, 48
meantone** 81:80 21.5062896 [ -4 4 -1] 12, 65,
43, 74, 31, 81, 50, 69, 88, 19, 45, 26
diaschismic 2048:2025 19.55256881 [ 11 -4 -2] 22, 78,
56, 90, 34, 80, 46, 58, 70, 12
x 78732:78125 13.39901073 [ 2 9 -7] 19, 84,
65, 46, 73
wuerschmidt 393216:390625 11.44528995 [ 17 1 -8] 28, 31,
96, 65, 99, 34, 71, 37
orwell 2109375:2097152 10.06099965 [-21 3 7] 31, 84,
53, 75, 97, 22
kleismic 15625:15552 8.107278862 [ -6 -5 6] 15, 49,
83, 34, 87, 53, 72, 91, 19, 23
x 1.22E+19:1.216E+19 7.015384276 [ 10 -40 23] 26, 99, 73
x 1224440064:1220703125 5.291731873 [ 8 14 -13] 19, 99,
80, 61, 42
schismic 32805:32768 1.953720788 [-15 8 1] 29, 41,
94, 53, 65, 77, 89, 12

(* Rameau major diesis)
(** syntonic comma)

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--- In tuning@y..., "monz" <joemonz@y...> wrote:

> excellent! thanks for that, Carl!
>
>
> i've corrected your table, inserted the missing ratios (easy
> to calculate from the vector), and added all the other vanishing
> commas for which Gene has supplied that data.

a lot of inconsistent ets in that table, for which the comma may not
really vanish (ooh, ooh, that fetish again). also, i reiterate my
desire for a 648:625 line, and maybe renaming 'diesic' to 'trefoil'
so 648:625 can be 'quatrefoil'.

also, the table should ideally include the *generator* of the linear
temperament, either in terms of the consistent ets, and/or the
optimal generator in cents -- since that's the most easily useful
single piece of data on a linear temperament.

I wrote,

> a lot of inconsistent ets in that table, for which the comma may
not
> really vanish (ooh, ooh, that fetish again). also, i reiterate my
> desire for a 648:625 line, and maybe renaming 'diesic' to 'trefoil'
> so 648:625 can be 'quatrefoil'.

though musically more meaningful terms would be 'augmented'
and 'diminished', respectively (the augmented scale is the 6-tone mos
of the 125;128 temperament, and the diminished scale is the 8-tone
mos of the 648;625 temperament).

>a lot of inconsistent ets in that table, for which the comma may
>not really vanish (ooh, ooh, that fetish again).

Right! I'll go through and remove them all.

>also, i reiterate my desire for a 648:625 line, and maybe
>renaming 'diesic' to 'trefoil' so 648:625 can be 'quatrefoil'.
>
>though musically more meaningful terms would be 'augmented'
>and 'diminished', respectively (the augmented scale is the
>6-tone mos of the 125;128 temperament, and the diminished scale
>is the 8-tone mos of the 648;625 temperament).

Yes, let's call these augmented and diminished.

//Generators on the table

Yes, I completely agree. Who can furnish rms optimums?

-Carl

--- In tuning@y..., "clumma" <carl@l...> wrote:

> Yes, I completely agree. Who can furnish rms optimums?

i bet gene can do this in a jiffy. maybe graham too. and oh, we need
the period as well as the generator.

--- In tuning@y..., "clumma" <carl@l...> wrote:

> Who can furnish rms optimums?

actually, gene already did this back in december. i'm making a graph
that includes these as well as the ets.

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning@y..., "clumma" <carl@l...> wrote:
>
> > Who can furnish rms optimums?
>
> actually, gene already did this back in december. i'm making a
graph
> that includes these as well as the ets.

well, i tried to, but the points get too crowded near the center for
me to label them. but it's easy to see the optimum point on the graph
on monz's page already. simply look at the line representing the
temperament you're interested in, and the point on that line that
comes closest to the center ('origin') of the graph is the optimal
one. so optimal meantone is near 50-equal, and optimal magic (in 5-
limit at least) is near 60-equal, etc.

--- In tuning@y..., "clumma" <carl@l...> wrote:
> >a lot of inconsistent ets in that table, for which the comma may
> >not really vanish (ooh, ooh, that fetish again).
>
> Right! I'll go through and remove them all.

hi carl,

have you done this and sent the result to monz? if so, his page isn't
reflecting it yet . . .

just hoping to avoid potential confusion for people,
paul

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning@y..., "clumma" <carl@l...> wrote:
> > >a lot of inconsistent ets in that table, for which the comma may
> > >not really vanish (ooh, ooh, that fetish again).
> >
> > Right! I'll go through and remove them all.
>
> hi carl,
>
> have you done this and sent the result to monz? if so, his page
isn't
> reflecting it yet . . .
>
> just hoping to avoid potential confusion for people,
> paul

just to emphasize the point, 65 is currently listed under meantone!