Monzo's lines

Here are more comma for the lines on Joe's graph at:

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

First, the 31-46-15 line is associated to 1990656/1953125, which is not the kleisma, so this should be changed.

The 59-71-12 line goes with 2^29 3^-8 5^-7 = 536870192/512578125.

The 72-84-12 line is of course the Pythagorean comma line,
2^-19 3^12.

The 23-37-45-12 line is associated to 6561/6250.

The magic line is associated to 3125/3072 already on the chart; this is the small diesis.

The kleisma = 15625/15552 is associated to the 19-72-53-34 line,which should get the name "kleismic".

The 34-99-65 line is associated to 393216/390625, which is Wuerschmidt's comma; I've already taken to calling the associated linear temperament the wuerschmidt.

The 37-56-75-94-19 line is associated to 2^-16 3^-6 5^11 =
48828125/4775744.

Finally the Orwell line of 22-75-53-84-31 is associated to the semicomma, which is 2^29 3^-8 5^-7 = 2109375/2097152.

> From: genewardsmith <genewardsmith@juno.com>
> To: <tuning@yahoogroups.com>
> Sent: Friday, February 15, 2002 12:12 AM
> Subject: [tuning] Monzo's lines
>
>
> Here are more comma for the lines on Joe's graph at:
>
> http://www.ixpres.com/interval/dict/eqtemp.htm

wow! thanks, Gene! this is awesome!

> The 23-37-45-12 line is associated to 6561/6250.

shouldn't that be "The 23-35-47-12 line ..." ?

> The 37-56-75-94-19 line is associated to 2^-16 3^-6 5^11 =
> 48828125/4775744.

that denominator is wrong -- you left a 7 out.
4775744 = 2^6 * 71^1 * 1051^1 . The correct value is
47775744.

so the 37-49-61-73-12 line is the only line i drew
that's still left unaccounted for.

take a look at the webpage now -- the graphic is
much better.

-monz

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--- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > From: genewardsmith <genewardsmith@j...>
> > To: <tuning@y...>
> > Sent: Friday, February 15, 2002 12:12 AM
> > Subject: [tuning] Monzo's lines
> >
> >
> > Here are more comma for the lines on Joe's graph at:
> >
> > http://www.ixpres.com/interval/dict/eqtemp.htm
>
>
> wow! thanks, Gene! this is awesome!
>
>
>
> > The 23-37-45-12 line is associated to 6561/6250.
>
>
> shouldn't that be "The 23-35-47-12 line ..." ?

gene probably couldn't read the numbers because of all the red lines!
i can't read the 31. the blue lines fade into the background. maybe
you can use yellow lines instead of red ones?

and you guys forgot about one of the most wonderful 5-limit commas of
all, the 250:243, upon which the stunning chord progression of herman
miller's 'mizarian porcupine overture' is based. he wrote it in 15,
but it would also work in 37, 59, 22, or 29.

--- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > From: genewardsmith <genewardsmith@j...>
> > To: <tuning@y...>
> > Sent: Friday, February 15, 2002 12:12 AM
> > Subject: [tuning] Monzo's lines
> >
> >
> > Here are more comma for the lines on Joe's graph at:
> >
> > http://www.ixpres.com/interval/dict/eqtemp.htm

hi monz.

in the hope of reducing clutter and increasing readability, and also
with the hope of getting 16-equal included, i've produced
another 'template' for you:

/tuning/files/perlich/monz.gif

again, i hope your added lines will be in yellow (or perhaps pink for
valentine's day), or at least be dotted lines . . .

16-equal should occur at the intersection of two very important
lines: the diminished scale line (comma 648:625) passing through 12,
28, and 16; and the pelog line (comma 135:128) passing though 23 and
16 (and 7 and 9, not on this graph but included in equaltemp2.jpg)

by the way, an important omission on your page: harald waage, in the
pages of _1/1_ and _interval_, has advocated 84-equal for 'just' 5-
limit harmony which preserves the familiar 'bicycle chain' circle of
fifths (since 84 = 12*7). i have such an _interval_ article here,
dated 1985-1986.

i wrote,

> and you guys forgot about one of the most wonderful 5-limit commas
of
> all, the 250:243, upon which the stunning chord progression of
herman
> miller's 'mizarian porcupine overture' is based. he wrote it in 15,
> but it would also work in 37, 59, 22, or 29.

this line has mysteriously appeared on monz's graph, alas in red and
with no name. perhaps we should call this 'porcupine' in honor of
herman's piece?

>in the hope of reducing clutter and increasing readability, and
>also with the hope of getting 16-equal included, i've produced
>another 'template' for you:
>
>/tuning/files/perlich/monz.gif

Any reason 16 was left out originally (anything else missing?)?

-Carl

----- Original Message -----
From: clumma <carl@lumma.org>
To: <tuning@yahoogroups.com>
Sent: Friday, February 15, 2002 6:02 PM
Subject: [tuning] Re: Monzo's lines

> >in the hope of reducing clutter and increasing readability, and
> >also with the hope of getting 16-equal included, i've produced
> >another 'template' for you:
> >
> >/tuning/files/perlich/monz.gif
>
> Any reason 16 was left out originally (anything else missing?)?

16, 10, 7 ... several of the low-cardinality EDOs which
don't really give all that great approximations to the
5-limit concordances. because the approximations aren't
very good, those EDOs lay pretty far from the center, and
when paul redid the diagram to make it clearer, he scaled
the JI axes a bit larger so the numbers would have more
space between them. in the process, the outlying EDOs
got cut out of the picture.

-monz

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here are a few more linear axes that i can see
on my adaptation of paul's graph of 5-limit ETs:

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

48-63-78-15

26-41-97-56-71-15

45-60-75-90-15

19-99-80-61-42

19-84-65-46-73

19-96-77-58-39

26-99-73

26-91-65

Gene, care to calculate those commas too?

unfortunately, i don't think i want to illustrate any
more lines on this diagram, because it's already too cluttered.

paul: thanks for making yet another version of the template
. . . but i'm afraid that i'll have to put considerable work
into using that to make a new diagram. in the meantime, i've
cleaned up the old one quite a bit.

-monz

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--- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> here are a few more linear axes that i can see
> on my adaptation of paul's graph of 5-limit ETs:
>
> http://www.ixpres.com/interval/dict/eqtemp.htm
>
>
> 48-63-78-15

2^-4 3^-15 5^12

> 26-41-97-56-71-15

2^-5 3^-10 5^9 = 1953125/1889568

> 45-60-75-90-15

2^-11 3^-15 5^15

> 19-99-80-61-42

2^8 3^14 5^-13

> 19-84-65-46-73

2^2 3^9 5^-7 = 78732/78125

This is the comma defining one of the better 5-limit temperaments, but I couldn't find a name for it, and jokingly suggested the "quite small diesis". Anone care to take a shot at it? If you have room to label one more line, you could label this one.

> 19-96-77-58-39

2^-2 3^13 5^-8

> 26-99-73

2^10 3^40 5^23

> 26-91-65

2^-11 3^26 5^-13

> 16, 10, 7 ... several of the low-cardinality EDOs which
> don't really give all that great approximations to the
> 5-limit concordances. because the approximations aren't
> very good, those EDOs lay pretty far from the center, and
> when paul redid the diagram to make it clearer, he scaled
> the JI axes a bit larger so the numbers would have more
> space between them. in the process, the outlying EDOs
> got cut out of the picture.

Aha! -C.

> From: paulerlich <paul@stretch-music.com>
> To: <tuning@yahoogroups.com>
> Sent: Friday, February 15, 2002 4:06 PM
> Subject: [tuning] Re: Monzo's lines
>
>
> i wrote,
>
> > and you guys forgot about one of the most wonderful 5-limit commas
> of
> > all, the 250:243, upon which the stunning chord progression of
> herman
> > miller's 'mizarian porcupine overture' is based. he wrote it in 15,
> > but it would also work in 37, 59, 22, or 29.
>
> this line has mysteriously appeared on monz's graph, alas in red and
> with no name. perhaps we should call this 'porcupine' in honor of
> herman's piece?

i'd accept that name for the family of temperaments.

i suggested calling that "comma" the "super-tripental great diesis"
because 250:243 is [2 3 5]^[1 -5 3], and so the name describes
the 5^3 part, which uniquely qualifies this diesis from the
others which fall in that generic interval size range but
have different exponents of 5. See
http://www.ixpres.com/interval/td/monzo/o483-26new5limitnames.htm

Rameau gave it the name "major diesis".

-monz

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

> paul: thanks for making yet another version of the template
> . . . but i'm afraid that i'll have to put considerable work
> into using that to make a new diagram.

that's a real shame, because the addition of 16 allows two of the
most important 5-limit commas to be included -- 648:625, which
defines the diminished/octatonic scale, central to some of my
favorite 20th century music by stravinsky, bartok, bloch, and a
great many jazz musicians since the 40s . . . and 135:128, which
defines the indonesian pelog scale if one hears it as supporting
5-limit harmony.

--- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > From: paulerlich <paul@s...>
> > To: <tuning@y...>
> > Sent: Friday, February 15, 2002 4:06 PM
> > Subject: [tuning] Re: Monzo's lines
> >
> >
> > i wrote,
> >
> > > and you guys forgot about one of the most wonderful 5-limit
commas
> > of
> > > all, the 250:243, upon which the stunning chord
progression of
> > herman
> > > miller's 'mizarian porcupine overture' is based. he wrote it
in 15,
> > > but it would also work in 37, 59, 22, or 29.
> >
> > this line has mysteriously appeared on monz's graph, alas in
red and
> > with no name. perhaps we should call this 'porcupine' in
honor of
> > herman's piece?
>
>
> i'd accept that name for the family of temperaments.

i don't see 'porcupine' on
http://www.ixpres.com/interval/dict/eqtemp.htm . . .

you know what would be cool? if you could click on one of the
commas/lines on this graph, and go to a page describing the
corresponding linear temperament, perhaps the optimal one in
the woolhouse sense, and a list of the associated mos scales.
for example, meantone would take you to a page mentioning the
696-cent generator and 1200-cent period as characterizing the
linear temperament, and a brief depiction of the 2-, 3-, 5-, 7-, 12-,
and 19-tone mos scales, perhaps all in a single horagram . . .