Tonality cube?

Another one of those "what do you call this" questions.

I make a square tonality diamond using all the numbers n = xy, where x and y are any integer between one and a set number; I'll use 16 in my example. This is the list of factors with all the redundancies removed (I hope there aren't any mistakes):

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 18 20 21 22 24 25 26 27 28 30 32 33 35 36 39 40 42 44 45 48 49 50 52 54 55 56 60 63 64 65 66 70 72 75 77 78 80 81 84 88 90 91 96 98 99 100 104 105 108 110 112 117 120 121 126 128 130 132 135 140 143 144 150 154 156 160 165 168 169 176 180 182 192 195 196 208 210 224 225 240 256

When octave-normalized, this produces a set of 507 ratios.

Is there a name for this? I'm thinking "cube" or "supersquare" or something.

This can also be extended to a "hypercube", using all the numbers n = xyz, where x, y and z are any integer between one and a certain number.

~Danny~

--- In tuning@yahoogroups.com, "Daniel A. Wier" <dawiertx@...> wrote:
>
> Another one of those "what do you call this" questions.
>
> I make a square tonality diamond using all the numbers
> n = xy, where x and y are any integer between one and
> a set number; I'll use 16 in my example. This is the
> list of factors with all the redundancies removed (I
> hope there aren't any mistakes):
>
> 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 18 20 21 22 24
> 25 26 27 28 30 32 33 ...
//
> When octave-normalized, this produces a set of 507 ratios.
>
> Is there a name for this? I'm thinking "cube" or
> "supersquare" or something.
>
> This can also be extended to a "hypercube", using all
> the numbers n = xyz, where x, y and z are any integer
> between one and a certain number.

Hi Danny,

How is this a tonality diamond? Looks more like a "cross
set" to me.

-Carl

Carl Lumma wrote:

> --- In tuning@yahoogroups.com, "Daniel A. Wier" <dawiertx@...> wrote:
>>
>> Another one of those "what do you call this" questions.
>>
>> I make a square tonality diamond using all the numbers
>> n = xy, where x and y are any integer between one and
>> a set number; I'll use 16 in my example. This is the
>> list of factors with all the redundancies removed (I
>> hope there aren't any mistakes):
>>
>> 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 18 20 21 22 24
>> 25 26 27 28 30 32 33 ...

[...]

> Hi Danny,
>
> How is this a tonality diamond? Looks more like a "cross
> set" to me.

I was going by what Scala calls a set of ratios A/B where A and B are identical, where octave normalization is optional; but the Tonalsoft and Wikipedia articles defines a tonality diamond as octave-normalized.

So it can be either. What I was describing can also be described as the sum of (A/B)*C and (A/B)/C, with A, B and C being identical sets again.

the diamond is a cross set (a set with its inversion) and this type of multiplication with cubes and hyper cubes was done or at least implied by by Euler
see http://anaphoria.com/Euler.PDF
--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

Kraig Grady wrote:

> the diamond is a cross set (a set with its inversion) and this type of
> multiplication with cubes and hyper cubes was done or at least implied
> by by Euler
> see http://anaphoria.com/Euler.PDF

Yeah, but that's an Euler genus in six dimensions, a 64-any (there's a big long Greek name for this, but I'm not going to try and guess it right now).

What I'm talking about is actually a hypercube, 4-D, not a cube as I said earlier. If a square tonality diamond is the set of ratios R = {1>n}/{1>n}, then my "square of a square" is R = ({1>n}/{1>n})/({1>n}/{1>n}). (I don't think I got the notation right.)

Or, it's the entire set of transpositions of a tonality diamond upon every ratio in the same diamond, then octave-normalized.

I *could* post the entire set of pitches, but there are 507 of them.

Hi Danny,

--- In tuning@yahoogroups.com, "Daniel A. Wier" wrote:
>
> Kraig Grady wrote:
>
> > the diamond is a cross set (a set with its inversion)
> > and this type of multiplication with cubes and hyper
> > cubes was done or at least implied by by Euler
> > see http://anaphoria.com/Euler.PDF
>
> Yeah, but that's an Euler genus in six dimensions, a
> 64-any (there's a big long Greek name for this, but I'm
> not going to try and guess it right now).
>
> What I'm talking about is actually a hypercube, 4-D, not
> a cube as I said earlier. If a square tonality diamond
> is the set of ratios R = {1>n}/{1>n},

How about writing instead I_n = {1..n} for the Initial
segment of the natural numbers bounded by and including
n? This is the set = {1, 2, ..., n} = {i: 1<=i<=n}.
Then your set R = R_n = I_n/I_n = {1..n}/{1..n}
= {i/j: 1<=i<=n, 1<=j<=n}.

> then my "square of a square" is
> R = ({1>n}/{1>n})/({1>n}/{1>n}).
> (I don't think I got the notation right.)

Using the same notation above, it's
R_n/R_n = (I_n/I_n)/(I_n/I_n)
= {ik/jl: 1<=i<=n, 1<=j<=n, 1<=k<=n, 1<=l<=n}.

Note that this R_n/R_n is a proper subset of R_(n^2).
Some members of R_(n^2), eg (n^2-1)/(n-1) = (n+1),
do not belong to R_n.

> Or, it's the entire set of transpositions of a
> tonality diamond upon every ratio in the same
> diamond, then octave-normalized.

The set notation above doesn't include the octave
normalisation step.

> I *could* post the entire set of pitches, but there
> are 507 of them.

To what point? Knowing their genesis and the
intervals between them would suffice for most
purposes, I think.

Regards,
Yahya

> >> Another one of those "what do you call this" questions.
> >>
> >> I make a square tonality diamond using all the numbers
> >> n = xy, where x and y are any integer between one and
> >> a set number; I'll use 16 in my example. This is the
> >> list of factors with all the redundancies removed (I
> >> hope there aren't any mistakes):
> >>
> >> 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 18 20 21 22 24
> >> 25 26 27 28 30 32 33 ...
>
> [...]
>
> > Hi Danny,
> >
> > How is this a tonality diamond? Looks more like a "cross
> > set" to me.
>
> I was going by what Scala calls a set of ratios A/B where
> A and B are identical, where octave normalization is
> optional;

What part of Scala (or its docs?) are you referring to here?

> but the Tonalsoft and Wikipedia articles defines a tonality
> diamond as octave-normalized.

I don't see any definition like yours on the Wikipedia page...

> So it can be either.

Strictly following Partch, tonality diamonds are octave-
normalized things. The term has been used more generally,
however.

> What I was describing can also be
> described as the sum of (A/B)*C and (A/B)/C, with
> A, B and C being identical sets again.

Sorry, you've lost me with these definitions. Can you give
an example of the ratios they produce?

-Carl

> > the diamond is a cross set (a set with its inversion)

This makes it sound like all cross sets are a set times
its inversion. A cross set is usually a set times itself.
A set times its inversion is indeed a 'tonality diamond',
though Partch apparently only used that term when the set
was a set of consecutive odd positive integers.

One can take a tonality diamond and cross it with
something (like the original set) and get a 3-D table
that's good for music. Wilson and Partch apparently
discussed this possibility. I've discussed it with
Denny Genovese, Kurt Bigler, and Jules Siegel, among
others.

-Carl

Carl Lumma wrote:

>> > Hi Danny,
>> >
>> > How is this a tonality diamond? Looks more like a "cross
>> > set" to me.
>>
>> I was going by what Scala calls a set of ratios A/B where
>> A and B are identical, where octave normalization is
>> optional;
>
> What part of Scala (or its docs?) are you referring to here?

It's the scale-generating function in Scala called "Square Tonality Diamond", where you enter a list of factors, and it produces a cross-product set. Octave normalizatin is optional and marked in a checkbox.

>> What I was describing can also be
>> described as the sum of (A/B)*C and (A/B)/C, with
>> A, B and C being identical sets again.
>
> Sorry, you've lost me with these definitions. Can you give
> an example of the ratios they produce?

I got my definition wrong, it should be (A/B)/(C/D).

I'll use a smaller set of numbers for my example: 7-limit, and I'll normalize it. The 7-limit diamond consists of the ratios:

1/1 8/7 7/6 6/5 5/4 4/3 7/5 10/7 3/2 8/5 5/3 12/7 7/4 2/1

If I was to transpose the diamond by every one of these intervals, add all the resulting intervals together, and normalize, I'd get this:

1/1 50/49 49/48 36/35 25/24 21/20 16/15 15/14 35/32 10/9 28/25 9/8 8/7 7/6 25/21 6/5 60/49 49/40 5/4 32/25 9/7 64/49 21/16 4/3 49/36 48/35 25/18 7/5 10/7 36/25 35/24 72/49 3/2 32/21 49/32 14/9 25/16 8/5 80/49 49/30 5/3 42/25 12/7 7/4 16/9 25/14 9/5 64/35 28/15 15/8 40/21 48/25 35/18 96/49 49/25 2/1

It's now a diamond using the factors 1 3 5 7 9 15 21 25 35 49.

> I'll use a smaller set of numbers for my example: 7-limit,
> and I'll normalize it. The 7-limit diamond consists of the
> ratios:
>
> 1/1 8/7 7/6 6/5 5/4 4/3 7/5 10/7 3/2 8/5 5/3 12/7 7/4 2/1

That looks right.

> If I was to transpose the diamond by every one of these
> intervals, add all the resulting intervals together, and
> normalize,

I'd call that the "cross set of the 7-limit tonality diamond".
YMMV.

-Carl

> see http://anaphoria.com/Euler.PDF

This raises the question, what 13-limit mapping is:

0 2 8 20 36 -1

Some kind of hemi-meantone?

-Carl

Carl Lumma wrote:
>> see http://anaphoria.com/Euler.PDF
> > This raises the question, what 13-limit mapping is:
> > 0 2 8 20 36 -1
> > Some kind of hemi-meantone?

I can't find anything like this; it's likely to be a rank 3 temperament with the odd-numbered keys representing a separate dimension. On the other hand, I did find this, with 22/105 generator (and no name):

[1, 2, 4, 7, 11, -3, 11]
[0, -2, -8, -20, -36, 32, -33]

> >> see http://anaphoria.com/Euler.PDF
> >
> > This raises the question, what 13-limit mapping is:
> >
> > 0 2 8 20 36 -1
> >
> > Some kind of hemi-meantone?
>
> I can't find anything like this; it's likely to be a rank 3
> temperament

It calls a half-fifth 16/13. With 13/8 in JI, that
makes for a fifth of 718 cents. Which is better than
the oft-used 15-tET, but which doesn't make for a
very good meantone third of 472 cents! I'm sure that
can be improved by tempering 13/8 and 2/1, but I doubt
it would ever be very accurate. Wilson's goal is
apparently JI with melodic sensibility, however. The
mapping only being used for the latter.

-Carl

> > >> see http://anaphoria.com/Euler.PDF
> > >
> > > This raises the question, what 13-limit mapping is:
> > >
> > > 0 2 8 20 36 -1
> > >
> > > Some kind of hemi-meantone?
> >
> > I can't find anything like this; it's likely to be
> > a rank 3 temperament
>
> It calls a half-fifth 16/13. With 13/8 in JI, that
> makes for a fifth of 718 cents. Which is better than
> the oft-used 15-tET, but which doesn't make for a
> very good meantone third of 472 cents! I'm sure that
> can be improved by tempering 13/8 and 2/1, but I doubt
> it would ever be very accurate. Wilson's goal is
> apparently JI with melodic sensibility, however. The
> mapping only being used for the latter.

I guess if the octave were tempered by 5 cents and the
13/8 by 8 cents, that could give us 472-74 = 398, or
slightly better than 12-tET thirds... and something
like 995 for the 7:4 and 586 for 11:8. But doing this
in my head to the nearest cent might be getting me in
trouble...

-C.