...continued...

Gene wrote...

>If T is a linear temperament, and T[n] a scale (within an octave) of
>n notes, then if the number of generator steps for an interval q times
>the number of periods in an octave is +-n, q is a chroma for T[n]. In
>terms of the programs I sent you, a7d(T,q)[1] = +-n.
>
>>Thanks. I believe that is the chromatic uv. There should only be
>>one for a given T[n].
>
>Any one of these, times a comma of T[n], will be another one; hence
>they are infinite in number.

Yeah, but this is true for any interval in the temperament.

What isn't clear to me is:

1.
The choice of a chroma q and pi(p)-1 commas (where p is the harmonic
limit) specifies a linear temperament, right? It does not specify
an n for T[n] or a tuning for T, but it does specify a family of maps
(and if we're lucky, a canonic map), right?

2.
Yet above it appears that changing n in T[n] changes the chroma (but
obviously not the temperament, T). Therefore, we have a problem,
unless changing n can only change the chroma among the family of
comma-transposed chroma for that T... ?

Finally, note that I'm still confused about prime- vs. odd-limit as
regards pi(p)-1. Obviously I'm assuming prime-limit here, but should
pi(p) be changed to ceiling(p/2)? That is, how many commas does a
9-limit linear temperament require? Paul?

-Carl

Just a note; for the purposes of the below message, and from now on,
I intend "chroma" = "chromatic unison vector" and "comma" = "commatic
unison vector".

-Carl

>Gene wrote...
>
>>If T is a linear temperament, and T[n] a scale (within an octave) of
>>n notes, then if the number of generator steps for an interval q times
>>the number of periods in an octave is +-n, q is a chroma for T[n]. In
>>terms of the programs I sent you, a7d(T,q)[1] = +-n.
>>
>>>Thanks. I believe that is the chromatic uv. There should only be
>>>one for a given T[n].
>>
>>Any one of these, times a comma of T[n], will be another one; hence
>>they are infinite in number.
>
>Yeah, but this is true for any interval in the temperament.
>
>What isn't clear to me is:
>
>1.
>The choice of a chroma q and pi(p)-1 commas (where p is the harmonic
>limit) specifies a linear temperament, right? It does not specify
>an n for T[n] or a tuning for T, but it does specify a family of maps
>(and if we're lucky, a canonic map), right?
>
>2.
>Yet above it appears that changing n in T[n] changes the chroma (but
>obviously not the temperament, T). Therefore, we have a problem,
>unless changing n can only change the chroma among the family of
>comma-transposed chroma for that T... ?
>
>
>Finally, note that I'm still confused about prime- vs. odd-limit as
>regards pi(p)-1. Obviously I'm assuming prime-limit here, but should
>pi(p) be changed to ceiling(p/2)? That is, how many commas does a
>9-limit linear temperament require? Paul?
>
>-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> The choice of a chroma q and pi(p)-1 commas (where p is the harmonic
> limit) specifies a linear temperament, right? It does not specify
> an n for T[n] or a tuning for T, but it does specify a family of maps
> (and if we're lucky, a canonic map), right?

No, the chroma has nothing to do with defining the temperament; it
defines the scale, given the temperament.

> Finally, note that I'm still confused about prime- vs. odd-limit as
> regards pi(p)-1. Obviously I'm assuming prime-limit here, but should
> pi(p) be changed to ceiling(p/2)? That is, how many commas does a
> 9-limit linear temperament require? Paul?

Exactly as many as a 7-limit linear temperament.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> The choice of a chroma q and pi(p)-1 commas (where p is the harmonic
> limit) specifies a linear temperament, right?

This should be pi(p)-2 commas, and no chroma, or 2 vals. In general
pi(p)-n commas, or n vals, specifies an (n-1)-temperament.

Carl Lumma wrote:

> 1.
> The choice of a chroma q and pi(p)-1 commas (where p is the harmonic
> limit) specifies a linear temperament, right? It does not specify
> an n for T[n] or a tuning for T, but it does specify a family of maps
> (and if we're lucky, a canonic map), right?

Gene says pi(p)-2 commas, which will be correct if you're counting 2. It does specify the n, but not the tuning for T. If you don't want n, you don't need the chroma.

> 2.
> Yet above it appears that changing n in T[n] changes the chroma (but
> obviously not the temperament, T). Therefore, we have a problem,
> unless changing n can only change the chroma among the family of
> comma-transposed chroma for that T... ?

No, there's no problem.

> Finally, note that I'm still confused about prime- vs. odd-limit as
> regards pi(p)-1. Obviously I'm assuming prime-limit here, but should
> pi(p) be changed to ceiling(p/2)? That is, how many commas does a
> 9-limit linear temperament require? Paul?

It generally goes by prime numbers, or more generally by prime intervals -- that is a set of intervals none of which can be arrived at by adding and subtracting the other ones. This is like linear independence. So the 5-limit (2-3-5) requires one comma. So does the 2-3-7 limit. And so would a system composed of octaves, fifths, and 7:5 tritones, although it uses 4 prime numbers.

Graham

>No, the chroma has nothing to do with defining the temperament; it
>defines the scale, given the temperament.

>Gene says pi(p)-2 commas, which will be correct if you're counting 2.
>It does specify the n, but not the tuning for T. If you don't want n,
>you don't need the chroma.

Thanks. Got it.

>This should be pi(p)-2 commas, and no chroma, or 2 vals. In general
>pi(p)-n commas, or n vals, specifies an (n-1)-temperament.

Can someone give the vals for 5-limit meantone?

>> Finally, note that I'm still confused about prime- vs. odd-limit as
>> regards pi(p)-1. Obviously I'm assuming prime-limit here, but should
>> pi(p) be changed to ceiling(p/2)? That is, how many commas does a
>> 9-limit linear temperament require? Paul?
>
>Exactly as many as a 7-limit linear temperament.

>... linear independence. So the 5-limit (2-3-5) requires one comma.
>So does the 2-3-7 limit. And so would a system composed of octaves,
>fifths, and 7:5 tritones, although it uses 4 prime numbers.

Ok, but what about stuff like (2-3-5-9) where we don't have linear
independence but wish to consider 9 as consonant as 3 or 5? How does
visualization in terms of blocks work on a lattice with a 9-axis?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> Can someone give the vals for 5-limit meantone?

Various choices are possible:

(1) Eytan, consisting of h12 = [12, 19, 28] and h7 = [7, 11, 16]

(2) Octave and fifth, consisting of oct = [1, 1, 0] and
fif = [0, 1, 4]

(3) Nominals and accidentals, consisting of h7 = [7, 11, 16] and
h5 = [5, 8, 12]

(4) Very meantone, h31 = [31, 49, 72] and h19 = [19, 30, 44]

And so forth. I suggest you try sticking each of these pairs into
my Maple routine "a5val" and seeing what comes forth.

> Ok, but what about stuff like (2-3-5-9) where we don't have linear
> independence but wish to consider 9 as consonant as 3 or 5? How does
> visualization in terms of blocks work on a lattice with a 9-axis?

If we have linearity, then if 9 is as consonant as 5, 3 will have to
be twice as consonant.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> How does
> visualization in terms of blocks work on a lattice with a 9-axis?

not very well, since every pitch will appear in an infinite number of
positions.

>>Can someone give the vals for 5-limit meantone?
>
>Various choices are possible:
>
>(1) Eytan, consisting of h12 = [12, 19, 28] and h7 = [7, 11, 16]
>
>(2) Octave and fifth, consisting of oct = [1, 1, 0] and
>fif = [0, 1, 4]
>
>(3) Nominals and accidentals, consisting of h7 = [7, 11, 16] and
>h5 = [5, 8, 12]
>
>(4) Very meantone, h31 = [31, 49, 72] and h19 = [19, 30, 44]
>
>And so forth. I suggest you try sticking each of these pairs into
>my Maple routine "a5val" and seeing what comes forth.

If I enter "a5val [19, 30, 44] [31, 49, 72];", it just spits back
the input.

>If we have linearity, then if 9 is as consonant as 5, 3 will have to
>be twice as consonant.

Why?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> >And so forth. I suggest you try sticking each of these pairs into
> >my Maple routine "a5val" and seeing what comes forth.
>
> If I enter "a5val [19, 30, 44] [31, 49, 72];", it just spits back
> the input.

a5val is a Maple function, so you need to enter

a5val([19, 30, 44], [31, 49, 72]);

> >If we have linearity, then if 9 is as consonant as 5, 3 will have to
> >be twice as consonant.
>
> Why?

Because if a is the approximation to 3, a^2 will be the approximation
to 9. Then the error log(a^2/9) will be twice that of log(a/3).

>>>And so forth. I suggest you try sticking each of these pairs
>>>into my Maple routine "a5val" and seeing what comes forth.
>>
>>If I enter "a5val [19, 30, 44] [31, 49, 72];", it just spits back
>>the input.
>
>a5val is a Maple function, so you need to enter
>
>a5val([19, 30, 44], [31, 49, 72]);

Ah, I'd tried that without the comma between the two lists.

Great. So they all give the comma for meantone. And I'm
beginning to understand vals.

>>>>>>Finally, note that I'm still confused about prime- vs. odd-limit
>>>>>>as regards pi(p)-1. Obviously I'm assuming prime-limit here,
>>>>>>but should pi(p) be changed to ceiling(p/2)? That is, how many
>>>>>>commas does a 9-limit linear temperament require? Paul?
>>>>>
>>>>>... linear independence. So the 5-limit (2-3-5) requires one
>>>>>comma. So does the 2-3-7 limit. And so would a system composed
>>>>>of octaves, fifths, and 7:5 tritones, although it uses 4 prime
>>>>>numbers.
>>>>
>>>>Ok, but what about stuff like (2-3-5-9) where we don't have linear
>>>>independence but wish to consider 9 as consonant as 3 or 5? How
>>>>does visualization in terms of blocks work on a lattice with a
>>>>9-axis?
>>>
>>>If we have linearity, then if 9 is as consonant as 5, 3 will have
>>>to be twice as consonant.
>>
>>Why?
>
>Because if a is the approximation to 3, a^2 will be the approximation
>to 9. Then the error log(a^2/9) will be twice that of log(a/3).

I'm not talking about error, but the assumption brought to the table,
namely that ratios of 9 are no less consonant than ratios of 3 in a
9-limit tuning, so that whatever errors exist should not be weighted
differently when optimizing the tuning.

Have you been including errors in ratios of 9 in optimizations?

I notice you never talk of the 9-limit. Only the 7- and 11- limits.
At the 7-limit, the 3-error is counted once. At the 11-limit, it
should be counted thrice.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> Have you been including errors in ratios of 9 in optimizations?

Of course; otherwise it would not be a 9-limit opitmization.

> I notice you never talk of the 9-limit. Only the 7- and 11- limits.

I do sometimes.