Tuning-Math Digests messages 6836 - 6860

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Message: 6836

Date: Fri, 30 May 2003 22:54:14

Subject: ...continued...

From: Carl Lumma

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


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Message: 6837

Date: Fri, 30 May 2003 23:02:02

Subject: Re: ...continued...

From: Carl Lumma

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


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Message: 6838

Date: Sat, 31 May 2003 11:40:30

Subject: Re: ...continued...

From: Carl Lumma

>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


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Message: 6842

Date: Sat, 31 May 2003 12:05:27

Subject: Re: ...continued...

From: Graham Breed

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


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