Tuning-Math Digests messages 10708 - 10732

This is an Opt In Archive . We would like to hear from you if you want your posts included. For the contact address see About this archive. All posts are copyright (c).

Contents Hide Contents S 11

Previous Next

10000 10050 10100 10150 10200 10250 10300 10350 10400 10450 10500 10550 10600 10650 10700 10750 10800 10850 10900 10950

10700 - 10725 -



top of page bottom of page down


Message: 10708

Date: Mon, 29 Mar 2004 14:55:17

Subject: Re: Some 13-limit microtemperaments

From: Graham Breed

Gene Ward Smith wrote:
> I took all the 13-limit superparticular commas with numerator greater
> than 2000 four at a time, giving 15 linear temperaments, listed below
> in TOP/Graham badness order. The first temperament on the list seems
> to be a standout, it has TM basis {1716/1715, 2080/2079, 3025/2024,
> 4096/4095}, a period of 1/2 octave and a 44/39 generator. Ets are 224,
> 270, and 494.

I take it you must have started with 6 commas, then.  Could you tell us 
what they are?  Then I could try duplicating the result.  Or you could 
even past them straight into the box at

Temperament Finder *

It'd help, anyway, if you were to paste commas into your messages 
instead of typing them.  It took me a while to work out that 3025/2024 
was supposed to be 3025/3024.

Oh yes, the standout temperament was already top of my 13- and 15-limit 
microtemperament lists.


                 Graham


________________________________________________________________________
________________________________________________________________________



------------------------------------------------------------------------
Yahoo! Groups Links

<*> To visit your group on the web, go to:
     Yahoo groups: /tuning-math/ *

<*> To unsubscribe from this group, send an email to:
     tuning-math-unsubscribe@xxxxxxxxxxx.xxx

<*> Your use of Yahoo! Groups is subject to:
     Yahoo! Terms of Service *


top of page bottom of page up down


Message: 10711

Date: Tue, 30 Mar 2004 09:51:23

Subject: Re: 98 out of 99

From: Carl Lumma

>>If you take a 5x5x5 chord cube in the 9-limit lattice of quintads, 
>>you get 153 notes to an octave. Reducing this by 99-et leads to
>>98 out of the 99 possible notes, the odd note out being 40, which
>>represents 250/189 (its TM reduced representative.) We may harmonize
>>this by adding the chord [1 -3 1], which is
>>25/21-250/189-125/84-250/147-125/63.  This is the 1-6/5-4/3-3/2-12/7
>>utonal quintad over 125/126.
>> 
>>All quintads [i j k] with absolute values less of i,j, and k less 
>>than 3, with the addition of [1 -3 1], is therefore one way to
>>harmonize everything in 99-et. This is a set of 126=5^3+1 quintads,
>>63 otonal and 63 utonal, each of which is distinct in 99-et. I may
>>try this for my next piece.
>
>Could you remind me how [a b c] represents a quintad in a 9-limit 
>lattice? (I know, I should look in the archives...)

Ok, I'll take a stab at this, and maybe Gene can step in later.
This is the lattice *of* quintads -- Z3, I believe.  Gene has usually
used done this with the dual to A3 in the 7-limit.  I think there may
have been a post at some point about how to get it to work in the
9-limit.... one can extend the chords indefinitely and still use Z3,
but not usually without leaving out certain modulations.  For example,
I'm not sure how Z3 can hold modulations by 9:5, 9:7, etc...

-Carl


top of page bottom of page up down


Message: 10713

Date: Tue, 30 Mar 2004 11:06:17

Subject: Re: 98 out of 99

From: Carl Lumma

>9/5 and 9/7 are simply 3*3/5 and 3*3/7; in other words I'm not 
>treating 9 any differently for this purpose, only using it when 
>constructing chords. This works fine for the 9-limit but obviously 
>not beyond.

I don't follow.  I can extend the scheme to the 11-limit and
not have modulations by any ratios of 11, even though the chords
contain 11-identities.  Why are ratios of 9 any different?

-Carl


top of page bottom of page up down


Message: 10714

Date: Tue, 30 Mar 2004 22:22:32

Subject: Re: Stoermer

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
> wrote:
> > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> > wrote:
> > 
> > > Great. Xenharmonikon readers might like to know: are there any 
23-
> > > limit superparticulars above 10,000,000?
> > 
> > Nope.
> 
> I spoke too soon, after only looking at the strict 23 limit table. 
For
> some reason probably connected to the fact that 19 is the larger of 
a
> twin prime pair, there are two 19-limit commas smaller than any
> strictly 23-limit comma. They are:
> 
> 5909761/5909760
> |-8 -5 -1 0 2 2 2 -1>
> 
> 11859211/11859210
> |-1 -4 -1 1 -4 1 0 4>

As expected, the former appears in XH17; the latter doesn't. So John 
Chalmers could inform the readership that if the 19-limit list is 
supplemented with 11859211/11859210, the "less than 10^7" 
qualification can be dropped, and the lists are complete. The total 
number of superparticulars in the 23-prime-limit would then be 241.


top of page bottom of page up down


Message: 10716

Date: Tue, 30 Mar 2004 18:57:48

Subject: Re: Musical harmony a fuzzy entropic characterization

From: Carl Lumma

>Vidyamurthy and Murty, Musical harmony a fuzzy entropic
>characterization, Fuzzy Sets and Systems 48 (1992) #2, 195-200

Boy does this look familiar, but I can't find the reference...

-Carl


top of page bottom of page up down


Message: 10719

Date: Wed, 31 Mar 2004 10:40:59

Subject: Re: Questions for Carl

From: Carl Lumma

>I was wondering if you could bring me up to speed regarding some
>gaps I have in my understanding of things that are posted on this
>list.

You may have me confused with someone who understands the things
posted on this list.  :)

>1. I notice that (sometimes)generators-to-primes "line up" with
>temperament values (like 12 19 28) when you invert a matrix of
>commas, and then multiply by the determinant. Is there a rule
>for this?

This sounds vaguely like Paul E.'s procedure to find the number
of notes in a periodicity block, which would correspond to the
number of pitches in an ET based on those commas (barring torsion).

In general, there are recurrence relations that give the number
of pitches in 'good' temperaments.  Some of them can be picked
off the Stern-Brocot tree.  Gene knows more about this.

>2. I still am not clear on how period values are calculated. (Using
>matrices). Using wedge products I see how they are calculated
>(from the wedgie) but I still am having trouble convincing myself
>why wedging period ^ generators leads to the same wedgie as you
>would obtain from monzo ^ monzo or value ^ value. (I know that
>monzo wedgie is backwards from value wedgie, etc)

I have yet to understand the techniques based on wedge products.
Maybe you can help explain them to me once you've mastered them!

>3. I get how periods may be part of an octave, when gcd(values)
>is not 1. Once again, is there a rule where these generators
>and periods "line up" with temperament values?

This sounds like your first question.  What exactly do you mean
by "temperament values"?  And "line up"?

-Carl


top of page bottom of page up down


Message: 10720

Date: Wed, 31 Mar 2004 10:47:36

Subject: Re: 98 out of 99

From: Carl Lumma

>>>Could you remind me how [a b c] represents a quintad in a 9-limit 
>>>lattice? (I know, I should look in the archives...)
>> 
>>Ok, I'll take a stab at this, and maybe Gene can step in later.
>>This is the lattice *of* quintads -- Z3, I believe.  Gene has 
>>usually used done this with the dual to A3 in the 7-limit.  I
>>think there may have been a post at some point about how to get
>>it to work in the 9-limit.... one can extend the chords
>>indefinitely and still use Z3, but not usually without leaving
>>out certain modulations.  For example, I'm not sure how Z3 can
>>hold modulations by 9:5, 9:7, etc...
>
>Do you mean the Z3 group?

I mean the cubic lattice.  Gene once told me it was called Z3.
Maybe it's the lattice that you get when you assume the symmetries
of the Z3 group??

>What's A3?

The FCC (usually 7-limit) lattice.

>Still don't see how the [i j k]
>represents a quintad...Thanks

Actually, if you follow the thread, you'll see I'm still waiting
for Gene to explain how he can get modulations of 9:5, 9:7, etc,
by moving only distance 1 on the lattice.  I thought the whole
point of the observation that the dual of the 7-limit lattice is
also a lattice was that you can represent all the possible
modulations as a single step (there are 6 possible modulations
in the 7-limit, and every point in the cubic lattice is connected
to 6 others).

-Carl


top of page bottom of page up down


Message: 10721

Date: Wed, 31 Mar 2004 19:55:20

Subject: Re: Questions for Carl

From: Graham Breed

Paul G Hjelmstad wrote:

> 1. I notice that (sometimes)generators-to-primes "line up" with
> temperament values (like 12 19 28) when you invert a matrix of
> commas, and then multiply by the determinant. Is there a rule
> for this? 

"Invert a matrix ... and then multiply by the determinant" gives the 
adjoint.  Once you know the word, it's easier to use it, because the 
inverse as such isn't that interesting.

Yes, the column of the adjoint corresponding to the row of the original 
matrix that represents the octave gives you a representative ET 
mapping/val/constant structure.  If you want to temper out all the 
commas, that's the equal temperament you were looking for.

If you don't temper out any columns, the constant structure refers to 
the periodicity block.  Fokker only worked with octave-equivalent 
matrices, so his determinant only told him how many notes there were. 
The octave-specific method gives you the mappings for the other primes, 
and also helps you find and remove torsion.

If you temper out some but not all commas, this is one equal temperament 
that's a special case of whatever dimensioned temperament you end up with.

> 2. I still am not clear on how period values are calculated. (Using
> matrices). Using wedge products I see how they are calculated
> (from the wedgie) but I still am having trouble convincing myself
> why wedging period ^ generators leads to the same wedgie as you
> would obtain from monzo ^ monzo or value ^ value. (I know that
> monzo wedgie is backwards from value wedgie, etc)

Do you mean the whole column mapping intervals to periods?  It's a 
fiddly calculation, and I'd need to check with my source code, which you 
have anyway.  But do it once and you can forget about it.

It happens that the period and generator mappings are both vals.  The 
generator mapping is special (and unique) in that it refers to an 
imaginary equal temperament with zero notes to the octave.  It must be a 
val, because you get it from the adjoint of the matrix of commas.  It's 
what you get when one of the "commas" you temper out is an octave.  You 
could get it by repeatedly subtracting more sensible vals, because the 
difference between vals is always a val (even if it isn't an equal 
temperament).

For temperaments like mystery that divide the octave into a middling 
number of steps, it's more obvious the period mapping is a val as well.

> 3. I get how periods may be part of an octave, when gcd(values)
> is not 1. Once again, is there a rule where these generators
> and periods "line up" with temperament values?

I don't see what you mean by this.

> 4. Geometry. I've got some questions... I'll discuss in the 
> relevant post
> 
> Thanks! Anyone else, feel free to chime in...

Yes, well, I've done so.  I don't know if you've given up on me yet.


              Graham


top of page bottom of page up

Previous Next

10000 10050 10100 10150 10200 10250 10300 10350 10400 10450 10500 10550 10600 10650 10700 10750 10800 10850 10900 10950

10700 - 10725 -

top of page