Yahoo Tuning Groups Ultimate Backup tuning-math 26 Hexachords and Cryptography

Since my 26 hexachord system uses A - Z, it could also be used
as a code (a la "Enigma" for example)

Since I am studying Crytography as part of my career, I hope
to combine the math in this with the math of hexachords.

One thing that will tie them together is the Finite Field.

I am going to see if I can find if anyone else has exploited
the 26 hexachord types as a code (monoalphabetical or
polyalphabetical code).

As usual, I don't have much specific to say.

Does anyone however, have an opinion/feedback for hexachords3.xls
which I uploaded a few weeks ago? Hopefully I have made my ideas
much clearer with improved notation and so forth.

I should have stated, though, that D4+ X S3+ is merely C4 X C3.
Otherwise, the notation is "unique".

I hope to add a sheet for grids of 0,1,2,3 weight and tritone count,
both with numbers and with the letters.

* * *

I am reading another good book on Sphere Packings and Error
Correcting Codes. There is so much to know! Also a good book
on Geometry and Symmetry which covers a lot of material.

Ultimately, I wish to master Exterior Algebra so I can get more
into the discussions on this newsgroup

Thanks,

PGH

🔗Graham Breed <gbreed@gmail.com>

🔗Paul G Hjelmstad <phjelmstad@msn.com>

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> Paul G Hjelmstad wrote:
>
> > Since I am studying Crytography as part of my career, I hope
> > to combine the math in this with the math of hexachords.
>
> Well, I don't know about that, but if you're doing
> cryptography you should know the inverse modulo function.
> It relates the complementary representations of an MOS.
>
>
> Graham
>

Thanks. Of course, the 26 letter thing is pure coincidence, so I was
kind of joking. I will look up the inverse modulo function. Does
this have anything to do with invese functions and inverse series?
Another thing of interest is the discrete logarithm.

I am afraid my hexachord theory is forming into a cul-de-sac, but
then again maybe not. In my paper "The Little Book of Hexachord
Theory" usng my hexagram system, I will have a chapter
called "hexagrams, hexachords, hexads, and hexanies" For example.

Good grief. Anyway, I am getting better at compartmentalization, and
not relating everything to everything, even though relating tuning
theory with musical set theory is something I have been striving for
for a long time.

Of course there is also the Golay Codes, perfect error-correcting
codes, sphere packings and lattices. I guess I should finish the
books I am reading before speculating on any theories of my own.

* * * *
Finally, here is one fun cipher with my hexachord system. Permute
as follows

(AEMY)(BNT)(CSX)(DLR)(JOV)(HQW)(GKFZ)(IPU)

One of many possible ciphers.
* * * *
Thanks again for the inverse modulo function "lead".

I am also trying to master R-modules and graded algebras, I wish I
could just take a week off so I could focus on this.

I am also looking at difference sets.

Of course is there was anything real striking to find relating
musical set theory to tuning theory, someone else would have found
it long ago....right?

PGH

🔗Graham Breed <gbreed@gmail.com>

Paul G Hjelmstad wrote:

> Thanks. Of course, the 26 letter thing is pure coincidence, so I was > kind of joking. I will look up the inverse modulo function. Does > this have anything to do with invese functions and inverse series? > Another thing of interest is the discrete logarithm. Oh, whatever the joke was it went way over my head!

Presumably inverse modulo is an inverse function. I don't know what an inverse series is. What's a discrete logarithm?

The point of inverse modulo is that modulo operations are a standard piece of number theory and usually come with programming languages. (For cryptographic purposes they have to work with big integers, of course, but they're becoming standard as well.) It's easier to calculate the principle value of one number modulo another than the inverse so it works as a basis for asymmetric encryption. Cryptographers care greatly about efficiecy so you can find efficient algorithms for inverse modulo. It's also in number theory textbooks.

For our purposes, inverse modulo relates different versions of an MOS like 7/12 or 7&5 for meantone. Carey and Clampitt give the formula. The numbers are always so small that you don't need to care much about efficiency. But I happen to know that the standard solution is the Extended Euclidian Algorithm and I implemented it for my temperament finding library.

> I am afraid my hexachord theory is forming into a cul-de-sac, but > then again maybe not. In my paper "The Little Book of Hexachord > Theory" usng my hexagram system, I will have a chapter > called "hexagrams, hexachords, hexads, and hexanies" For example. I've never understood what you were doing but I don't think it would interest me.

> Good grief. Anyway, I am getting better at compartmentalization, and > not relating everything to everything, even though relating tuning > theory with musical set theory is something I have been striving for > for a long time.

That should work.

<snip>

> Thanks again for the inverse modulo function "lead". > > I am also trying to master R-modules and graded algebras, I wish I > could just take a week off so I could focus on this. > > I am also looking at difference sets. > > Of course is there was anything real striking to find relating > musical set theory to tuning theory, someone else would have found > it long ago....right?

I wouldn't count on it. I think we found some really neat results to do with temperament searching and nobody else was even looking at it. I still don't have any citations for magic temperament before 2001!

I usually follow a path of trying to use mathematics to solve problems that come up in my theory. That's in opposition to researching interesting mathematics and then looking for musical applications. What the temperament searches are is simultaneous Diophantine approxiations so I'd like to know more about them but don't know where to look. Another field I think is fruitful is statistics -- some statistical functions happened to come out of my error/complexity research but I'd like to ask questions like "is this set of 10 best temperaments 90% likely to be complete?"

Graham

🔗Paul G Hjelmstad <phjelmstad@msn.com>

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> Paul G Hjelmstad wrote:
>
> > Thanks. Of course, the 26 letter thing is pure coincidence, so I
was
> > kind of joking. I will look up the inverse modulo function. Does
> > this have anything to do with invese functions and inverse
series?
> > Another thing of interest is the discrete logarithm.
>
> Oh, whatever the joke was it went way over my head!

Just typical Hjelmstad numerology I guess
>
> Presumably inverse modulo is an inverse function. I don't
> know what an inverse series is. What's a discrete logarithm?

I was thinking inverse series/inverse limit, nothing to do with this.
But inverse modulo or inverse modular function is close to the
discrete logarithm: They are group-theoretical analogs of the
logarithm.
>
> The point of inverse modulo is that modulo operations are a
> standard piece of number theory and usually come with
> programming languages. (For cryptographic purposes they
> have to work with big integers, of course, but they're
> becoming standard as well.) It's easier to calculate the
> principle value of one number modulo another than the
> inverse so it works as a basis for asymmetric encryption.
> Cryptographers care greatly about efficiecy so you can find
> efficient algorithms for inverse modulo. It's also in
> number theory textbooks.
>
> For our purposes, inverse modulo relates different versions
> of an MOS like 7/12 or 7&5 for meantone. Carey and Clampitt
> give the formula. The numbers are always so small that you
> don't need to care much about efficiency. But I happen to
> know that the standard solution is the Extended Euclidian
> Algorithm and I implemented it for my temperament finding
> library.

This is real interesting, I will pursue this angle.

> > I am afraid my hexachord theory is forming into a cul-de-sac, but
> > then again maybe not. In my paper "The Little Book of Hexachord
> > Theory" usng my hexagram system, I will have a chapter
> > called "hexagrams, hexachords, hexads, and hexanies" For example.
>
> I've never understood what you were doing but I don't think
> it would interest me.

It's mostly Polya theory. It grew out of necklace theory, and I have
put my own twist on it. There isn't much beyond the M5 relation on
the inverse M5 relation in my tables, do you have any interest in
that? (reversing ^1 and ^5 in the interval vector, also (17)(39)(5,11)
generator....)

> > Good grief. Anyway, I am getting better at compartmentalization,
and
> > not relating everything to everything, even though relating
tuning
> > theory with musical set theory is something I have been striving
for
> > for a long time.
>
> That should work.

What would work? I still haven't found it. It's like oil and water...

>
> <snip>
>
> > Thanks again for the inverse modulo function "lead".
> >
> > I am also trying to master R-modules and graded algebras, I wish
I
> > could just take a week off so I could focus on this.
> >
> > I am also looking at difference sets.
> >
> > Of course is there was anything real striking to find relating
> > musical set theory to tuning theory, someone else would have
found
> > it long ago....right?
>
> I wouldn't count on it. I think we found some really neat
> results to do with temperament searching and nobody else was
> even looking at it. I still don't have any citations for
> magic temperament before 2001!

Well at least I am talking with those on the cutting edge....

>
> I usually follow a path of trying to use mathematics to
> solve problems that come up in my theory. That's in
> opposition to researching interesting mathematics and then
> looking for musical applications. What the temperament
> searches are is simultaneous Diophantine approxiations so
> I'd like to know more about them but don't know where to
> look. Another field I think is fruitful is statistics --
> some statistical functions happened to come out of my
> error/complexity research but I'd like to ask questions like
> "is this set of 10 best temperaments 90% likely to be complete?"
>
>
> Graham
>

Thanks. I hate statistics, but I recognize its importance. I like
clear cut mathematics, which is one reason I find complex analysis
so attractive...

PGH

🔗John Gilbert <preciousatonement@gmail.com>

🔗Paul G Hjelmstad <phjelmstad@msn.com>

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:

<snip>

> The point of inverse modulo is that modulo operations are a
> standard piece of number theory and usually come with
> programming languages. (For cryptographic purposes they
> have to work with big integers, of course, but they're
> becoming standard as well.) It's easier to calculate the
> principle value of one number modulo another than the
> inverse so it works as a basis for asymmetric encryption.
> Cryptographers care greatly about efficiecy so you can find
> efficient algorithms for inverse modulo. It's also in
> number theory textbooks.
>
> For our purposes, inverse modulo relates different versions
> of an MOS like 7/12 or 7&5 for meantone. Carey and Clampitt
> give the formula. The numbers are always so small that you
> don't need to care much about efficiency. But I happen to
> know that the standard solution is the Extended Euclidian
> Algorithm and I implemented it for my temperament finding
> library.

Graham, I am trying to find this formula on the net but unless
I buy a book (which I could I suppose) it is not available.

Would it be possible to post Carey & Clampitt's formula, and show
how this relates the complementary representations of the MOS
you list (And explain again what 7/12 and 7&5 mean)

Thanks,

PGH

🔗Graham Breed <gbreed@gmail.com>

Paul G Hjelmstad wrote:
> --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
> > <snip>
> >>The point of inverse modulo is that modulo operations are a >>standard piece of number theory and usually come with >>programming languages. (For cryptographic purposes they >>have to work with big integers, of course, but they're >>becoming standard as well.) It's easier to calculate the >>principle value of one number modulo another than the >>inverse so it works as a basis for asymmetric encryption. >>Cryptographers care greatly about efficiecy so you can find >>efficient algorithms for inverse modulo. It's also in >>number theory textbooks.
>>
>>For our purposes, inverse modulo relates different versions >>of an MOS like 7/12 or 7&5 for meantone. Carey and Clampitt >>give the formula. The numbers are always so small that you >>don't need to care much about efficiency. But I happen to >>know that the standard solution is the Extended Euclidian >>Algorithm and I implemented it for my temperament finding >>library.
> > > Graham, I am trying to find this formula on the net but unless
> I buy a book (which I could I suppose) it is not available.

The formula seems to be d = -g^{-1}_{Mod N} in ASCII. Which doesn't mean much unless you know what in inverse modulo should look like. (They call it a multiplicative inverse modulo N.) I think d is the number of notes in the octave for one of the representative ETs, N is the number of notes in the octave for one of the larger, representative ETs, and g is the number of steps one of the generators in the N note scale.

> Would it be possible to post Carey & Clampitt's formula, and show
> how this relates the complementary representations of the MOS
> you list (And explain again what 7/12 and 7&5 mean) Here's the extended Eucidian algorithm in Python:

def extendedEuclid(f, g):
assert f>0 and g>0
lastR, nextR = f, g
lastS, lastT, nextS, nextT = 1, 0, 0, -1

while nextR:
q, r = divmod(lastR, nextR)
nextR, lastR = r, nextR
nextS, lastS = lastS - q*nextS, nextS
nextT, lastT = lastT - q*nextT, nextT

return lastT, lastS, lastR

Well, I got my numbers wrong. Start with 7&5 which means the number of steps to an octave in either 7- or 5-note scales. Together these define the melodic structure of meantone, which can have either 7 or 5 notes. Call the function

extendedEucid(5,7)

That returns (2, 3, 1). That tells you that the generator is 2 steps of the pentatonic or 3 steps of the diatonic scale, and that the period equals the octave.

Another way of defining meantone is as 12&19. So

extendedEuclid(12,19) = (5,8,1)

The generator is 5 steps of the 12 note scale or 8 steps of the 19 note scale. By contrast, schismatic is 12&17, so

extendedEuclid(12,17) = (-5, -7, 1)

The generator is the same size on the 12 note scale, but only 7 steps on the 17 note scale. (The signs are fairly arbitrary. In my code, I add periods to them to become positive so in this case they'd be fifths of 7 and 10 steps.)

For pajara,

extendedEuclid(12, 22) = (1, 2, 2)

The last 2 tells you there are 2 periods to the octave.

There's also away to start with the generator sizes and get the octave sizes but I can't find that now.

Graham

🔗Paul G Hjelmstad <phjelmstad@msn.com>

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> Paul G Hjelmstad wrote:
> > --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@> wrote:
> >
> > <snip>
> >
> >>The point of inverse modulo is that modulo operations are a
> >>standard piece of number theory and usually come with
> >>programming languages. (For cryptographic purposes they
> >>have to work with big integers, of course, but they're
> >>becoming standard as well.) It's easier to calculate the
> >>principle value of one number modulo another than the
> >>inverse so it works as a basis for asymmetric encryption.
> >>Cryptographers care greatly about efficiecy so you can find
> >>efficient algorithms for inverse modulo. It's also in
> >>number theory textbooks.
> >>
> >>For our purposes, inverse modulo relates different versions
> >>of an MOS like 7/12 or 7&5 for meantone. Carey and Clampitt
> >>give the formula. The numbers are always so small that you
> >>don't need to care much about efficiency. But I happen to
> >>know that the standard solution is the Extended Euclidian
> >>Algorithm and I implemented it for my temperament finding
> >>library.
> >
> >
> > Graham, I am trying to find this formula on the net but unless
> > I buy a book (which I could I suppose) it is not available.
>
> The formula seems to be d = -g^{-1}_{Mod N} in ASCII. Which
> doesn't mean much unless you know what in inverse modulo
> should look like. (They call it a multiplicative inverse
> modulo N.) I think d is the number of notes in the octave
> for one of the representative ETs, N is the number of notes
> in the octave for one of the larger, representative ETs, and
> g is the number of steps one of the generators in the N note
> scale.
>
> > Would it be possible to post Carey & Clampitt's formula, and show
> > how this relates the complementary representations of the MOS
> > you list (And explain again what 7/12 and 7&5 mean)
>
> Here's the extended Eucidian algorithm in Python:
>
> def extendedEuclid(f, g):
> assert f>0 and g>0
> lastR, nextR = f, g
> lastS, lastT, nextS, nextT = 1, 0, 0, -1
>
> while nextR:
> q, r = divmod(lastR, nextR)
> nextR, lastR = r, nextR
> nextS, lastS = lastS - q*nextS, nextS
> nextT, lastT = lastT - q*nextT, nextT
>
> return lastT, lastS, lastR
>
> Well, I got my numbers wrong. Start with 7&5 which means
> the number of steps to an octave in either 7- or 5-note
> scales. Together these define the melodic structure of
> meantone, which can have either 7 or 5 notes. Call the function
>
> extendedEucid(5,7)
>
> That returns (2, 3, 1). That tells you that the generator
> is 2 steps of the pentatonic or 3 steps of the diatonic
> scale, and that the period equals the octave.
>
> Another way of defining meantone is as 12&19. So
>
> extendedEuclid(12,19) = (5,8,1)
>
> The generator is 5 steps of the 12 note scale or 8 steps of
> the 19 note scale. By contrast, schismatic is 12&17, so
>
> extendedEuclid(12,17) = (-5, -7, 1)
>
> The generator is the same size on the 12 note scale, but
> only 7 steps on the 17 note scale. (The signs are fairly
> arbitrary. In my code, I add periods to them to become
> positive so in this case they'd be fifths of 7 and 10 steps.)
>
> For pajara,
>
> extendedEuclid(12, 22) = (1, 2, 2)
>
> The last 2 tells you there are 2 periods to the octave.
>
>
> There's also away to start with the generator sizes and get
> the octave sizes but I can't find that now.
>
>
> Graham

Thanks. As you know, I also am interested in your temperament
from unison vectors, and particularly "mapping by steps". It
always works for a generalized keyboard, not always for a black-white
linear system, I also notice that for example, meantone with
81/80, mapping by steps always works out perfectly for any
meantone temperament if you determine the proportions of the
step sizes, (1x, 3y or whatever). Of course you could say
it always works for a black white system in an abstract sense...

Do you suppose you could give the code for mapping by steps also?

Thanks for the nice explanation, makes me want to play around
with Python again.

PGH
>

🔗Graham Breed <gbreed@gmail.com>

Paul G Hjelmstad wrote:

> Thanks. As you know, I also am interested in your temperament
> from unison vectors, and particularly "mapping by steps". It
> always works for a generalized keyboard, not always for a black-white
> linear system, I also notice that for example, meantone with
> 81/80, mapping by steps always works out perfectly for any
> meantone temperament if you determine the proportions of the
> step sizes, (1x, 3y or whatever). Of course you could say
> it always works for a black white system in an abstract sense...

Yes, it has a direct application to a generalized keyboard. But also to a black-white keyboard. You set the smaller scale to the white keys and the larger scale to all the keys. Or maybe the larger scale to the white keys and the smaller scale to the black keys.

Yes, again, the mapping by steps is a complete definition of the temperament class.

> Do you suppose you could give the code for mapping by steps also?

You mean the code for finding the mapping by steps from the unison vectors? That's not so simple, and it would tend to drag the whole multivector code along with it. You may as well download the full source code yourself. Try

http//x31eq.com/temper

> Thanks for the nice explanation, makes me want to play around
> with Python again.

Have at it!

Graham

🔗Carl Lumma <carl@lumma.org>

🔗Graham Breed <gbreed@gmail.com>

🔗Carl Lumma <carl@lumma.org>

🔗Graham Breed <gbreed@gmail.com>

🔗Carl Lumma <carl@lumma.org>

🔗Graham Breed <gbreed@gmail.com>

🔗Paul G Hjelmstad <phjelmstad@msn.com>

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> Paul G Hjelmstad wrote:
>
> > Thanks. As you know, I also am interested in your temperament
> > from unison vectors, and particularly "mapping by steps". It
> > always works for a generalized keyboard, not always for a black-
white
> > linear system, I also notice that for example, meantone with
> > 81/80, mapping by steps always works out perfectly for any
> > meantone temperament if you determine the proportions of the
> > step sizes, (1x, 3y or whatever). Of course you could say
> > it always works for a black white system in an abstract sense...
>
> Yes, it has a direct application to a generalized keyboard.
> But also to a black-white keyboard. You set the smaller
> scale to the white keys and the larger scale to all the
> keys. Or maybe the larger scale to the white keys and the
> smaller scale to the black keys.
>
> Yes, again, the mapping by steps is a complete definition of
> the temperament class.
>
> > Do you suppose you could give the code for mapping by steps also?
>
> You mean the code for finding the mapping by steps from the
> unison vectors? That's not so simple, and it would tend to
> drag the whole multivector code along with it. You may as
> well download the full source code yourself. Try
>
> http//x31eq.com/temper
>
> > Thanks for the nice explanation, makes me want to play around
> > with Python again.
>
> Have at it!
>
>
> Graham
>
Thanks again. I only have one little issue. Mapping by steps doesn't
always give a physically possible black-white keyboard. Of course,
I might not be doing it right. How do you know its all-steps/white
steps or white steps/black steps? Don't the black keys "move around"
based on what prime you are mapping? Etc.

PGH

🔗Herman Miller <hmiller@IO.COM>

Graham Breed wrote:
> Paul G Hjelmstad wrote:
> >> Thanks. As you know, I also am interested in your temperament
>> from unison vectors, and particularly "mapping by steps". It
>> always works for a generalized keyboard, not always for a black-white
>> linear system, I also notice that for example, meantone with
>> 81/80, mapping by steps always works out perfectly for any
>> meantone temperament if you determine the proportions of the
>> step sizes, (1x, 3y or whatever). Of course you could say
>> it always works for a black white system in an abstract sense...
> > Yes, it has a direct application to a generalized keyboard. > But also to a black-white keyboard. You set the smaller > scale to the white keys and the larger scale to all the > keys. Or maybe the larger scale to the white keys and the > smaller scale to the black keys.

That's more or less what I do with my 24-note keyboard layout for lemba: the 10-note scale is on the black keys, with the other 6 notes of the larger 16-note scale on the white keys. There aren't enough keys for a full 26-note scale, but the remaining white keys fill in most of the gaps. It works out pretty well, since I rarely go to the far extremes of the temperament, but occasionally have need for a note outside the 16-note scale. If I could rearrange the black and white keys, I could put 10 notes on the white keys and 6 on the black keys, or for the larger scale, 16 white keys and 10 black keys.

One of my early father temperament experiments (with a MIDI illustration at http://www.io.com/~hmiller/music/ex/yulung.mid) used the white keys (with two duplicated notes) for the 5-note scale, and the black keys to fill in the other 3 notes. If all you've got is black and white keys, this sort of arrangement seems to work out pretty well for focusing on the characteristic scales of the temperament while allowing a degree of chromaticism and modulation.

🔗Graham Breed <gbreed@gmail.com>

Paul G Hjelmstad wrote:

> Thanks again. I only have one little issue. Mapping by steps doesn't
> always give a physically possible black-white keyboard. Of course,
> I might not be doing it right. How do you know its all-steps/white > steps or white steps/black steps? Don't the black keys "move around"
> based on what prime you are mapping? Etc. You don't know. It depends on how many keys you want. Say you want a normal Halberstadt keyboard. You can either define it as 7&5 or 7 from 12.

Probably 7 from 12 is easier. Then you can define the white key pattern as a maximally even 7 from 12 scale. The mapping by steps tells you how many white keys and how many keys total to count for each interval.

The temperament-from-two-ets script works on maximally even subsets as well. So if you search for 7 from 31 it won't give you the meantone diatonic. Maximal evenness is a good model for how the temperaments are chosen.

The keys don't move around. The mapping tells you how to map primes to keys.

Graham

🔗Paul G Hjelmstad <phjelmstad@msn.com>

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> Paul G Hjelmstad wrote:
>
> > Thanks again. I only have one little issue. Mapping by steps
doesn't
> > always give a physically possible black-white keyboard. Of course,
> > I might not be doing it right. How do you know its all-
steps/white
> > steps or white steps/black steps? Don't the black keys "move
around"
> > based on what prime you are mapping? Etc.
>
> You don't know. It depends on how many keys you want. Say
> you want a normal Halberstadt keyboard. You can either
> define it as 7&5 or 7 from 12.
>
> Probably 7 from 12 is easier. Then you can define the white
> key pattern as a maximally even 7 from 12 scale. The
> mapping by steps tells you how many white keys and how many
> keys total to count for each interval.

I see. I need to review "maximally even"
>
> The temperament-from-two-ets script works on maximally even
> subsets as well. So if you search for 7 from 31 it won't
> give you the meantone diatonic. Maximal evenness is a good
> model for how the temperaments are chosen.

Because 7 from 31 is not maximally even? Do you mean 7 whites
and 31 total?

> The keys don't move around. The mapping tells you how to
> map primes to keys.

I don't disagree, I just can't find the right distribution

For example, take 64/63 and 128/125:

unison vectors

128:125
64:63
calculated

unison vectors

64:63
126:125

1/2, 89.0 cent generator

basis:
(0.333333333333, 0.0741308071121)

mapping by period and generator:
[(3, 0), (5, -1), (7, 0), (8, 2)]

mapping by steps:
[(3, 3), (4, 5), (7, 7), (10, 8)]

highest interval width: 3
complexity measure: 9 (12 for smallest MOS)
highest error: 0.011405 (13.686 cents)
unique

Assuming this works with 12-tET, you get x+3y, but to move
from 7,7 to 10,8 you move 3 black keys but only one white.

Thanks

PGH

🔗Graham Breed <gbreed@gmail.com>

Paul G Hjelmstad wrote:
> --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:

>>Probably 7 from 12 is easier. Then you can define the white >>key pattern as a maximally even 7 from 12 scale. The >>mapping by steps tells you how many white keys and how many >>keys total to count for each interval.
> > I see. I need to review "maximally even"

The maximally even d from m scale is notes

[(m*n)//d for n in range(d+1)]

>>The temperament-from-two-ets script works on maximally even >>subsets as well. So if you search for 7 from 31 it won't >>give you the meantone diatonic. Maximal evenness is a good >>model for how the temperaments are chosen.
> > Because 7 from 31 is not maximally even? Do you mean 7 whites
> and 31 total?

Yes. If you want a keyboard with 7 white notes and a profusion of split keys to give 31 in total, you ask for a 7 from 31 scale and the mapping will be for a neutral third scale, not the normal diatonic. And the temperament you'll get describe isn't meantone (if it even qualifies as a temperament). That's because 7&31 is contorted in the 5-limit.

>>The keys don't move around. The mapping tells you how to >>map primes to keys.
> > I don't disagree, I just can't find the right distribution
> > For example, take 64/63 and 128/125:
> 1/2, 89.0 cent generator
> > basis:
> (0.333333333333, 0.0741308071121)
> > mapping by period and generator:
> [(3, 0), (5, -1), (7, 0), (8, 2)]
> > mapping by steps:
> [(3, 3), (4, 5), (7, 7), (10, 8)]
> > highest interval width: 3
> complexity measure: 9 (12 for smallest MOS)
> highest error: 0.011405 (13.686 cents)
> unique
> > Assuming this works with 12-tET, you get x+3y, but to move
> from 7,7 to 10,8 you move 3 black keys but only one white.
> > ?

Um, yes, sorry. The rule only tells you how to find intervals that exist between two white notes. To tune the black notes you need to choose a bigger MOS. Then each black note has to "belong" to a white note. For example, C# belongs to C for meantone. So 25:24 should map as no white keys and one black key. And that has to be C-C# and not C-Db.

There are also going to be intervals that can't be played on the keyboard, or not from the note you want to start on. 1 white and 3 black is an example of this, for a normal halberstadt.

You can think of this as a meantone halberstadt having 7 large and 5 small steps to the octave. But they layout itself doesn't tell you which are large and which are small. How could it when there are different ways of choosing the gamut? All you know is that two adjacent white keys are a large step, and two white keys with a black key in the middle are a large plus a small step. And that the pattern of large and small steps should be an MOS.

Graham