--- In tuning-math@yahoogroups.com, "manuphonic" <manuphonic@...> wrote:
>
> In his essay "The 17-tone Puzzle -- And the Neo-medieval Key That Unlocks It"
George Secor wrote:
>
> "Is 17-WT, then, the final step in this alternate history of tuning and
temperament? At this point I think not. Once the resources of the 17-tone system
were fully exploited, we could expect that other options with better intonation
would be sought. I have tried a number of the progressions that we have
discovered in 17-WT in other tuning systems, and there is a near-just 13-limit
system (that includes rations of 5) into which virtually everything that we have
tried can be transferred; the progressions not only work, but they sound even
better than in 17-WT! Hopefully, this will be the topic for a follow-up
article."
>
> George, or anyone here, can you say more now about near-just 13-limit
temperaments that sound better than 17-WT?
>
> Also, for any such "better" temperament, can you devise a keyboard or
buttonboard layout (or layouts) for concertina, accordion, bayan, bandoneon,
symphonetta or some other acoustic free reed instrument?
>
> Just curious!
> ==
> Manu

Aha!  This message was posted nearly 2 months ago and just showed up today!

"Manu" & I have been discussing this off-list, and I'm now in the process of
working out several buttonboard designs for microtonal concertinas.  For the
benefit of any others who may be curious, the following is the reply I gave
regarding the question in the subject line:

<< I intended to write the follow-up article for the next issue of
Xenharmonikon, but unfortunately there won't be any more.  I wrote to John
Chalmers in October 2006 suggesting that a Xenharmonikon website could be set up
so that future articles could be published online (and possibly some past
articles made available), but nothing has materialized.  ...

But to answer your question: the near-just tuning is my 29-tone high-tolerance
temperament.  It was first described in XH3, but I mentioned it a couple of
times on the tuning lists.  See:
http://launch.groups.yahoo.com/group/MakeMicroMusic/message/6889
which references an earlier message (with .scl data) and contains a link (no
longer good) to an mp3 file that you can now temporarily download from here:
http://tech.groups.yahoo.com/group/tuning-math/files/secor/improv29.mp3

The 29-HTT basically consists of 3 open chains of tempered fifths (such that 9
fifths stacked exactly equal 63/52 plus 5 octaves); one chain of fifths (8
tones) contains C=1/1, the second (7 tones) contains an exact E\!=5/4, and the
third (13 tones) contains an exact B!!!)=7/4.  The first chain has prime 3
relationships; the second chain has prime 5 relationships (relative to chain 1),
and the third chain has primes 7, 11, & 13 (with 9:13 exact, and 9:11 almost
exact).  Thus (with octave-equivalence) it's a 28-tone 4-D tuning that maps to
the 29-division of the octave.  The hole at Eb is filled by adding a tone that
makes fifths of equal size with Bb and Ab, almost exactly 19/16 relative to C. 
The result is 16 harmonics in 6 different keys with maximum error <3.25 cents;
at the 7-odd limit the max error is half of that.

Listen to the above sound file and tell me if it doesn't sound like JI. >>

The sound file is still there -- if anyone else is still curious.

--George

--- In tuning-math@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Your moderators were all asleep at the switch.  Sorry!

And I'm still waking up.  Sorry I approved that intelligent
design thing.  -Carl

Hi aerisselicious,

Was your message below truncated?  It's missing an endquote
apparently.  Also, who are you?

-Carl

At 04:57 PM 8/1/2009, you wrote:
>Here is my compilation of data for diaschismic and kleismic temperaments, two
of my favourite classes of all limits and my most favourite in the 5-limit. I
did this for the Tonalsoft encyclopedia because I didn't see data for either of
those temperament families at all there.
>I omitted poptimal generators from most of the temperaments listed, because I
lack the tools to calculate them quickly enough, and same goes for some of the
TM basis sets shown here.
>If you or I find any errors in the information here, then I will correct them
as soon as I find out.
>If you find any poptimal generators or TM basis sets, then I will check them
and add the valid ones to my data file.
>
>~Diaschismic family~
>Family name: Diaschismic
>Period: Demioctave (Half octave; 45:32)
>Generator: Semitone (Half of 9:8; 16:15)
>
>5-limit
>Comma: Diaschisma, 2048:2025 |11.-4.-2>
>Mapping: [2 | 0][3 | 1][5 | -2]
>Poptimal generator: 10/114
>Optimal minimax generator: 1/3 of 6:5; approx. 105,21 cents (5:4 and 3:2 are
1/6 diaschisma wide)
>TOP period/generator: [599,55 | 104,70] cents
>TOP-RMS period/generator: [599,41 | 104,80] cents
>Poptimal MOS cardinalities: 10 \ 12 \ 22 \ 34 \ 46 \ 80 \ 114
>TOP MOS cardinalities: 10 \ 12 \ 22 \ 34 \ 46 \ 80 \ 126
>TOP-RMS MOS cardinalities: 10 \ 12 \ 22 \ 34 \ 46 \ 80 \ 126
>
>7-limit, 9-limit
>~Pajara~ 10&12
>Other names: "Paultone", "Twintone"
>TM Basis: {50:49 \ 64:63}
>Comma sequence: [2048:2025 | 50:49]
>Wedgie: << 2'-4'-4"-11'-12" 2||
>Mapping: [2 | 0][3 | 1][5 | -2][6 | -2]
>7-limit poptimal*: 2/22
>9-limit poptimal: 5/56
>TOP-RMS period and generator: [598,86 | 106,84]
>7-limit poptimal MOS cardinalities: 10 \ 12 \ 22
>9-limit poptimal MOS cardinalities: 10 \ 12 \ 22 \ 34 \ 56
>TOP-RMS MOS cardinalities: 10 \ 12 \ 22 \ 34 \ 56
>*This value is based on inclusion of the entire mimimax range, which includes
2/22.
>
>~Starlidiaschismic~ 12&46
>This title comes from the 126:125 unison of Starling temperament
>Other names: "Stand

Your moderators were all asleep at the switch.  Sorry!

-Carl

In his essay "The 17-tone Puzzle -- And the Neo-medieval Key That Unlocks It"
George Secor wrote:

"Is 17-WT, then, the final step in this alternate history of tuning and
temperament? At this point I think not. Once the resources of the 17-tone system
were fully exploited, we could expect that other options with better intonation
would be sought. I have tried a number of the progressions that we have
discovered in 17-WT in other tuning systems, and there is a near-just 13-limit
system (that includes rations of 5) into which virtually everything that we have
tried can be transferred; the progressions not only work, but they sound even
better than in 17-WT! Hopefully, this will be the topic for a follow-up
article."

George, or anyone here, can you say more now about near-just 13-limit
temperaments that sound better than 17-WT?

Also, for any such "better" temperament, can you devise a keyboard or
buttonboard layout (or layouts) for concertina, accordion, bayan, bandoneon,
symphonetta or some other acoustic free reed instrument?

Just curious!
==
Manu

--- In tuning-math@yahoogroups.com, "rick_ballan" <rick_ballan@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul H" <phjelmstad@> wrote:
> >
> > --- In tuning-math@yahoogroups.com, rick ballan <rick_ballan@> wrote:
> > >
> > > Hello everyone,
> > >
> > > My name's Rick and I'm a new member. Forgive me for starting a new thread
but I haven't got my 'daily digest' yet and couldn't find an answer in your
archives.
> > >
> > > I just wanted to run an idea by someone who understands Gaussian integer
maths better than I (which wouldn't be difficult). I've had this vague thought
that
> > > the concept of gcd frequency, which helps to explain basic tonality, might
be generalised to the (sometimes)
> > > multiple gcd's of Gaussian integers (or some other Ring).  The first
> > > task is to see how, if at all, this might translate into waves.  The
> > > second is to see if new harmonies can be generated.
> > >
> > > In fact the first task seems quite credible and straightforward. For eg,
> > > take e^i(a + bi)t, where it is observed that this is more than the
> > > Euler identity since the frequency is now complex. We then have i(a +
> > > bi) = ai - b = - (b - ai). Therefore, this suggests a wave of the form
> > > e^-bt
> > > . e^iat = e^-bt (cos at +isin at), which appears to be a sine wave
> > > under the force of damping, or if the coefficient of b is positive,  a
> > > forced oscillation.
> > >
> > > Given another wave of complex frequency (c + id), the addition of the two
would be:
> > > e^-bt
> > > (cos at +isin at) + e^-dt (cos ct +isin ct), the sum of two waves with
> > > not necessarily equal damping/forcing. Now since periodicity is
> > > independent of amplitude, then the gcd of the waves should be just the
> > > usual integer result where a/c = p/q, p and q are  relatively prime and
resultant
> > > frequency is g = a/p2pi = c/q2pi.  But here's the rub: (a + bi)/(c +
> > > di) = ((ac + bd) - i(ad + bc)) / c^2 + d^2 often gives rise to more
> > > than one gcd. If we can think of how these others could "fit in" somehow,
a
> > > whole new area of tonality might open up. Or if it doesn't fit in at
> > > all and is complete nonsense, then this too might be something.
> > >
> > > Thanks
> >
> > Rick,
> >
> > This looks really interesting. I am into algebra (groups, rings), and am
learning more about complex analysis. I would be fun to start
> > discussing these matters here, with respect to Fourier analysis,
> > and so forth.
> >
> > I know the Euclidean algorithm comes up a lot, and of course, gcd, lcm, and
the like, in these discussions, you might want to check out the work by people
on this newsgroup (at the Home page). There isn't as much using complex analysis
but that would be a great area to pursue, seeing its application to
acoutics/physics/soundwaves.
> >
> > Wish I knew more about it...
> >
> > PGH
> >
> Actually Paul, there is one other problem that came up on the alternate
tunings list which I can't find a math's solution for. And that is, how can the
lcm be interpreted in terms of waves? For eg, given the freq's p and q, p > q,
and p/q = a/b which are relatively prime, then as you know this gives two other
equations, gcd = p/a = q/b and lcm = pb = qa. The first gives the period of the
wave, proved but adding whole numbered multiples of the gcd period (T = Na/p =
Nb/q), but I can't for the life of me find how we could hear the lcm. Is it the
time between smallest wave-crests or something?
>
> Rick
>
> PS: I did look up the messages and there's allot their.

I can't seem to get to a lot of the links at the top. (Resources).
I would say try to get to Gene's stuff and Graham's stuff to start.
And don't worry about the physical waves too much (crests and troughs, phase,
period, etc.) at first, just the theories involving
tuning, Euclidean algorithm and such. These ideas come into play
with respect to linear tunings, multilinear algebra, and the like.

Another interesting property is the Farey sequence, and the Stern-Brocat tree. I
think it would be fun to bring in complex analysis still though...

PGH

--- In tuning-math@yahoogroups.com, "Paul H" <phjelmstad@...> wrote:
>
> --- In tuning-math@yahoogroups.com, rick ballan <rick_ballan@> wrote:
> >
> > Hello everyone,
> >
> > My name's Rick and I'm a new member. Forgive me for starting a new thread
but I haven't got my 'daily digest' yet and couldn't find an answer in your
archives.
> >
> > I just wanted to run an idea by someone who understands Gaussian integer
maths better than I (which wouldn't be difficult). I've had this vague thought
that
> > the concept of gcd frequency, which helps to explain basic tonality, might
be generalised to the (sometimes)
> > multiple gcd's of Gaussian integers (or some other Ring).  The first
> > task is to see how, if at all, this might translate into waves.  The
> > second is to see if new harmonies can be generated.
> >
> > In fact the first task seems quite credible and straightforward. For eg,
> > take e^i(a + bi)t, where it is observed that this is more than the
> > Euler identity since the frequency is now complex. We then have i(a +
> > bi) = ai - b = - (b - ai). Therefore, this suggests a wave of the form
> > e^-bt
> > . e^iat = e^-bt (cos at +isin at), which appears to be a sine wave
> > under the force of damping, or if the coefficient of b is positive,  a
> > forced oscillation.
> >
> > Given another wave of complex frequency (c + id), the addition of the two
would be:
> > e^-bt
> > (cos at +isin at) + e^-dt (cos ct +isin ct), the sum of two waves with
> > not necessarily equal damping/forcing. Now since periodicity is
> > independent of amplitude, then the gcd of the waves should be just the
> > usual integer result where a/c = p/q, p and q are  relatively prime and
resultant
> > frequency is g = a/p2pi = c/q2pi.  But here's the rub: (a + bi)/(c +
> > di) = ((ac + bd) - i(ad + bc)) / c^2 + d^2 often gives rise to more
> > than one gcd. If we can think of how these others could "fit in" somehow, a
> > whole new area of tonality might open up. Or if it doesn't fit in at
> > all and is complete nonsense, then this too might be something.
> >
> > Thanks
>
> Rick,
>
> This looks really interesting. I am into algebra (groups, rings), and am
learning more about complex analysis. I would be fun to start
> discussing these matters here, with respect to Fourier analysis,
> and so forth.
>
> I know the Euclidean algorithm comes up a lot, and of course, gcd, lcm, and
the like, in these discussions, you might want to check out the work by people
on this newsgroup (at the Home page). There isn't as much using complex analysis
but that would be a great area to pursue, seeing its application to
acoutics/physics/soundwaves.
>
> Wish I knew more about it...
>
> PGH
>
Actually Paul, there is one other problem that came up on the alternate tunings
list which I can't find a math's solution for. And that is, how can the lcm be
interpreted in terms of waves? For eg, given the freq's p and q, p > q, and p/q
= a/b which are relatively prime, then as you know this gives two other
equations, gcd = p/a = q/b and lcm = pb = qa. The first gives the period of the
wave, proved but adding whole numbered multiples of the gcd period (T = Na/p =
Nb/q), but I can't for the life of me find how we could hear the lcm. Is it the
time between smallest wave-crests or something?

Rick

PS: I did look up the messages and there's allot their.

--- In tuning-math@yahoogroups.com, "Paul H" <phjelmstad@...> wrote:
>
> --- In tuning-math@yahoogroups.com, rick ballan <rick_ballan@> wrote:
> >
> > Hello everyone,
> >
> > My name's Rick and I'm a new member. Forgive me for starting a new thread
but I haven't got my 'daily digest' yet and couldn't find an answer in your
archives.
> >
> > I just wanted to run an idea by someone who understands Gaussian integer
maths better than I (which wouldn't be difficult). I've had this vague thought
that
> > the concept of gcd frequency, which helps to explain basic tonality, might
be generalised to the (sometimes)
> > multiple gcd's of Gaussian integers (or some other Ring).  The first
> > task is to see how, if at all, this might translate into waves.  The
> > second is to see if new harmonies can be generated.
> >
> > In fact the first task seems quite credible and straightforward. For eg,
> > take e^i(a + bi)t, where it is observed that this is more than the
> > Euler identity since the frequency is now complex. We then have i(a +
> > bi) = ai - b = - (b - ai). Therefore, this suggests a wave of the form
> > e^-bt
> > . e^iat = e^-bt (cos at +isin at), which appears to be a sine wave
> > under the force of damping, or if the coefficient of b is positive,  a
> > forced oscillation.
> >
> > Given another wave of complex frequency (c + id), the addition of the two
would be:
> > e^-bt
> > (cos at +isin at) + e^-dt (cos ct +isin ct), the sum of two waves with
> > not necessarily equal damping/forcing. Now since periodicity is
> > independent of amplitude, then the gcd of the waves should be just the
> > usual integer result where a/c = p/q, p and q are  relatively prime and
resultant
> > frequency is g = a/p2pi = c/q2pi.  But here's the rub: (a + bi)/(c +
> > di) = ((ac + bd) - i(ad + bc)) / c^2 + d^2 often gives rise to more
> > than one gcd. If we can think of how these others could "fit in" somehow, a
> > whole new area of tonality might open up. Or if it doesn't fit in at
> > all and is complete nonsense, then this too might be something.
> >
> > Thanks
>
> Rick,
>
> This looks really interesting. I am into algebra (groups, rings), and am
learning more about complex analysis. I would be fun to start
> discussing these matters here, with respect to Fourier analysis,
> and so forth.
>
> I know the Euclidean algorithm comes up a lot, and of course, gcd, lcm, and
the like, in these discussions, you might want to check out the work by people
on this newsgroup (at the Home page). There isn't as much using complex analysis
but that would be a great area to pursue, seeing its application to
acoutics/physics/soundwaves.
>
> Wish I knew more about it...
>
> PGH
>
Hi Paul,

Thanks for getting back. At this stage I don't know much about complex numbers
either so you're not alone there. Unlike reals, I can't 'picture' how one
complex number of cycles/periods would divide into another and, consequently,
how we could interpret the gcd's in terms of waves. Another doubt is that if
they give damped/forced oscillations as I mentioned above, then periodicity
(harmony) would be independent of amplitude i.e. the e^-bt part. So even if we
could find some physical interpretation for the other gcd's, they mightn't
effect the harmony. OTOH,as you said the Euclidean algorithm comes up a lot in
discussions of harmony and there just might be something I/we haven't thought
of. For eg, I once Googled 'negative frequency' and found it has a physical
interpretation (which might come into complex frequencies??).

(Excuse my ignorance but how do you get back to the homepage?)

Wish I knew more about it too...

Rick

--- In tuning-math@yahoogroups.com, rick ballan <rick_ballan@...> wrote:
>
> Hello everyone,
>
> My name's Rick and I'm a new member. Forgive me for starting a new thread but
I haven't got my 'daily digest' yet and couldn't find an answer in your
archives.
>
> I just wanted to run an idea by someone who understands Gaussian integer maths
better than I (which wouldn't be difficult). I've had this vague thought that
> the concept of gcd frequency, which helps to explain basic tonality, might be
generalised to the (sometimes)
> multiple gcd's of Gaussian integers (or some other Ring).  The first
> task is to see how, if at all, this might translate into waves.  The
> second is to see if new harmonies can be generated.
>
> In fact the first task seems quite credible and straightforward. For eg,
> take e^i(a + bi)t, where it is observed that this is more than the
> Euler identity since the frequency is now complex. We then have i(a +
> bi) = ai - b = - (b - ai). Therefore, this suggests a wave of the form
> e^-bt
> . e^iat = e^-bt (cos at +isin at), which appears to be a sine wave
> under the force of damping, or if the coefficient of b is positive,  a
> forced oscillation.
>
> Given another wave of complex frequency (c + id), the addition of the two
would be:
> e^-bt
> (cos at +isin at) + e^-dt (cos ct +isin ct), the sum of two waves with
> not necessarily equal damping/forcing. Now since periodicity is
> independent of amplitude, then the gcd of the waves should be just the
> usual integer result where a/c = p/q, p and q are  relatively prime and
resultant
> frequency is g = a/p2pi = c/q2pi.  But here's the rub: (a + bi)/(c +
> di) = ((ac + bd) - i(ad + bc)) / c^2 + d^2 often gives rise to more
> than one gcd. If we can think of how these others could "fit in" somehow, a
> whole new area of tonality might open up. Or if it doesn't fit in at
> all and is complete nonsense, then this too might be something.
>
> Thanks

Rick,

This looks really interesting. I am into algebra (groups, rings), and am
learning more about complex analysis. I would be fun to start
discussing these matters here, with respect to Fourier analysis,
and so forth.

I know the Euclidean algorithm comes up a lot, and of course, gcd, lcm, and the
like, in these discussions, you might want to check out the work by people on
this newsgroup (at the Home page). There isn't as much using complex analysis
but that would be a great area to pursue, seeing its application to
acoutics/physics/soundwaves.

Wish I knew more about it...

PGH

--- In tuning-math@yahoogroups.com, "Paul H" <phjelmstad@...> wrote:
>
> p. 154: Projective Planes and Difference Sets
>
> This Chapter is exciting to me for a couple reasons. They discuss
> 7-tET, 13t-ET and 31-tET. 43-tET unfortunately is not a projective plane,
although 43 is k^2-k+1 when k=7.
>
> 1. Difference Sets in these projective planes have geometric duals.
> And these duals of difference sets end up being the inversion of the
> first difference set! So here we have a completely different application of
duals (where points and lines are interchanged) from
> the kind discussed here more often (between commas and vals).
>
> The theme of this chapter of the book here is finding what sets are fruitful
for composition, and which temperaments, etc.
>
> A FLID is a Difference Set, where every interval is used once.
> 12-tET only has one, a tetrachord and its Z-relation (111111)
> which is (0,1,4,7) and (0,1,3,6) in canonical form. This is
> the "first Z-relation" in 12-tET. Pentachords have only 3 of them,
> and Hexachords have 15, which are also complementary sets.
>
>
> 2. I was excited to see them use the 13-point plane (13 points and
> 13 lines in the 26-node diagram) for 13-tET and its difference sets.
> This is exciting because it is the same diagram used for "An Elementary
Approach to the Monster" in constructing the Bi-Monster (Y555 or "M666", the
Beast Group)which connects down to the Monster by
> means of M X M. (Subgroup of M | 2, Wreath group of the Monster (Bi-Monster))
This also relates to Conway's M13 game, etc.
>
> There is a composer who has composed a piece based on the Monster group.
>
> Definitions:
>
> FLID: Flat line Interval Distribution.
>
> Difference Set: When all the differences in a chord, are...different!
>
> Z-relation: When 2 or more sets of different Tn/TnI type have
> the same interval vector.
>
> * * *
>
> I have mapped the Z-relation for 31-tET chords, but stopped at
> hexachords because the numbers and patterns became overwhelming.
> But there are definite patterns. I am not yet certain of the
> relationship between Z-related sets, difference sets, and
> the affine relation. They might be 3 completely independent
> ideas? I do know that the Z-relation "carries through" in
> the affine relationship. And that (more obvious) the affine
> relationship also preserves difference sets, therefore I guess
> it is safe to say that:
>
> "A FLID or Difference Set operated upon by the Affine Group
> produces a Z-related Set (Isomeric Set) or is just trivial
> (Inversion, or Transposition of the Set or both (Same Tn/TnI Type).
>
> So in this case all three properties are related. Somehow this
> relates Affine and Projective geometry too.
>
> PGH

A better way to state this would be to say:

A Z-related difference set (pair, or triple, or whatever) operated
upon by the affine group action will produce another Z-related difference set
(along with its pair or triple, or whatever)

Any Z-related collection of sets operated upon by the affine
group will produce another Z-related collection, doesn't have to
be a difference set. If it is not, you will get a scrambled interval
vector.

Any difference set  operated upon by the affine action will create
another difference set, which will be Z-related if the first set
(collection) is, or will not be if the first set is not.

PGH

Hi all,

try it out and improve the site:
http://www40.wolframalpha.com/examples/Music.html

bye
A.S.

Difference Set: Sorry, when all Differences are the Same. The other
case, such as the Diatonic Collection <1,4,3,2,5,0> is another matter.

--- In tuning-math@yahoogroups.com, "Paul H" <phjelmstad@...> wrote:
>
> p. 154: Projective Planes and Difference Sets
>
> This Chapter is exciting to me for a couple reasons. They discuss
> 7-tET, 13t-ET and 31-tET. 43-tET unfortunately is not a projective plane,
although 43 is k^2-k+1 when k=7.
>
> 1. Difference Sets in these projective planes have geometric duals.
> And these duals of difference sets end up being the inversion of the
> first difference set! So here we have a completely different application of
duals (where points and lines are interchanged) from
> the kind discussed here more often (between commas and vals).
>
> The theme of this chapter of the book here is finding what sets are fruitful
for composition, and which temperaments, etc.
>
> A FLID is a Difference Set, where every interval is used once.
> 12-tET only has one, a tetrachord and its Z-relation (111111)
> which is (0,1,4,7) and (0,1,3,6) in canonical form. This is
> the "first Z-relation" in 12-tET. Pentachords have only 3 of them,
> and Hexachords have 15, which are also complementary sets.
>
>
> 2. I was excited to see them use the 13-point plane (13 points and
> 13 lines in the 26-node diagram) for 13-tET and its difference sets.
> This is exciting because it is the same diagram used for "An Elementary
Approach to the Monster" in constructing the Bi-Monster (Y555 or "M666", the
Beast Group)which connects down to the Monster by
> means of M X M. (Subgroup of M | 2, Wreath group of the Monster (Bi-Monster))
This also relates to Conway's M13 game, etc.
>
> There is a composer who has composed a piece based on the Monster group.
>
> Definitions:
>
> FLID: Flat line Interval Distribution.
>
> Difference Set: When all the differences in a chord, are...different!
>
> Z-relation: When 2 or more sets of different Tn/TnI type have
> the same interval vector.
>
> * * *
>
> I have mapped the Z-relation for 31-tET chords, but stopped at
> hexachords because the numbers and patterns became overwhelming.
> But there are definite patterns. I am not yet certain of the
> relationship between Z-related sets, difference sets, and
> the affine relation. They might be 3 completely independent
> ideas? I do know that the Z-relation "carries through" in
> the affine relationship. And that (more obvious) the affine
> relationship also preserves difference sets, therefore I guess
> it is safe to say that:
>
> "A FLID or Difference Set operated upon by the Affine Group
> produces a Z-related Set (Isomeric Set) or is just trivial
> (Inversion, or Transposition of the Set or both (Same Tn/TnI Type).
>
> So in this case all three properties are related. Somehow this
> relates Affine and Projective geometry too.
>
> PGH
>

p. 154: Projective Planes and Difference Sets

This Chapter is exciting to me for a couple reasons. They discuss
7-tET, 13t-ET and 31-tET. 43-tET unfortunately is not a projective plane,
although 43 is k^2-k+1 when k=7.

1. Difference Sets in these projective planes have geometric duals.
And these duals of difference sets end up being the inversion of the
first difference set! So here we have a completely different application of
duals (where points and lines are interchanged) from
the kind discussed here more often (between commas and vals).

The theme of this chapter of the book here is finding what sets are fruitful for
composition, and which temperaments, etc.

A FLID is a Difference Set, where every interval is used once.
12-tET only has one, a tetrachord and its Z-relation (111111)
which is (0,1,4,7) and (0,1,3,6) in canonical form. This is
the "first Z-relation" in 12-tET. Pentachords have only 3 of them,
and Hexachords have 15, which are also complementary sets.


2. I was excited to see them use the 13-point plane (13 points and
13 lines in the 26-node diagram) for 13-tET and its difference sets.
This is exciting because it is the same diagram used for "An Elementary Approach
to the Monster" in constructing the Bi-Monster (Y555 or "M666", the Beast
Group)which connects down to the Monster by
means of M X M. (Subgroup of M | 2, Wreath group of the Monster (Bi-Monster))
This also relates to Conway's M13 game, etc.

There is a composer who has composed a piece based on the Monster group.

Definitions:

FLID: Flat line Interval Distribution.

Difference Set: When all the differences in a chord, are...different!

Z-relation: When 2 or more sets of different Tn/TnI type have
the same interval vector.

* * *

I have mapped the Z-relation for 31-tET chords, but stopped at
hexachords because the numbers and patterns became overwhelming.
But there are definite patterns. I am not yet certain of the
relationship between Z-related sets, difference sets, and
the affine relation. They might be 3 completely independent
ideas? I do know that the Z-relation "carries through" in
the affine relationship. And that (more obvious) the affine
relationship also preserves difference sets, therefore I guess
it is safe to say that:

"A FLID or Difference Set operated upon by the Affine Group
produces a Z-related Set (Isomeric Set) or is just trivial
(Inversion, or Transposition of the Set or both (Same Tn/TnI Type).

So in this case all three properties are related. Somehow this
relates Affine and Projective geometry too.

PGH

THEORY IN A NUTSHELL

1. The 132 Steiner hexads comprise 1/7 of the 924 total hexads (hexachords), in
fact they comprise 1/7 in every BW (Black and White) category:

0B/6W 1B/5W 2B/4W 3B/3W 4B/2W 5B/1W

7 105 350 350 105 7 = 924 (Totals)
1 15 50 50 15 1 = 132 (Steiner Hexads)

2. A Steiner System is a Block Design such that each of the 132 Hexads contains
6 Pentads. In fact, every 792 Pentad is contained
exactly once, by some Hexad. No two Steiner Hexads contain the same Pentad.

3. (Technical) M12 is constructed from the Exceptional Outer Automorphism of the
Symmetric(6) Group, using duads and triads. This is modelled by a
ProjectiveGeometry(3,2). M12 is the Symmetry Group (Automorphism Group) of
Steiner(5,6,12), a Block design as mentioned previously.

4. By Diatonic Transposition of the Whites, we can span all 924 hexachords,
together and in each BW category. Open: Can Seven independent Steiner Systems be
constructed? The remaining hexads
from 924-132=792 comprise a Sextuple Steiner System. The total is a trivial
Septuple Steiner System.

5. The Black keys used are passively determined from this construction, they are
merely the remaining combinations in each BW category. However, there might be a
pattern there also.

6. Definition of Diatonic Transposition: 024->245->457, ie CDE->DEF->EFG etc. 
so with the 350 3B3W, 50 SS hexads -> 350 hexads, each 35 3W is used 10 times,
and there are exactly 10 3B to use with each of these. 3W is C(7,3) 3B is C(5,3)

7. Ultimately, I will show that 35 hexachord types of D4 X S3 X S2 also have
this pattern. Every hexachord type can be found
from 3W and 3B keys. (Known)  Basically, I will show that the 35 hexachord types
are merely 5 x 7, generated by the Steiner System above.

Note:

D4 X S3 X S2 is a Fourfold Symmetry whose hexachords can be found
from 80 + 20 + 8 + 32 = 140, 140/4 = 35. The separate parts are
found from Polya Necklace Theory.

Here is an interesting find, a paper on "David Lewin and Maximally Even Sets".
As a "pleasant by-product" of Lewin's work with the FFT (Fast Fourier
Transform), Amiot also presents a short proof of the Hexachord Theorem. (Theorem
1.9). It's only two lines. (p. 4 bottom p. 5 top)

http://canonsrythmiques.free.fr/4journalOfM&M/lewinMESets.pdf

Here's a one-page description of the Hexachord Theorem:

http://myweb.lsbu.ac.uk/~whittyr/MathSci/TheoremOfTheDay/MusicAndArt/Hexachord/T\
otDHexachord.pdf

Of course, Maximally-Even Sets, MOS, Myhill's Propriety (and maybe
Rothenberg Propriety?) are all related. It's fun to see this tie-in
with Hexachord Theory, which is my baby I guess, the Hexachord Theorem
leads to the Z-relation (Isomeric Sets) and the main issue of
"complementability" or the S2-symmetry. (Symmetry Types of Periodic
Sequences, Gilbert and Riordan, Ill. Journ. of Mus. 1961)

More tie-ins:

1. Z-relation, Flat-Line Interval Distribution, Difference Sets, Projective
Spaces, etc.

2. M5-relation, Affine Group, Necklace Theory, D4 X S3 for example.

The jump between Z-relation and FLIDs is a bit tenous but if I review
Jon Wild's paper I think it is in there.

Another interesting tangent would go between Projective Spaces,
Steiner Systems, M12, and ultimately, the Monster group.

The musical aspect of M12 and S(5,6,12) will be covered in my
paper on M12 and the Piano Keyboard Layout, where Diatonic Transposition spans
all 924 hexachords, seeded from 132 Steiner Hexads. (Seven
fold expansion.)

And that leads to Diatonic Set Theory, I guess.

PGH

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
I agree. This is exactly the type of math I am looking for.
I was daunted at first by the 123 pages, but actually, it's kind
of a Powerpoint style piece, and doesn't take too long to get through.

I especially liked the diagram of the staircases going in four
directions, what a great visualization of Christoffel words.
It's a very generous and straight-to-the-point contribution overall.

- Paul

>
> A great find.  The author is apparently unaware of the music
> connection, citing biology and such but not music theory.
> I like the graphic method.  I bet Erv would appreciate that.
>
> -Carl
>
> At 07:02 PM 4/21/2009, you wrote:
> >Here's a presentation on the mathematical equivalent of MOS
> >scales:
> >
> >http://www-irma.u-strasbg.fr/~kassel/ChristoffelNJ0407.pdf
> >
> >So, they're old, and they're known to relate to continued
> >fractions.
> >
> >The Fibonacci sequence makes a special guest appearance!
> >
> >
> >                  Graham
>

A great find.  The author is apparently unaware of the music
connection, citing biology and such but not music theory.
I like the graphic method.  I bet Erv would appreciate that.

-Carl

At 07:02 PM 4/21/2009, you wrote:
>Here's a presentation on the mathematical equivalent of MOS
>scales:
>
>http://www-irma.u-strasbg.fr/~kassel/ChristoffelNJ0407.pdf
>
>So, they're old, and they're known to relate to continued
>fractions.
>
>The Fibonacci sequence makes a special guest appearance!
>
>
>                  Graham

Here's a presentation on the mathematical equivalent of MOS
scales:

http://www-irma.u-strasbg.fr/~kassel/ChristoffelNJ0407.pdf

So, they're old, and they're known to relate to continued
fractions.

The Fibonacci sequence makes a special guest appearance!


                   Graham

This may be of interest:

http://www.science-bookmarks.com/2009/04/very-short-history-of-mathematics.html

I don't see direct citations, and I don't quite agree
they got the right events since 1950, but the early stuff
is interesting, including the development of logarithms
in 400 BC.

-Carl

I've had Kraig Grady post a simple Excel spreadsheet that approximates
intervals in cents by ratios by means of continued fractions at
http://anaphoria.com/journal.html. It is based on a decimal to ratio
converter written by Kardi Teknomo. Use Sheet 2 for your computations.

In additions to continued fraction convergents, the spreadsheet also
computes string lengths and frequencies using several common pitch bases
such as 440, 261.63, 256, and 264 hz.

I apologize in advance for cross-posting, but I think this program will
be of interest to members of all three lists and will help limit future
   conflicts over representing scales in decimals, cents or ratios.

--John

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:
>
> Note that 50/49 and 49/48 are not included in the list
> of accidentals. Since these are so close together
> (2401/2400 apart), it would be convenient not to need
> both of them. The easiest way to do this would be to
> use a temperament that tempers out 2401/2400.
>

Or, to the contrary, we have the option to keep those accidentals. The
2401/2400 ratio is a very small interval, around 0.7212 cents, but it
can be audible. Near 440 Hz it creates a beat rate of about one every
few (five to seven?) seconds. If say we propose a 55-tone 7-limit
scale where the 2401/2400 step size occurs four times in every octave
then our tonal resources include four different near-unisons with four
different slow beat rates per octave. As programmers often say, that's
not a bug, it's a feature!

WOLFGANG VON SCHWEINITZ
REDCAT January 24th, 2009

http://redcat.org/season/0809/mus/schweinitz.php

"[Plainsound Glissando Modulation is] one of the most outstanding
works of the middle generation of composers today." Manfred Karullus,
MusikTexte

The German composer has been long celebrated for the impassioned
eloquence and rare lyrical beauty of his music. More recently, von
Schweinitz has turned to a profound exploration of microtonal
phenomena in a body of work whose newest opus is this evening's
sublime Plainsound Glissando Modulation, a raga in just intonation for
violin and double bass. Receiving its U.S. debut, the monumental
composition is played with consummate artistry by violinist Helge
Slaatto and bassist Frank Reinecke, the virtuoso duo who last year
recorded the 75-minute raga for Bayerische Rundfunk. The concert opens
with Erika Duke-Kirkpatrick's performance of Plainsound-Litany, for
solo cello. Von Schweinitz, whose extensive oeuvre includes the
visionary 1989 opera Patmos-a setting of the Book of Revelation-holds
the Roy E. Disney Family Chair in Musical Composition at CalArts.

January 24th, 8:30 PM
WOLFGANG VON SCHWEINITZ

WHERE
REDCAT (The Roy and Edna Disney/CalArts Theater) is located at the
corner of W. 2nd St. and S. Hope St., inside the Walt Disney Concert
Hall complex.
TICKETS
Performance is scheduled for Saturday, January 24th, 2009, at 8:30
p.m. Ticket prices range from $16-$20, with discounts available.
Seating is general admission. Buy tickets at the REDCAT box office
located at the corner of 2nd and Hope Streets, by calling 213.237.2800
, or at http://www.redcat.org. <http://www.redcat.org./> Please plan
on arriving at least 30 minutes before curtain time. Seating at REDCAT
is unreserved, and late seating is not guaranteed.
PARKING
Parking is available in the Walt Disney Concert Hall parking garage.
Enter from 2nd St. and proceed to level P3 for direct access to
REDCAT. The evening event rate is $8 after 5 PM, $4 after 8 PM. Before
5 PM, the maximum daytime rate is $17.

But I guess what I am saying is, is it not interesting that the
Steiner hexads comprise exactly 1/7 in each black white category?

Colors: 0B6W,1B5W,2B4W,3B3W,4B2W,5B1W
Totals 7,105,350,350,105,7
Steiner 1,15,50,50,35,1

--- In tuning-math@yahoogroups.com, "Paul H" <phjelmstad@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul H" <phjelmstad@> wrote:
> >
> > Have been studying Myhill's Property and also Rothenberg Scales.
> >
> > Interesting that the diatonic and the pentatonic scales have
> > Myhill's property. So I thought I would study hexachords based
> > on black and white keys. Subsets of each have the property,
> > combined together, i don't know....
> >
> > Here is how all 924 hexachords break down
> >
> > White Keys Black Keys Hexachords
> >
> > 6  0   7
> > 5  1  105
> > 4  2  350
> > 3  3  350
> > 2  4  105
> > 1  5  7
> >
> > Hold C fixed
> >
> > 6  0  6
> > 5  1  75
> > 4  2  200
> > 3  3  150
> > 2  4  30
> > 1  5  1
> >
> > Notice subtracting Column Three Lower from Column Three Upper
> produces
> >
> > 1,30,150,200,75,6 which is just the lower set reversed. Also
> > 1+30+200=6+75+150=231 which is exactly half of 462 and 1/4 of 924..
> >
> > Now I have found that all 80 hexachord types can be expressed
> > with no more than 2 black keys.
> >
> > Now to study Rothenberg Scales and their hexachord subsets.
> >
> > PGH
>
> Now, of course the combinations for white and black keys are based on
> 7 * 1, 21 * 5, 35 * 10, 35 * 10, 21 * 5, 7 * 1 which is simple
> combinations. This gives
>
> 7,105,350,350,105,7. Now I was excited to find that the Steiner
> System I mention in my other post (Rothenberg Scales) is literally
> 1/7 of this in every combination: 1,15,50,50,15,1.
>
> I studied this a bit, and found that of course, of the 66 Steiner
> Hexads and their complements, all five blacks are used in each row.
> (If a hexad has 2 blacks, its complement will have 3, etc)
>
> Therefore 66*5=330 blacks used. Now of course one would expect
> equal usage, and its true, each black is used 66 times.
>
> Now its possible to derive an algorithm which will split the 924
> hexads into seven systems. (The first a Steiner System as given
> and the last six systems a Sextuple Steiner System, and possibly
> six independent systems?) Now for example with no blacks,
> we have one in SS, and of course 7 possible.
>
> With one black there are 15 sets, and of course each black is
> used 3 times (3 3 3 3 3) and in the Septuple system (all hexads)
> it's clear that each black is used 21 times, for each 21 white
> 5-sets (Binom(7,5)). One might expect that there are just 3 white
> 5-sets used, which is 1/7 of the 21 in the full system, so
> that you obtain a one-to-one with each black, for each one, but
> it is more subtle, there are 15 white 5-sets, and the 6 not
> used determine the key. It gets more elaborate with 3W 3B but
> the same principles apply.
>
> Finding Seven systems will determine a matching to the 35
> lines of PG(3,2), which is used to give the construction of the
> Steiner System (5,6,12) using Picture A in 3 combinations
> (15*3) and Pictures B and C (10 + 10) and a base set (012345)
> this makes 66, with their complements makes 132 hexads.
>
> I also feel that this will match up one to one with the 35
> hexachord types based on interval vector, that is D12 X S2
> where S2 is 2-complementation.
>
> PGH
>

Let's get this out of the other chain:)

Question:

Has anyone here studied any of these lattices, in relationship to
musical tuning?

1. A2.12 sublattice of the Leech Lattice (one of 23 Neimeier lattices)
2. A1.24 sublattice of the Leech Lattice
3. K12 Lattice
4. Barnes-Walls Lattice
5. Coxeter-Todd Lattice

There first two are based on M12 and M24 respectively, and of course
Leech is just a lattice of S(5,8,24) blown up, K12 I think is the
same as Coxeter-Todd?

Finally, is the A4 or D4 lattice discussed here based on the same
nomenclature as the Dynkin diagrams? (Used for Lie Algebras, et c)

Thanks

PGH

Question:

Has anyone here studied any of these lattices, in relationship to
musical tuning?

1. A2.12 sublattice of the Leech Lattice (one of 23 Neimeier lattices)
2. A1.24 sublattice of the Leech Lattice
3. K12 Lattice
4. Barnes-Walls Lattice
5. Coxeter-Todd Lattice

There first two are based on M12 and M24 respectively, and of course
Leech is just a lattice of S(5,8,24) blown up, K12 I think is the
same as Coxeter-Todd?

Finally, is the A4 or D4 lattice discussed here based on the same
nomenclature as the Dynkin diagrams? (Used for Lie Algebras, et c)

Thanks

PGH

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:

> This suggests something around 53-ET as a first approximation. 53-ET
> has some useful properties,....

how about
http://launch.groups.yahoo.com/group/tuning/message/81808
as second approxiamtion with other useful properites?

Represented in
extended Bosanquet-Helmholtz notation
by the additional accidentials:

'/' one comma sharper up
'//' two commata sharper up '+'
and respectively
'\' one comma flattend down
'\\' two commata flattend down '-'

Begin with:
0: C- 1 C\\ absolute fundamental root 1/1 == 3^0/2^0 @ start
1: G- 3 G\\ 3^1
2: D- 9 D\\ 3^2
3: A- 27 A\\ 3^3
4: E- 81 E\\ 3^4
5: B- 243 B\\ 3^5
6: GB 729 Gb\ 3^6 last just Pythagorean 3-limit-smooth ratio 729/512
7: DB(547 1094 2188 4376>) 4375 (>4374 2187 = 3^7) poppy-seed ragisma
8: AB 1641 := 547*3 Ab\
9: EB 1231 2646 4924 (> 4923 := 1641*3) Eb\
10: BB 1847 3694 (> 3693 := 1231*3 ) Bb\
11: F\ 2771 5542 (> 5541 := 1847*3 )
12: C\ 4157 8314 (> 8313)
13: G\ 1559 3118 6236 12472 (> 12471)
14: D\ 1169 ... 4678 (> 4677)
15: A\ 877 ... 3508 (> 3507) absolute normal-pitch A\_4=438.5 Hz
16: E\ 329 ... 2632 (> 2631)
17: B\ 987
18: Gb 2961
19: Db 2221 4442 8884 (> 8883 := 2961*3)
20: Ab 833 ... 6664 (> 6663 := 2221*3)
21: Eb 625 1250 2500 (> 2499 = 833*3) = 5^4 = 5*5*5*5 first 5-smooth
22: Bb 1875 = 5^4*3
23: F_ 703 1406 2812 5624 (<5625 = 5^4*3^2)
24: C_ 527 middele_C4 263.5 Hz
25: G_ 1581
26: D_ 2371 4742 (< 4743)
27: A_ 889 ... 7112 (< 7113) ~@ Herbert Karajan's pitch a'=444.5Hz
28: E_ 2667
29: B_ 125 = 5^3 = 5*5*5 first schisma ready versus 21:
30: F# 375
31: C# 1125
32: G# 1687 3374 (< 3375 = 5^3*3^3 )
33: D# 1265 ... 5065 (< 5061)
34: A# 1897 3794 (< 3795)
35: F/ 2845 5690 (< 5691)
36: C/ 4267 8534 (< 8535)
37: G/ 25 50 100 200 400 800 1600 3200 6400 12800(<12801) 5^2=5*5
38: D/ 75 second schisma ready on 37:
39: A/ 225
40: E/ 675
41: B/ 253 ... 2024 (< 2025 = 5^2*3^4)
42: F& 759 F#/
43: C& 569 1138 2276 (< 2277) C#/
44: G& 1707 G#/
45: D& 5 ... 5120 (< 5121) D#/ third schisma ready;simplest 5-smoothie
46: A& 15 A#/
47: F+ 45 F//
48: C+ 135 C//
49: G+ 405 G//
50: D+ 1215 D+
51: A+ 911 1822 3644 (<3645 = 5*3^6) A// attack the last schisma
52: E+ 683 1366 2732 (<2733) E//
53: B+ 1 ... 2048 (< 2049) B//=C\\ returned back to root C-_-4 = 1/1

or lined up in ascending order
as scala-file ratios ! vs. note-names in absolute-pitches

! poppy_seed53tone.scl
!
Sparschuh's 7-lim.dyadic 53-tone ragismatic 4375:4374 poppy-seed-comma
!
! 1/1 ___ ! @ 00: C- 512Hz C\\ tenor-C\\5, 9-octave above the root 1/1
4157/4096 ! A 01: C\ 519.625
527/512 !_! B 02: C_ 527
4267/4096 ! C 03: C/ 533.375
135/128 !_! D 04: C+ 540 C//
4375/4096 ! E 05: DB 546.875 Db\ := 4375/8 = 7*5^3/2^3
2221/2048 ! F 06: Db 555.25
1125/1024 ! G 07: D# 562.5
569/512 !_! H 08: C& 569 C#\
9/8 !_____! I 09: D- 576 D\\
1169/1024 ! J 10: D\ 584.5
2371/2048 ! K 11: D_ 592.75
75/64 !___! L 12: D/ 600
1215/1024 ! M 13: D+ 607.5 D//
1231/1024 ! N 14: EB 615.5 Eb\
625/512 !_! O 15: Eb 625 = 5^4
1265/1024 ! P 16: D# 630.5
5/4 !_____! Q 17: D& 640 D#/
81/64 !___! R 18: E- 648 E\\
329/256 !_! S 19: E\ 658
2667/2048 ! T 20: E_ 666.75
675/512 !_! U 21: E/ 675
683/512 !_! V 22: E+ 683 E// = F\\ = F- temper out Mercator's comma
2771/2048 ! W 23: F\ 692.75
703/512 !_! X 24: F_ 703 F
2485/2048 ! Y 25: F/ 711.25
45/32 !___! Z 26: F+ 720 F//
729/512 !_! a 27: GB 729 Gb\
2961/2048 ! b 28: Gb 740.25
375/256 !_! c 29: F# 750
759/512 !_! d 30: F& 759 F#/
3/2 !_____! e 31: G- 768 G\\ quinte
1559/1024 ! f 32: G\ 779.5
1581/1024 ! g 33: G_ 790.5
25/16 !___! h 34: G/ 800 Mersenne, Sauveur & Werckmeister's CammerThon
405/256 !_! i 35: G+ 810 G//
1641/1024 ! j 36: AB 820.5 Ab\
833/512 !_! k 37: Ab 833
1687/1024 ! l 38: G# 843.5
1707/1024 ! m 39: G& 853.5 G#/
27/32 !___! n 40: A- 864 A\\
877/512 !_! o 41: A\ 877 = 2*438.5Hz @ 90 MetronomeBeats/min vs.440Hz
889/512 !_! p 42: A_ 889
225/128 !_! q 43: A/ 900
911/512 !_! r 44: A+ 911 A//
1847/1024 ! s 45: BB 923.5 Bb\
1875/1024 ! t 46: Bb 937.5
1897/1024 ! u 47: A# 948.5
15/8 !____! v 48: A& 960 A#/
243/128 !_! w 49: B- 972 B\\
987/512 !_! x 50: B\ 987
125/64 !__! y 51: B_ 1000 = 10^3 = 1kHz psycho-acustical normal-pitch
253/128 !_! z 52: B/ 1016
2/1 !_____! @'53: B+ 1024 = 2^11 B//_5 = C\\_6 = C-_6 the sopran-C\\
!
!
So far my 'theoretically-notation',
but practically i do recommend
out of that theory the plain absolute fixed pitch-frequencies alone

@ 00: C- 512Hz root
A 01: C\ 519.625
B 02: C_ 527
C 03: C/ 533.375
D 04: C+ 540
E 05: DB 546.875
F 06: Db 555.25
G 07: D# 562.5
H 08: C& 569
I 09: D- 576
J 10: D\ 584.5
K 11: D_ 592.75
L 12: D/ 600
M 13: D+ 607.5
N 14: EB 615.5
O 15: Eb 625
P 16: D# 630.5
Q 17: D& 640 third
R 18: E- 648
S 19: E\ 658
T 20: E_ 666.75
U 21: E/ 675
V 22: E+ 683
W 23: F\ 692.75
X 24: F_ 703
Y 25: F/ 711.25
Z 26: F+ 720
a 27: GB 729
b 28: Gb 740.25
c 29: F# 750
d 30: F& 759
e 31: G- 768 quinte
f 32: G\ 779.5
g 33: G_ 790.5
h 34: G/ 800
i 35: G+ 810
j 36: AB 820.5
k 37: Ab 833
l 38: G# 843.5
m 39: G& 853.5
n 40: A- 864
o 41: A\ 877 = 2*438.5Hz @ 90 MetronomeBeats/min vs.440Hz
p 42: A_ 889
q 43: A/ 900
r 44: A+ 911
s 45: BB 923.5
t 46: Bb 937.5
u 47: A# 948.5
v 48: A& 960
w 49: B- 972
x 50: B\ 987
y 51: B_ 1000
z 52: B/ 1016
@'53: B+ 1024 or sopran_C-

Labeled here more restrictive
only in one- or two-letter note-name nomenclauture,
in order to keep my rational-53 basic concept more simple.

bye
A.S.

As a first attempt, I'm starting with the assumption that the notation
system I'm looking for will have names for 4/3 and 3/2 -- for simplicity
I'll call the 1/1 "D" so that I can use "G" for 4/3, "A" for 3/2.
Pitches near these notes will use accidentals. 7/5 can be notated G +
21/20 or A - 15/14, so it won't be necessary to include it in the basic
scale unless the scale has enough notes that the gap between 4/3 and 3/2
is too large.

So, with a list of intervals likely to be used in 7-limit scales, I can
identify 81/80, 64/63, 36/35, 28/27, 25/24, 21/20, 135/128, 16/15, and
15/14 as some of the more important accidentals that will be needed. The
importance of superparticular ratios is one of the ideas that I've
established for Zireen music, so I'll want to take another look at that
135/128.

21/20 * 225/224 = 135/128

Since the difference is relatively small, the 225/224 could be
represented as an extra mark (e.g. a hook or a slash) added to the main
21/20 accidental. Note that the pairs (28/27 25/24) and (16/15 15/14)
are also separated by 225/224.

Note that 50/49 and 49/48 are not included in the list of accidentals.
Since these are so close together (2401/2400 apart), it would be
convenient not to need both of them. The easiest way to do this would be
to use a temperament that tempers out 2401/2400.

49/48 is the difference between 8/7 and 7/6. 50/49 seems to be mainly
needed for ratios containing 49 (e.g. 49/40 * 50/49 = 5/4).

Not counting 49/48, you've got 5 basic size classes of accidentals.

81/80, 64/63
36/35
28/27, 25/24
21/20, 135/128
16/15, 15/14

This suggests something around 53-ET as a first approximation. 53-ET has
some useful properties, but tempering out 2401/2400 isn't one of them.
Another approach is back to 171-ET, which does temper out 2401/2400 and
has a step size near 225/224.

Graham Breed wrote:
> Herman Miller wrote:
>> As a simplification, you really only need a notation for the pitches
>> from 1/1 to 3/2. The rest of the pitches can be reached by inverting the
>> notation down from 2/1, or transposing up by 3/2.
>
> I don't follow this.  Decimal notation, for example, doesn't
> have a 4/3 from the root.

Well, that's an example of symmetrical notation. You can go 4 steps up
and 5 steps down (or vice versa).

>> 5/4 and 6/5 are also pretty basic intervals, but you don't need both of
>> them: the difference between them is 25/24, which is a useful accidental
>> to have. E.g., 25/18 = 4/3 * 25/24. You could represent these in other
>> ways, e.g. 5/4 as 4/3 * 15/16, 6/5 as 7/6 * 36/35, but at least one of
>> these would be useful to have a basic notation for. (There's always the
>> option of the chain of fifths notation using 81/80 as an accidental, but
>> I'm looking for alternative notations.)
>
> You do need both of them to get a major triad, which is nice
> to have.

You don't need both of them to be notated from the same root without an
accidental.

>> An essential thing for a 7-limit notation is being able to represent 7/6
>> and 8/7 reasonably well. 7/6 could be 6/5 * 35/36, 8/7 could be 9/8 *
>> 64/63, but it would be nice to have one of these as a basic pitch of the
>> notation system, with 49/48 as an accidental to notate the other one.
>
> 8:7 is useful because it breaks down into 15:14 and 16:15 --
> two secors or toes.

Okay, that's a point in its favor. Either 15:14 could be another basic
step of the notation, or it could be an accidental.

>> Ideally, the kind of scale I'm looking for should have all relatively
>> simple ratios, largest/smallest step size ratio less than 2, and
>> strictly proper.
>>
>> This is the sort of scale that might be appropriate, but there doesn't
>> seem to be anything obvious or special about it. I'm hoping to find
>> something better.
>>
>> 1/1 15/14 8/7 5/4 4/3 7/5 3/2 8/5 12/7 15/8 2/1
>
> Move 5/4 and 4/3 down a quomma and you have a decimal scale.

Hmm, well I suppose I could give that a look.

Another thing to try is finding notations for scales that I've already
used in Zireen music and see what fits. Here's one possibility for
lemba[26]:

(+3, -6)  5/4 / 225/224   56/45
(+4, -6)  7/4 *  64/63    16/9
(+2, -5) 10/9 /  16/15    25/24
(+3, -5)  3/2 /  36/35    35/24
(+2, -4) 10/9 *  21/20     7/6
(+3, -4)  7/4 /  21/20     5/3
(+2, -3)  4/3              4/3
(+3, -3)  7/4 *  16/15    28/15
(+1, -2) 10/9 /  25/24    16/15
(+2, -2)  3/2 *  25/24    25/16
(+1, -1)  5/4              5/4
(+2, -1)  7/4              7/4
(+0, +0)  1/1              1/1
(+1, +0)  3/2 /  15/14     7/5
(+0, +1) 10/9 *  36/35     8/7
(+1, +1)  3/2 *  16/15     8/5
(+0, +2)  4/3 /  64/63    21/16
(+1, +2)  7/4 *  15/14    15/8
(-1, +3) 10/9 /  28/27    15/14
(+0, +3)  3/2              3/2
(-1, +4)  5/4 /  25/24     6/5
(+0, +4)  7/4 /  49/48    12/7
(-2, +5)  1/1 * 225/224  225/224
(-1, +5)  3/2 /  16/15    45/32
(-2, +6) 10/9 *  81/80     9/8
(-1, +6)  3/2 *  15/14    45/28

Probably not the notation I'll end up using, but this should give me an
idea of which nominals and accidentals are most productive.