Yahoo Tuning Groups Ultimate Backup tuning-math Connecting up ET's

Another monologue, to be sure :)

My goal with my studies of M12, M22 and M24 Mathieu Groups (I suppose I could throw in M11 and M23) and their related Steiner systems from which they are built, is to connect 22-tET, 24-tET and 22-tET in a fluid way. (I prefer these designations to 22-EDO, 24-EDO and 22-EDO)

So, 12-tET is grand central station, as is 24-tET, being diaschismic, schismic, meantone, misty, etc. and 22-tET is magic, porcupine, diaschismic, orwell. 11-tET is macrotemperament, similar to 22-tET, and 23-tET is pelogic and schismic. So that's start with that.

Anyway, my goal with the 12-22-24 complex is to consider Steiner systems (which have some amazing properties wrt tuning), and look first at 5,6,12; 3,6,22; and 5,8,24 Steiner systems, (well and also 4,5,11 and 4,7,23 systems). And let's link up pentads in 12-tET and 24-tET (Easy to do, consider ALL pentads in each system), and then also link up hexads in 12-tET and 22-tET (a little harder; these are the Stein BIBD (blocks)), and then work with all the fun symmetries such as S3 X S4, and those that are maximal subgroups (or related to them) in M12, M22 and M24.

But with regards to tuning, let's link up the 12-22-24 complex, and this cannot involve porcupine for example. So I get schismic as the only possibility. How do these related to the hook-ups with pentads and hexads? (12-(6)-22) and (12-(5)-24). The more interesting one of course is the first one, the building up from M12 to M24 is well-known, although not known to Mathieu himself! So relating hexads in M12 to M22 is key; and this might involve the schismic tuning.

Here is what is in Xenharmonic Wiki:

The 5-limit parent comma for the diaschismic family is 2048/2025, the diaschisma. Its monzo is |11 -4 -2>, and flipping that yields <<2 -4 -11|| for the wedgie for 5-limit diaschismic, or srutal, temperament. This tells us the period is half an octave, the GCD of 2 and -4, and that the generator is a fifth. Three periods gives 1800 cents, and decreasing this by two fifths gives the major third. 34edo is a good tuning choice, with 46edo, 56edo, 58edo or 80edo being other possibilities. Both 12edo and 22edo support it, and retuning them to a MOS of diaschismic gives two scale possibilities

* * *

So of course four fifths plus two major thirds match an octave. Now consider the set (in 12-tET) of 0,7,2,9,4,8 and use this as a baseline to build both Steiner systems in M12 and M22, (fundamental set) and by geometric processes construct each Steiner system.

This connects the two systems, both in terms of tuning and set theoretical processes.

Damn, too late for APRIL FOOLS!!!!!!!!! But maybe not. I kind of like where this is going. And maybe the building up of M22 (through M23) to M24 and then back down to M12 might be relevant here.

PGH

🔗Graham Breed <gbreed@gmail.com>

🔗Paul <phjelmstad@msn.com>

--- In tuning-math@yahoogroups.com, "Paul" <phjelmstad@...> wrote:
>
> Another monologue, to be sure :)
>
> My goal with my studies of M12, M22 and M24 Mathieu Groups (I suppose I could throw in M11 and M23) and their related Steiner systems from which they are built, is to connect 22-tET, 24-tET and 22-tET in a fluid way. (I prefer these designations to 22-EDO, 24-EDO and 22-EDO)
>
> So, 12-tET is grand central station, as is 24-tET, being diaschismic, schismic, meantone, misty, etc. and 22-tET is magic, porcupine, diaschismic, orwell. 11-tET is macrotemperament, similar to 22-tET, and 23-tET is pelogic and schismic. So that's start with that.
>
> Anyway, my goal with the 12-22-24 complex is to consider Steiner systems (which have some amazing properties wrt tuning), and look first at 5,6,12; 3,6,22; and 5,8,24 Steiner systems, (well and also 4,5,11 and 4,7,23 systems). And let's link up pentads in 12-tET and 24-tET (Easy to do, consider ALL pentads in each system), and then also link up hexads in 12-tET and 22-tET (a little harder; these are the Stein BIBD (blocks)), and then work with all the fun symmetries such as S3 X S4, and those that are maximal subgroups (or related to them) in M12, M22 and M24.
>
> But with regards to tuning, let's link up the 12-22-24 complex, and this cannot involve porcupine for example. So I get schismic as the only possibility. How do these related to the hook-ups with pentads and hexads? (12-(6)-22) and (12-(5)-24). The more interesting one of course is the first one, the building up from M12 to M24 is well-known, although not known to Mathieu himself! So relating hexads in M12 to M22 is key; and this might involve the schismic tuning.
>
> Here is what is in Xenharmonic Wiki:
>
> The 5-limit parent comma for the diaschismic family is 2048/2025, the diaschisma. Its monzo is |11 -4 -2>, and flipping that yields <<2 -4 -11|| for the wedgie for 5-limit diaschismic, or srutal, temperament. This tells us the period is half an octave, the GCD of 2 and -4, and that the generator is a fifth. Three periods gives 1800 cents, and decreasing this by two fifths gives the major third. 34edo is a good tuning choice, with 46edo, 56edo, 58edo or 80edo being other possibilities. Both 12edo and 22edo support it, and retuning them to a MOS of diaschismic gives two scale possibilities
>
> * * *
>
> So of course four fifths plus two major thirds match an octave. Now consider the set (in 12-tET) of 0,7,2,9,4,8 and use this as a baseline to build both Steiner systems in M12 and M22, (fundamental set) and by geometric processes construct each Steiner system.
>
> This connects the two systems, both in terms of tuning and set theoretical processes.
>
> Damn, too late for APRIL FOOLS!!!!!!!!! But maybe not. I kind of like where this is going. And maybe the building up of M22 (through M23) to M24 and then back down to M12 might be relevant here.
>
> PGH
>

Actually, I should read "connect 12-tET, 22-tET, and 24-tET in a fluid way"

The idea in brief, here is that you can kind of linearize or divide up the fabric of the musical space (texture) with Steiners, and there are great patterns, in 22-tET and 12-tET it works based on 7 key Steiners in 22-tET, and 11 in the second, so that you can use these as a basis to figure out the whole set-theoetical space. In 12-tET, I am using the Elkies double Steiner system, and there are 11 key sets (along with their 12 transpositions, and inverses and inverses of transpositions) but you can find the pentad space easily with just 11 forward and 11 backwards based on pentad class. And then you can find the remaining hexads in each row based on this fabric ---

Without going into the details much, one can take 0,7,2,9,4,8, hexad, and either generate all the Steiners, or merely compare this set with Steiners that are "sisters' to it, (share a pentad), and you end up finding 10 of the 22 Steiners, but in brief, the main point is that it cuts up the space in an aesthetically please way --- and also the permutation to get to the base set to the others is meaningful, moving end nodes 1,1,3,3,4,4,5,5,6,6 semitones based on 0,7,2,9,4,8 as nodes and 1,6 ; 4; 3,3 ; 4; 1,6 ; 5,5; so odd nodes have one sister steiners and the evens have two, of course there are other Steiners (12 left) which can be found using 0,1,2,3,4,8, and assume the tritone as half ---

Now diaschismic of course is 2048/2025, one implication being that 5/4 * 9/8, 5/4 * 9/8 matches the octave, so the tritone has to be equal, the key set is based on this (3,3,3,3,5,5 just multiplication, ignoring "2", Long story short, the idea is the canvas and configure the space, finding key Steiners maps the pentads, and then back again to the hexads remaining...

I made some fun Dynkin style diagrams (although nothing like a Dynkin diagram), based on 3-3-3-3 | 5 | 5 in various tinkertoy combinations, looking like chemicals

PGH

🔗Paul <phjelmstad@msn.com>

--- In tuning-math@yahoogroups.com, "Paul" <phjelmstad@...> wrote:
>
>
>
> --- In tuning-math@yahoogroups.com, "Paul" <phjelmstad@> wrote:
> >
> > Another monologue, to be sure :)
> >
> > My goal with my studies of M12, M22 and M24 Mathieu Groups (I suppose I could throw in M11 and M23) and their related Steiner systems from which they are built, is to connect 22-tET, 24-tET and 22-tET in a fluid way. (I prefer these designations to 22-EDO, 24-EDO and 22-EDO)
> >
> > So, 12-tET is grand central station, as is 24-tET, being diaschismic, schismic, meantone, misty, etc. and 22-tET is magic, porcupine, diaschismic, orwell. 11-tET is macrotemperament, similar to 22-tET, and 23-tET is pelogic and schismic. So that's start with that.
> >
> > Anyway, my goal with the 12-22-24 complex is to consider Steiner systems (which have some amazing properties wrt tuning), and look first at 5,6,12; 3,6,22; and 5,8,24 Steiner systems, (well and also 4,5,11 and 4,7,23 systems). And let's link up pentads in 12-tET and 24-tET (Easy to do, consider ALL pentads in each system), and then also link up hexads in 12-tET and 22-tET (a little harder; these are the Stein BIBD (blocks)), and then work with all the fun symmetries such as S3 X S4, and those that are maximal subgroups (or related to them) in M12, M22 and M24.
> >
> > But with regards to tuning, let's link up the 12-22-24 complex, and this cannot involve porcupine for example. So I get schismic as the only possibility. How do these related to the hook-ups with pentads and hexads? (12-(6)-22) and (12-(5)-24). The more interesting one of course is the first one, the building up from M12 to M24 is well-known, although not known to Mathieu himself! So relating hexads in M12 to M22 is key; and this might involve the schismic tuning.
> >
> > Here is what is in Xenharmonic Wiki:
> >
> > The 5-limit parent comma for the diaschismic family is 2048/2025, the diaschisma. Its monzo is |11 -4 -2>, and flipping that yields <<2 -4 -11|| for the wedgie for 5-limit diaschismic, or srutal, temperament. This tells us the period is half an octave, the GCD of 2 and -4, and that the generator is a fifth. Three periods gives 1800 cents, and decreasing this by two fifths gives the major third. 34edo is a good tuning choice, with 46edo, 56edo, 58edo or 80edo being other possibilities. Both 12edo and 22edo support it, and retuning them to a MOS of diaschismic gives two scale possibilities
> >
> > * * *
> >
> > So of course four fifths plus two major thirds match an octave. Now consider the set (in 12-tET) of 0,7,2,9,4,8 and use this as a baseline to build both Steiner systems in M12 and M22, (fundamental set) and by geometric processes construct each Steiner system.
> >
> > This connects the two systems, both in terms of tuning and set theoretical processes.
> >
> > Damn, too late for APRIL FOOLS!!!!!!!!! But maybe not. I kind of like where this is going. And maybe the building up of M22 (through M23) to M24 and then back down to M12 might be relevant here.
> >
> > PGH
> >
>
> Actually, I should read "connect 12-tET, 22-tET, and 24-tET in a fluid way"
>
> The idea in brief, here is that you can kind of linearize or divide up the fabric of the musical space (texture) with Steiners, and there are great patterns, in 22-tET and 12-tET it works based on 7 key Steiners in 22-tET, and 11 in the second, so that you can use these as a basis to figure out the whole set-theoetical space. In 12-tET, I am using the Elkies double Steiner system, and there are 11 key sets (along with their 12 transpositions, and inverses and inverses of transpositions) but you can find the pentad space easily with just 11 forward and 11 backwards based on pentad class. And then you can find the remaining hexads in each row based on this fabric ---
>
> Without going into the details much, one can take 0,7,2,9,4,8, hexad, and either generate all the Steiners, or merely compare this set with Steiners that are "sisters' to it, (share a pentad), and you end up finding 10 of the 22 Steiners, but in brief, the main point is that it cuts up the space in an aesthetically please way --- and also the permutation to get to the base set to the others is meaningful, moving end nodes 1,1,3,3,4,4,5,5,6,6 semitones based on 0,7,2,9,4,8 as nodes and 1,6 ; 4; 3,3 ; 4; 1,6 ; 5,5; so odd nodes have one sister steiners and the evens have two, of course there are other Steiners (12 left) which can be found using 0,1,2,3,4,8, and assume the tritone as half ---
>
> Now diaschismic of course is 2048/2025, one implication being that 5/4 * 9/8, 5/4 * 9/8 matches the octave, so the tritone has to be equal, the key set is based on this (3,3,3,3,5,5 just multiplication, ignoring "2", Long story short, the idea is the canvas and configure the space, finding key Steiners maps the pentads, and then back again to the hexads remaining...
>
> I made some fun Dynkin style diagrams (although nothing like a Dynkin diagram), based on 3-3-3-3 | 5 | 5 in various tinkertoy combinations, looking like chemicals
>
> PGH
>

To be very brief, take diaschismic to mean 3,3,3,3,5,5 in terms of
primes. Now with various tinkertoy arrangments (the most obvious being 3,3,3,3,5,5; 3,3,3,5,3,5; 3,3,5,3,3,5 etc) we can canvas all the Steiners (there are 22). I will show the most difficult one.

(From there you get all the pentads, twice, since it's a double Steiner system, and the you can get all hexads back by considering
Pentad + Element member of (Complementary(Steiner(Hexad))) after working out the displacement of the complementary hexads)

0,1,2,3,4,6 is the hardest to construct. Connect 0,4 and 2,6 with Maj3rd. Connect 0,2 with two P5ths (one missing) Connect 1,3 the same way. Viola, 4 uses of 3 and 2 of 5 (3^4 5^2 shape). Of course, more than just Steiners can be constructed this way, but this gives an elegant way to canvass all pentads... and then back to hexads, (assuming the correct fifth or third remains). A way around this is to make sure than complementary Steiners are hooked together with one fifth or major third ... to locate them ... and work with that. Or just hook up the one you need to take ... to an element in the pentad on the first side.