Yahoo Tuning Groups Ultimate Backup tuning PERMOS

I've not gotten any replies on FB so I'm moving discussion of permutations of MOS here.

Suppose you have a MOS, notated in terms of large and small steps, for example LsLLLsLLLsLL. If L-s is added to the s of an adjacent Ls or sL pair, then Ls is converted to sL, and sL to Ls. The MOS has be acted on by an adjacent transposition: http://planetmath.org/SymmetricGroupIsGeneratedByAdjacentTranspositions.html
Similarly, if L-s is subtracted from L in Ls or sL, then they are converted to sL and Ls. These lead to MODMOS; since every permutation is generated by adjacent transpositions, permutations of the steps of a MOS lead to MODMOS. On the other hand, not all MODMOS arise in this way (eg, the harmonic minor scale) since you can add L-s to L, for instance.

We might call a MODMOS which is derived from a MOS by a circular permutation (modal transposition) followed by at minimum k adjacent transpositions a PERMOS of inversion number k (though that fudges the definition of inversion number, so maybe something else?)

One interesting aspect is that we can generalize Fokker blocks in this way. For instance, the centaur scale is fully characterized by this:

august: sLLLLsLLLsLL; dominant: sLsLssLsLsLs; meantone: sLsLLsLsLsLL

That's a PERMOS and two MOS, and you need to define the temperaments they go with, as above.

🔗Mike Battaglia <battaglia01@...>

On Sun, Nov 25, 2012 at 3:46 PM, genewardsmith <genewardsmith@...>
wrote:
>
> Suppose you have a MOS, notated in terms of large and small steps, for
> example LsLLLsLLLsLL. If L-s is added to the s of an adjacent Ls or sL pair,
> then Ls is converted to sL, and sL to Ls. The MOS has be acted on by an
> adjacent transposition:
> http://planetmath.org/SymmetricGroupIsGeneratedByAdjacentTranspositions.html

Am I supposed to understand the 2-tuples here to represent inversions?

> We might call a MODMOS which is derived from a MOS by a circular
> permutation (modal transposition) followed by at minimum k adjacent
> transpositions a PERMOS of inversion number k (though that fudges the
> definition of inversion number, so maybe something else?)

Does "adjacent transposition" here mean the same as an inversion?
- If so, how is calling it "inversion number k" fudging the definition
of inversion number? It seems exactly the same to me.
- If not, what is it?

> One interesting aspect is that we can generalize Fokker blocks in this
> way. For instance, the centaur scale is fully characterized by this:
>
> august: sLLLLsLLLsLL; dominant: sLsLssLsLsLs; meantone: sLsLLsLsLsLL
>
> That's a PERMOS and two MOS, and you need to define the temperaments they
> go with, as above.

Do you know if these Fokker blocks have any special properties vs ones
generated by MODMOS and MOS in general?

-Mike

🔗genewardsmith <genewardsmith@...>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Sun, Nov 25, 2012 at 3:46 PM, genewardsmith <genewardsmith@...>
> wrote:
> >
> > Suppose you have a MOS, notated in terms of large and small steps, for
> > example LsLLLsLLLsLL. If L-s is added to the s of an adjacent Ls or sL pair,
> > then Ls is converted to sL, and sL to Ls. The MOS has be acted on by an
> > adjacent transposition:
> > http://planetmath.org/SymmetricGroupIsGeneratedByAdjacentTranspositions.html
>
> Am I supposed to understand the 2-tuples here to represent inversions?

2-tuples are just 2-tuples.

> > We might call a MODMOS which is derived from a MOS by a circular
> > permutation (modal transposition) followed by at minimum k adjacent
> > transpositions a PERMOS of inversion number k (though that fudges the
> > definition of inversion number, so maybe something else?)
>
> Does "adjacent transposition" here mean the same as an inversion?

A transposition is (a, b), where a and b are not necessariy adjacent: so, (2,5) interchanges 2 and 5. An adjacent transposition is (a, a+1). Both are involutions (self-inverse.)

> Do you know if these Fokker blocks have any special properties vs ones
> generated by MODMOS and MOS in general?

They temper to MODMOS, if that counts.

🔗genewardsmith <genewardsmith@...>

🔗Mike Battaglia <battaglia01@...>

On Sun, Nov 25, 2012 at 9:27 PM, genewardsmith <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> >
> > On Sun, Nov 25, 2012 at 3:46 PM, genewardsmith <genewardsmith@...>
> > >
> > > http://planetmath.org/SymmetricGroupIsGeneratedByAdjacentTranspositions.html
> >
> > Am I supposed to understand the 2-tuples here to represent inversions?
>
> 2-tuples are just 2-tuples.

It seems like the 2-tuples here are representing these:
http://en.wikipedia.org/wiki/Inversion_(discrete_mathematics)

Is this not correct? If not, what do they represent?

> > > We might call a MODMOS which is derived from a MOS by a circular
> > > permutation (modal transposition) followed by at minimum k adjacent
> > > transpositions a PERMOS of inversion number k (though that fudges the
> > > definition of inversion number, so maybe something else?)
> >
> > Does "adjacent transposition" here mean the same as an inversion?
>
> A transposition is (a, b), where a and b are not necessariy adjacent: so,
> (2,5) interchanges 2 and 5. An adjacent transposition is (a, a+1). Both are
> involutions (self-inverse.)

I guess you want this then:
http://en.wikipedia.org/wiki/Damerau%E2%80%93Levenshtein_distance

> > Do you know if these Fokker blocks have any special properties vs ones
> > generated by MODMOS and MOS in general?
>
> They temper to MODMOS, if that counts.

This isn't true of non-PERMOS MODMOS?

-Mike

🔗Mike Battaglia <battaglia01@...>

🔗genewardsmith <genewardsmith@...>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> I guess you want this then:
> http://en.wikipedia.org/wiki/Damerau%E2%80%93Levenshtein_distance

Except I don't want insertions, deletions, or substitutions. I want the Damerau part without the Levenshtein part. For now we could try calling that Damerau distance.

You can use "Damerau distance" to expand on the idea of domes and arenas. Given a rank n note group N, JI or formal temperament, we can define a certain symmetric regular graph; let's assume p-limit JI with p=pi(N) to make this easier. We assume an edo-type val v, so that v[1] = v(2) = m is the number m of notes of the scale, that we have m-1 rank two temperaments which are a Fokker basis for v, and that for each such temperament we have m-note MOS. The vertices of the graph are lists of m-1 PERMOS of these MOS, and there is an edge between any two vertices with Damerau distance 1.

🔗Mike Battaglia <battaglia01@...>

I wish you'd answered my question about the tuples, because I still
don't know what that notation means.

On Mon, Nov 26, 2012 at 7:31 AM, genewardsmith <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > I guess you want this then:
> > http://en.wikipedia.org/wiki/Damerau%E2%80%93Levenshtein_distance
>
> Except I don't want insertions, deletions, or substitutions. I want the
> Damerau part without the Levenshtein part.

OK, but you're only using the DL distance over a restricted domain
which doesn't allow for insertions, deletions, or substitutions
anyway.

> For now we could try calling that Damerau distance.

OK.

All I have to add is that it seems like what you want is the DL
distance, but restricted to the domain Sym(S)^2, the set of ordered
pairs of elements from the symmetric group on the set S (where S is a
scale). DL distance also happens to be defined over greater domains,
like Sym(S) X Sym(T) or something, which are cases you don't care
about. I think these cases might be useful though, and we've looked
into them previously
https://xenharmonic.wikispaces.com/Warped+diatonic

> You can use "Damerau distance" to expand on the idea of domes and arenas.
> Given a rank n note group N, JI or formal temperament, we can define a
> certain symmetric regular graph; let's assume p-limit JI with p=pi(N) to
> make this easier. We assume an edo-type val v, so that v[1] = v(2) = m is
> the number m of notes of the scale, that we have m-1 rank two temperaments
> which are a Fokker basis for v

If a Fokker basis for v means a basis for the Fokker group of bivals
built around v, then I don't understand how m-1 rank-2 temperaments
serve as a Fokker basis for v.

For instance, this would imply that we have 11 rank-2 temperaments
which are a bival Fokker basis for <12 19 28|, but the Fokker group
built around <12 19 28| has a two-dimensional basis generated by [<<1
4 4||, <<3 0 -7||].

Do you mean N-1 rank 2 temperaments instead of m-1 rank 2 temperaments here?

I have to understand this point to understand the rest of what you
wrote, but this is all starting to remind me to that part III of
Rothenberg's paper that nobody ever talks about, "Part III: The Graph
Embedding Of Pitch Structures." I've only skimmed this in the past but
maybe it's time to dive in and see what he's saying.

There's a metric he's defined on scales called "image distance" as
well; I think it's also defined on a wider set of scales than just
those which are permutations of one another. I'm not sure how it
differs from DL distance or if it's similar at all.

The key thing is he devises a whole graph out of the thing, much like
what you just did. See
http://www.lumma.org/tuning/rothenberg/AModelForPatternPerception.pdf

-Mike

🔗genewardsmith <genewardsmith@...>

🔗Brofessor <kraiggrady@...>

if you look at the Marwa permutation http://anaphoria.com/xenmar9.PDF
page 3 titled fig.1

You will see one perMOS in fig.e B1. and the other in fig. C 1

It must be reminded that MOS is just not about steps, it is about every interval class and so see no reason to restrict the permutations just to the step level.

if you apply it to having thirds for instance in only two step sizes this will include at least 3 other scales that occur in B.
M=major m=minor

B1. M m m M M m m which we can say on the 1 and 2 step level has only two interval sizes but not on a 3 step

2. m M m M M m m

5. m M M m M m m

It seems once one enters the place where the generator chain, in this instance can occur in 3 different sizes then i see no reason no to allow the scales shown here caused by the permutations . These are some of the most Classic scales of India and hence we have a wonderful historical precedent to use them. like the sub-moments of the MOS , it appears they never occur in more than 4 different sizes.

I am in the process of writing something on the Marwa permutations which I hope will be useful. Some note here to move the specific to the general .

If we replace the specific generator of a fourth with a generic one which we will use the letter G and likewise with the disjunction which we will use D fig d A1 will appear as D G G G G G G

With B1 we have to add new symbols D G' D G G G G.and

G'= 2G-D. In other words however much the disjunction deviates from the generator is compensated in G'. We can generate variations by replacing G' D with any G G as long as we don't have two D D at the end of the chain.

All i can tell so far is that the possibilities are predicted by the how many occurrences we have of G" D either solo or in a chain but another rule needs to be worked out

Here are the possibilities of a pentatonic

D G G G G

D G' D G G

D G G' D G
2 forms

with 7 tones we have

EXAMPLE
D G G G G G G F B E A D G C
D G' D G G G G F B Eb A D G C
D G G' D G G G F B E Ab Db G C 3 different intervals
D G G G' D G G F B E A Db G C 3 different intervals
D G G G G' D G F B E A D F# C 3 different intervals
D G' D G' D G G F B Eb A Db G C
D G G' D G' D G F B E Ab D Gb C 3 different intervals

2 forms

It seems from the above that the particular form of having the same scale sizes is found only in the form
of D G' D or expanded chain of them. They also don't work where you have two D at the end that you would if you look at the last underlined example above.

the nine tone scale would add
D G' D G' D G' D G G

I haven't completely tested these yet in enough different cases or with other 7 tone scales such as meta mavila so not sure this holds.

As far as the addition of note to an MOS it is difficult to determine if it a scale or just a limited sample of a larger MOS.

When such things occur in classical music we just accept that they are passing tones just as we don't call melodies that only use 6 of the 7 notes in major or minor a new scale.
Boomliter and Creel also provided us with the notion of extended reference where a tone is taken temporarily as a pivot point. One can apply this even within the tempered 12. We also find the same type of extended references in much mid-east music. In both these cultures, there has been no need to define as a new scale.

🔗Mike Battaglia <battaglia01@...>

🔗gedankenwelt94 <gedankenwelt94@...>

Hi,

first of all, the link doesn't seem to work, you probably meant http://anaphoria.com/xen9mar.PDF ;)

--- In tuning@yahoogroups.com, "Brofessor" <kraiggrady@...> wrote:
> It must be reminded that MOS is just not about steps, it is about every interval class and so see no reason to restrict the permutations just to the step level.

I think that's a very good point! Let's take a look at the structure of the 7-note MOS's:

2L5s (sLsssLs), 4T3t (TtTtTtT), 6F1f (FFFfFFF) [antidiatonic scale]
5L2s (LsLLLsL), 3T4t (tTtTtTt), 1F6f (fffFfff) [diatonic scale]
3L4s (sLsLsLs), 6T1t (TTTtTTT), 2F5f (fFfffFf)
4L3s (LsLsLsL), 1T6t (tttTttt), 5F2f (FfFFFfF)
6L1s (LLLsLLL), 5T2t (TtTTTtT), 4F3f (FfFfFfF)
1L6s (sssLsss), 2T5t (tTtttTt), 3F4f (fFfFfFf)

Here, T and t denote the greater and lesser 2-step interval ("major and minor third"), and F and f the greater and lesser 3-step interval ("augmented and perfect fourth").

As we can see, there are basically only three different patterns: 1M6m (mmmMmmm), 2M5m (mMmmmMm) and 3M4m (mMmMmMm), plus inversions (MMMmMMM instead of mmmMmmm, and so on...), and each MOS has all of these patterns if we look at different interval classes.
(M and m denote the greater and lesser interval of a generic interval class here, resp.)

So if LsLLLLs is a PERMOS (on the step level), why don't we generalize the concept and call TtTTTTt or FfFFFFf a PERMOS (on the 2- or 3- step interval level, resp.), as well?
This would be especially useful for MOS's like sssLsss, because otherwise the concept of PERMOS couldn't be applied there at all (any adjacent transposition leads to a transposed version of the original MOS).

In the case of 8- or 9-tone MOS's with period = interval of equivalence, the patterns repeat in a similar way if we ignore step numbers that are not coprime to the number of notes:

8 notes:

5L3s (LLsLLsLs), 7F1f (FFFFFFFf)
3L5s (ssLssLsL), 1F7f (fffffffF)
7L1s (LLLLLLLs), 5F3f (FFfFFfFf)
1L7s (sssssssL), 3F5f (ffFffFfF)

-> patterns: 1M6m (mmmmmmmM), 3M5m (mmMmmMmM)

9 notes:
(Q and q = 4-step intervals)

2L7s (ssLsssLss), 4T5t (tTtTtTtTt), 8Q1q (QQQQqQQQQ)
7L2s (LLsLLLsLL), 5T4t (TtTtTtTtT), 1Q8q (qqqqQqqqq)
4L5s (sLsLsLsLs), 8T1t (TTTTtTTTT), 7Q2q (QQqQQQqQQ)
5L4s (LsLsLsLsL), 1T8t (ttttTtttt), 2Q7q (qqQqqqQqq)
8L1s (LLLLsLLLL), 7T2t (TTtTTTtTT), 5Q4q (QqQqQqQqQ)
1L8s (ssssLssss), 2T7t (ttTtttTtt), 4Q5q (qQqQqQqQq)

-> patterns: 1M8m (mmmmMmmmm), 2M7m (mmMmmmMmm), 4M5m (mMmMmMmMm)

...I assume this applies to MOS's with any number of notes, but that's pure speculation. Of course the above constraints aren't necessary when defining PERMOS, I just wanted to point out those symmetric relations.

🔗gedankenwelt94 <gedankenwelt94@...>

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
> One interesting aspect is that we can generalize Fokker blocks in this way. For instance, the centaur scale is fully characterized by this:
>
> august: sLLLLsLLLsLL; dominant: sLsLssLsLsLs; meantone: sLsLLsLsLsLL
>
> That's a PERMOS and two MOS, and you need to define the temperaments they go with, as above.

With 'centaur scale', you're refering to this scale, correct?

http://www.anaphoria.com/centaur.html

So its form is mXsXLmXsXmXL, with s = 28/27, m = 21/20, L = 16/15 and X = 15/14.

Can you explain in which way the scale is characterized by the PERMOS and the two MOS? I had a guess what you might mean, but it didn't work out (it almost did, though).

🔗genewardsmith <genewardsmith@...>

--- In tuning@yahoogroups.com, "gedankenwelt94" <gedankenwelt94@...> wrote:

> With 'centaur scale', you're refering to this scale, correct?
>
> http://www.anaphoria.com/centaur.html
>
> So its form is mXsXLmXsXmXL, with s = 28/27, m = 21/20, L = 16/15 and X = 15/14.
>
> Can you explain in which way the scale is characterized by the PERMOS and the two MOS? I had a guess what you might mean, but it didn't work out (it almost did, though).

If you temper centaur in an optimized tuning of each of the above three temperaments, you get the pattern of steps shown. You can reverse the process by brute force, by requiring each step have the right small-large type, though algebra will also do the job for you. Applying august, dominant, meantone to 28/27 gives L,s,s; to 21/20 gives s,s,s; to 16/15 gives L,s,L; to 15/14 gives L, L, L.

🔗gedankenwelt94 <gedankenwelt94@...>

> If you temper centaur in an optimized tuning of each of the above three temperaments, you get the pattern of steps shown. You can reverse the process by brute force, by requiring each step have the right small-large type, though algebra will also do the job for you. Applying august, dominant, meantone to 28/27 gives L,s,s; to 21/20 gives s,s,s; to 16/15 gives L,s,L; to 15/14 gives L, L, L.

This way it works, but is it really alright to associate 28/27 (the smallest step) with L,s,s, and 21/20 (which is larger) with s,s,s? Wouldn't that mean the scale that you really characterized is a variant of the centaur scale, which has the two smallest steps swapped? Shouldn't the centaur scale being characterized by LLsLLLLsLLLL (PERMOS of LLsLLLLLsLLL), sLsLssLsLsLs (dominant) and sLsLLsLsLsLL (meantone), instead?

Sorry if what I'm writing here doesn't make much sense!

🔗genewardsmith <genewardsmith@...>

--- In tuning@yahoogroups.com, "gedankenwelt94" <gedankenwelt94@...> wrote:
>
> > If you temper centaur in an optimized tuning of each of the above three temperaments, you get the pattern of steps shown. You can reverse the process by brute force, by requiring each step have the right small-large type, though algebra will also do the job for you. Applying august, dominant, meantone to 28/27 gives L,s,s; to 21/20 gives s,s,s; to 16/15 gives L,s,L; to 15/14 gives L, L, L.
>
> This way it works, but is it really alright to associate 28/27 (the smallest step) with L,s,s, and 21/20 (which is larger) with s,s,s?

The difference between 28/27 and 21/20 is 81/80. In dominant and meantone that gets tempered to 1, but in august it's even worse--it's smaller than 1. It's the same size as 64/63, but going the wrong way; it's 63/64.

🔗kraiggrady <kraiggrady@...>

PERMOS & different interval classes
</tuning/topicId_105380.html#105462;_ylc=X3oDMTJxbXR1ZWcyBF9TAzk3MzU5NzE1BGdycElkAzcwNjA1BGdycHNwSWQDMTcwNTg5Nzc1MwRtc2dJZAMxMDU0NjIEc2VjA2Rtc2cEc2xrA3Ztc2cEc3RpbWUDMTM1NDQzNjcyMA-->

Sat Dec 1, 2012 8:58 am (PST) . Posted by:

"gedankenwelt94" gedankenwelt94
<mailto:gedankenwelt94@...?subject=Re%3A%20PERMOS%20%26%20different%20interval%20classes>

Hi,

first of all, the link doesn't seem to work, you probably meant http://anaphoria.com/xen9mar.PDF <http://anaphoria.com/xen9mar.PDF> ;)
thanks for that, i thought that is what i posted

-
...> wrote:
> It must be reminded that MOS is just not about steps, it is about every interval class and so see no reason to restrict the permutations just to the step level.

I think that's a very good point! Let's take a look at the structure of the 7-note MOS's:

Here, T and t denote the greater and lesser 2-step interval ("major and minor third"), and F and f the greater and lesser 3-step interval ("augmented and perfect fourth").

~~~~Thank you for the examples. I think thought this notation might complicated the issue though I don't see any reason not to call them all s and L regardless of the level. The point you make is shown on page 16 of http://anaphoria.com/mos.PDF which has a recent update, but this page was included a bit back and may not be in the version you might have.
If you start adding new symbols for every level, when you get to a 14 tone MOS you will run out of alphabetical notes :) 14x2=28

So if LsLLLLs is a PERMOS (on the step level), why don't we generalize the concept and call TtTTTTt or FfFFFFf a PERMOS (on the 2- or 3- step interval level, resp.), as well?
------------------------------------------------
~This is what i was proposing since this is what some of the scales is the Marwa Permutation show and i tried to point out as examples.

One will note that it is possible to have 2 step sizes on even two layers and only one might have 3 sizes. An Interesting Category might be how many layer retain 2 steps or notor if only certain sets were possible. One would have to look at larger sets.

One point i would like to accent about the Marwa Permutations is not only do you have a great variety of scales but you also have them arrange in a sequence of modulations by only changing a single note. Erv came up with this at a time he was collaborating with Amiya Dasgupta on a book on North Indian Ragas used for Amiya's lessons at Cal Arts in California. What is unknown if there is a something similar in North Indian Theory or not. If not it surely adds to it as it explains many of it scales.
-----------------------------------

This would be especially useful for MOS's like sssLsss, because otherwise the concept of PERMOS couldn't be applied there at all (any adjacent transposition leads to a transposed version of the original MOS).

In the case of 8- or 9-tone MOS's with period = interval of equivalence, the patterns repeat in a similar way if we ignore step numbers that are not coprime to the number of notes:

8 notes:

🔗kraiggrady <kraiggrady@...>

The simplest way to see Centaur converted to s and L is the 2120 and 28/27 as small and the 15/14 and 16/15 as L
this way it relates directly to an MOS. if i start on A it is easier to see.
s L L s L s L L s L s L

It is not uncommon for constant structures in general to mimic MOS patterns . One odd piece is that Erv found a 10 cycle in it where 10/9 is the largest step. I assume he discovered this via viggo brun's algorithm but not positive. The 7 tone subset out of 10 gives some interesting subset which should work well in tempered versions.
--
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🔗gedankenwelt94 <gedankenwelt94@...>

--- In tuning@yahoogroups.com, kraiggrady <kraiggrady@...> wrote:
> ~~~~Thank you for the examples. I think thought this notation might
> complicated the issue though I don't see any reason not to call them all
> s and L regardless of the level. The point you make is shown on page 16
> of http://anaphoria.com/mos.PDF which has a recent update, but this page
> was included a bit back and may not be in the version you might have.
> If you start adding new symbols for every level, when you get to a 14
> tone MOS you will run out of alphabetical notes :) 14x2=28

Yes, it's good to have a general notation independent from the level; that's why I used 'M' and 'm' to denote general patterns. I thought using 's' and 'L' might be confusing, because then it's not clear if we're talking about general patterns, or the single step level in particular. On the other hand, 'm' might be ambigous because it is also used when having three different step sizes (s,m,L), but I guess that's not a problem as long as there are 'M's as well.

And thanks for the hint about the pdf, seems I didn't have the updated version! :)

Running out of alphabetical notes doesn't happen that fast when using upper/lower case, and if there are n notes the structure on the k-step level is the same as on the (n-k)-step level, so we can ignore one of them. The problem is rather running out of meaningful letters fast, and I admit I'm not really comfortable with calling a 2-step interval a "third". ^^

> One will note that it is possible to have 2 step sizes on even two
> layers and only one might have 3 sizes.

Unless I'm wrong, it's even possible to have 2 step sizes on all layers except for the layer containing the generator, even for arbitrary MOS with period = interval of equivalence. Just imagine a generator chain with n notes on positions 1 to n+2 in the chain, except for positions 2 and n+1 where there is no note. Something like the melodic minor scale ('X' = empty space in the chain):

F X G D A E B X C#

The intervals that occur are the same as in the continuous generator chain F C G D A E B, except for a third interval pair on the generator level (the augmented fifth F - C# / diminished fourth C# - F).

> [...]An Interesting Category might be
> how many layer retain 2 steps or notor if only certain sets were
> possible. One would have to look at larger sets.

Yes, I thought about something similar, too. What is clear is that it's never possible to have a PERMOS with 2 intervals on the generator level, since the generator level of an MOS always has the form 1M(n-1)m, or (n-1)M1m. I'd also assume that interval levels of the form xMym are much more "fruitful" / uncomplicated if x and y are closer together (e.g. 8M9m, as opposed to 2M15m).

🔗cityoftheasleep <igliashon@...>

🔗genewardsmith <genewardsmith@...>

🔗Brofessor <kraiggrady@...>

--- In tuning@yahoogroups.com, "gedankenwelt94" <gedankenwelt94@...> wrote:

>
> > [...]An Interesting Category might be
> > how many layer retain 2 steps or notor if only certain sets were
> > possible. One would have to look at larger sets.
>
> Yes, I thought about something similar, too. What is clear is that it's never possible to have a PERMOS with 2 intervals on the generator level, since the generator level of an MOS always has the form 1M(n-1)m, or (n-1)M1m. I'd also assume that interval levels of the form xMym are much more "fruitful" / uncomplicated if x and y are closer together (e.g. 8M9m, as opposed to 2M15m).
>
This is also true of the Marwa Permutations, One can do them on any level except the level of the generator unless it has been alerted as in my earlier example but i had hope to have shown that changing one of the other levels can result in getting a layer with 3 sizes to play with on the generator one. That one can branch off of any of the leyer in general is an exciting compositional point of departure one can return to or work toward as opposed to away from. If samples of all the possible directions are represented, it seems that withholding the parent scale till the end might potentially be quite dramatic
I don't follow the temperament application because fundamentally i am working towards an opposite goal. To have as many different sizes intervals and commas as possible. Still one likes to have something that holds together as a self propelling structure which MOS and ETs can do structurally. How far one can go will always be a fuzzy border, but near the border seems one of the more interesting places.

🔗Brofessor <kraiggrady@...>

🔗gedankenwelt94 <gedankenwelt94@...>

Ok, I think I figured it out by now...

--- In tuning@yahoogroups.com, "gedankenwelt94" <gedankenwelt94@...> wrote:
> This way it works, but is it really alright to associate 28/27 (the smallest step) with L,s,s, and 21/20 (which is larger) with s,s,s? Wouldn't that mean the scale that you really characterized is a variant of the centaur scale, which has the two smallest steps swapped? Shouldn't the centaur scale being characterized by LLsLLLLsLLLL (PERMOS of LLsLLLLLsLLL), sLsLssLsLsLs (dominant) and sLsLLsLsLsLL (meantone), instead?

So if I want to characterize the centaur scale like I suggested above, it would work with LLsLLLLsLLLL (pajara), since 50/49 and 64/63 (and therefore also 225/224) are tempered out - correct?

P.S.: It is clear to me now that it doesn't have to be done that way, but I think it *might* be nice to map 28/27 to s,s,s, 21/20 to L,s,s, 16/15 to L,s,L and 15/14 to L,L,L.

🔗genewardsmith <genewardsmith@...>

🔗Brofessor <kraiggrady@...>

--- In tuning@yahoogroups.com, "gedankenwelt94" <gedankenwelt94@...>
wrote:
>
> Ok, I think I figured it out by now...

>Hello Gedankenwelt.
An easy way to see the various way to see where centaur can go in with a
tempered or untempered state is via the scale tree. if we look at page
11 of http://anaphoria.com/sctree.PDF <http://anaphoria.com/sctree.PDF>
and if centaur is mimicking a 7/12 MOS scale we know then it leads to
either a 17 or 19. from there we can see
the 19 tone scale can lead to 26 or 31 tones scales and the 17 to a 22
and 29. These too can each be taken in all the division one want with
this entire page being possible from where we started. These cover the
parent Ls small patterns you use in your example with just the 4 i
mentioned.

Another unrelated thought is that since Viggo Bruns Algorithm does not
designate which order the subsequent divisions are placed, one has the
opportunity there to start with a Permos by how one places it and to
construct each layer likewise. this makes the step of finding the MOS
first and then changing it an unnecessary step.