Temperamental subgroups and sub-MOS's

Every temperament defines more then one series of MOS. For example,
meantone defines the MOS series that you get if you use the 3/2 as a
generator, but it also defines the MOS series that you get if you use
9/8 as a generator. This is a subgroup of the original temperament.

This subgroup can be particularly useful in generating useful scales
that have the MOS property while reducing in complexity some of the
useful higher-complexity intervals from the original temperament. I'll
call the MOS that's generated by the doubled generator the "2nd-order
sub-MOS," the one by tripled generator the "3rd-order sub-MOS," etc.
Some examples:

1) 4L3s in 11-equal, which defies categorization (is it a messed up
Hanson? Is it a messed up Keemun? Is it "orgone" temperament?), makes
the most sense being viewed as a 2nd-order sub-MOS of porcupine (think
about it.)
2) "Machine" temperament, which I've been referring to as a 2.7.9.11
subgroup temperament, can be thought of us as a subgroup of superpyth
temperament. So machine's MOS's make the most sense if viewed as
2nd-order sub-MOS's of superpyth.
3) "Deutone" temperament is a subgroup of flattone, and hence its
MOS's are 2nd-order sub-MOS of flattone.
4) Meantone temperament is a subgroup of mohajira, and hence its MOS's
are 2nd-order sub-MOS of mohajira.

There are no doubt others that are worth working out, and I haven't
even started looking at 3rd-order sub-MOS's of anything. I think that
miracle's sub-MOS's will likely be rather useful, owing to the overall
high complexity of the temperament.

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> Every temperament defines more then one series of MOS. For example,
> meantone defines the MOS series that you get if you use the 3/2 as a
> generator, but it also defines the MOS series that you get if you use
> 9/8 as a generator. This is a subgroup of the original temperament.

This is practically identical to the MOS series of the following 2.5.7.9 temperament, right?:
http://x31eq.com/cgi-bin/rt.cgi?ets=6_19&limit=2_5_7_9

> This subgroup can be particularly useful in generating useful scales
> that have the MOS property while reducing in complexity some of the
> useful higher-complexity intervals from the original temperament. I'll
> call the MOS that's generated by the doubled generator the "2nd-order
> sub-MOS," the one by tripled generator the "3rd-order sub-MOS," etc.
> Some examples:
>
> 1) 4L3s in 11-equal, which defies categorization (is it a messed up
> Hanson? Is it a messed up Keemun? Is it "orgone" temperament?), makes
> the most sense being viewed as a 2nd-order sub-MOS of porcupine (think
> about it.)

Same here. The way I see it you're not really talking about "porcupine", but about this 2.(5/3).9 temperament:
http://x31eq.com/cgi-bin/rt.cgi?ets=4_11&limit=2_5%2F3_9

> 2) "Machine" temperament, which I've been referring to as a 2.7.9.11
> subgroup temperament, can be thought of us as a subgroup of superpyth
> temperament. So machine's MOS's make the most sense if viewed as
> 2nd-order sub-MOS's of superpyth.

A 2.(5/3).7.9.11 temperament:
http://x31eq.com/cgi-bin/rt.cgi?ets=11+17p&limit=2.5%2F3.7.9.11

Keenan

On Fri, Aug 5, 2011 at 7:29 PM, Keenan Pepper <keenanpepper@gmail.com> wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> >
> > Every temperament defines more then one series of MOS. For example,
> > meantone defines the MOS series that you get if you use the 3/2 as a
> > generator, but it also defines the MOS series that you get if you use
> > 9/8 as a generator. This is a subgroup of the original temperament.
>
> This is practically identical to the MOS series of the following 2.5.7.9 temperament, right?:
> http://x31eq.com/cgi-bin/rt.cgi?ets=6_19&limit=2_5_7_9

Right.

> > 1) 4L3s in 11-equal, which defies categorization (is it a messed up
> > Hanson? Is it a messed up Keemun? Is it "orgone" temperament?), makes
> > the most sense being viewed as a 2nd-order sub-MOS of porcupine (think
> > about it.)
>
> Same here. The way I see it you're not really talking about "porcupine", but about this 2.(5/3).9 temperament:
> http://x31eq.com/cgi-bin/rt.cgi?ets=4_11&limit=2_5%2F3_9

Who's to say it's not porcupine? You could use that scale even in
porcupine temperament, and then modulate by 9/8 or 10/9 if you want.

> > 2) "Machine" temperament, which I've been referring to as a 2.7.9.11
> > subgroup temperament, can be thought of us as a subgroup of superpyth
> > temperament. So machine's MOS's make the most sense if viewed as
> > 2nd-order sub-MOS's of superpyth.
>
> A 2.(5/3).7.9.11 temperament:
> http://x31eq.com/cgi-bin/rt.cgi?ets=11+17p&limit=2.5%2F3.7.9.11

Right, well what you're doing is to figure out what JI subgroup
corresponds to the temperamental subgroup. Nothing wrong with that,
and I'm sure we could formalize the process. And likewise, I guess
magic's second-order sub-MOS would be in the 2.9.25 subgroup. To me,
it's more conceptually useful to simply define these as a new class of
scales that one can use while playing in some temperament, rather than
to change the limit that we're working within. There is, of course,
nothing stopping you from going outside of the scale if you want, and
doing some mind-boggling Petr Parizek-style comma pumps around the
lattice that incorporates the generators you're skipping over.

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> > > 1) 4L3s in 11-equal, which defies categorization (is it a messed up
> > > Hanson? Is it a messed up Keemun? Is it "orgone" temperament?), makes
> > > the most sense being viewed as a 2nd-order sub-MOS of porcupine (think
> > > about it.)
> >
> > Same here. The way I see it you're not really talking about "porcupine", but about this 2.(5/3).9 temperament:
> > http://x31eq.com/cgi-bin/rt.cgi?ets=4_11&limit=2_5%2F3_9
>
> Who's to say it's not porcupine? You could use that scale even in
> porcupine temperament, and then modulate by 9/8 or 10/9 if you want.

You could certainly use it in porcupine. Modulating by 10/9 makes perfect sense because then you moved just 1 generator and now you're in the other of the two equivalent subscales.

But modulating by 9/8 doesn't make much sense to me because that's 6 porcupine generators, so you stay in the same subscale. Modulating by 4/3 would make more sense.

> > > 2) "Machine" temperament, which I've been referring to as a 2.7.9.11
> > > subgroup temperament, can be thought of us as a subgroup of superpyth
> > > temperament. So machine's MOS's make the most sense if viewed as
> > > 2nd-order sub-MOS's of superpyth.
> >
> > A 2.(5/3).7.9.11 temperament:
> > http://x31eq.com/cgi-bin/rt.cgi?ets=11+17p&limit=2.5%2F3.7.9.11
>
> Right, well what you're doing is to figure out what JI subgroup
> corresponds to the temperamental subgroup. Nothing wrong with that,
> and I'm sure we could formalize the process. And likewise, I guess
> magic's second-order sub-MOS would be in the 2.9.25 subgroup. To me,
> it's more conceptually useful to simply define these as a new class of
> scales that one can use while playing in some temperament, rather than
> to change the limit that we're working within. There is, of course,
> nothing stopping you from going outside of the scale if you want, and
> doing some mind-boggling Petr Parizek-style comma pumps around the
> lattice that incorporates the generators you're skipping over.

This makes sense, and I think it's a useful concept.

I propose using the word "index" rather than "order" though, because that jives with the use in abstract algebra. So we can say machine is an index-2 subtemperament of superpyth, for example.

BTW, your guess about magic happens to be incorrect. The harmonic 3 is represented by an odd number of generators, so it gets thrown out and replaced with 9, and harmonic 5 is also an odd number of generators so it gets replaced by 25. Their *ratio* 5/3, however, is an even number of generators, so it is represented in the index-2 subtemperament. Therefore the subtemperament is this 2.(5/3).9-limit guy:
http://x31eq.com/cgi-bin/rt.cgi?ets=11%2C19&limit=2.5%2F3.9

Since meantone is basically the only temperament that will be instantly recognized and appreciated by non-xenharmonic types, note that a special position is occupied by its small-index supertemperaments, such as semaphore (or "godzilla"??) and mohajira. If you're playing in a large enough mohajira scale, for example, you can stick to its meantone subset and play nice familiar diatonic melodies for a while, but then suddenly modulate by a neutral third and freak everyone out.

Finally I note that, in rank 2, and assuming you don't alter the period, the number of distinct index-n subtemperaments of a given temperament is simply equal to n. So in meantone you have semaphore and mohajira for index 2, and for index 3 you have: a weird temperament that resembles porcupine but is not porcupine; mothra; and liese.

Keenan

On Wed, Aug 10, 2011 at 11:27 AM, Keenan Pepper <keenanpepper@gmail.com> wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> >
> > Who's to say it's not porcupine? You could use that scale even in
> > porcupine temperament, and then modulate by 9/8 or 10/9 if you want.
>
> You could certainly use it in porcupine. Modulating by 10/9 makes perfect sense because then you moved just 1 generator and now you're in the other of the two equivalent subscales.
>
> But modulating by 9/8 doesn't make much sense to me because that's 6 porcupine generators, so you stay in the same subscale. Modulating by 4/3 would make more sense.

I meant to write 4/3 there, my bad.

> > Right, well what you're doing is to figure out what JI subgroup
> > corresponds to the temperamental subgroup. Nothing wrong with that,
> > and I'm sure we could formalize the process. And likewise, I guess
> > magic's second-order sub-MOS would be in the 2.9.25 subgroup. To me,
> > it's more conceptually useful to simply define these as a new class of
> > scales that one can use while playing in some temperament, rather than
> > to change the limit that we're working within. There is, of course,
> > nothing stopping you from going outside of the scale if you want, and
> > doing some mind-boggling Petr Parizek-style comma pumps around the
> > lattice that incorporates the generators you're skipping over.
>
> This makes sense, and I think it's a useful concept.
>
> I propose using the word "index" rather than "order" though, because that jives with the use in abstract algebra. So we can say machine is an index-2 subtemperament of superpyth, for example.

OK, and are we going to say that machine[11] is an index-2 subMOS of superpyth?

> BTW, your guess about magic happens to be incorrect. The harmonic 3 is represented by an odd number of generators, so it gets thrown out and replaced with 9, and harmonic 5 is also an odd number of generators so it gets replaced by 25. Their *ratio* 5/3, however, is an even number of generators, so it is represented in the index-2 subtemperament. Therefore the subtemperament is this 2.(5/3).9-limit guy:
> http://x31eq.com/cgi-bin/rt.cgi?ets=11%2C19&limit=2.5%2F3.9

OK, sure. Yes, 6/5 is an even number of generators, so we can throw
that in there too. I wasn't trying to get into rational subgroups in
my basic examples, because I don't really even want to deal with
subgroups at all. My whole point with this is to start the process of
getting away from having a near-infinite amount of temperaments in
crazy subgroups like 2.3.9/7.13/11.9' and to start looking at those as
higher-index subtemperaments of what we already know. This may mean
giving up my divinely inspired "machine" name in favor of just calling
it index-2 superpyth. but it's probably worth it. At some point I need
to go back and finish up my stuff on "spectral complexity" too, which
is related.

> Since meantone is basically the only temperament that will be instantly recognized and appreciated by non-xenharmonic types, note that a special position is occupied by its small-index supertemperaments, such as semaphore (or "godzilla"??) and mohajira. If you're playing in a large enough mohajira scale, for example, you can stick to its meantone subset and play nice familiar diatonic melodies for a while, but then suddenly modulate by a neutral third and freak everyone out.

Right. The same applies to gamelan and miracle, or mohajira and
miracle, or meantone and cynder/mothra. This might make a temperament
like miracle more usable.

> Finally I note that, in rank 2, and assuming you don't alter the period, the number of distinct index-n subtemperaments of a given temperament is simply equal to n. So in meantone you have semaphore and mohajira for index 2, and for index 3 you have: a weird temperament that resembles porcupine but is not porcupine; mothra; and liese.

It seems like you're using the terminology "index-n subtemperament"
differently than you are above. Is meantone an index-2 subtemperament
of semaphore, or is semaphore an index-2 subtemperament of meantone?

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> OK, and are we going to say that machine[11] is an index-2 subMOS of superpyth?

Makes perfect sense to me.

> OK, sure. Yes, 6/5 is an even number of generators, so we can throw
> that in there too. I wasn't trying to get into rational subgroups in
> my basic examples, because I don't really even want to deal with
> subgroups at all. My whole point with this is to start the process of
> getting away from having a near-infinite amount of temperaments in
> crazy subgroups like 2.3.9/7.13/11.9' and to start looking at those as
> higher-index subtemperaments of what we already know. This may mean
> giving up my divinely inspired "machine" name in favor of just calling
> it index-2 superpyth. but it's probably worth it. At some point I need
> to go back and finish up my stuff on "spectral complexity" too, which
> is related.

I'm generally in favor of this.

> > Finally I note that, in rank 2, and assuming you don't alter the period, the number of distinct index-n subtemperaments of a given temperament is simply equal to n. So in meantone you have semaphore and mohajira for index 2, and for index 3 you have: a weird temperament that resembles porcupine but is not porcupine; mothra; and liese.
>
> It seems like you're using the terminology "index-n subtemperament"
> differently than you are above. Is meantone an index-2 subtemperament
> of semaphore, or is semaphore an index-2 subtemperament of meantone?

Achh, sorry, it should say "supertemperaments" above. The number of index-n *supertemperaments* of a given temperament is n. (The number of subtemperaments of index n is, of course, just 1, because it's uniquely defined.)

So, meantone is the index-2 subtemperament of semaphore, because if you take every other note of the chain of generators in semaphore, you get meantone. Thus semaphore is an index-2 supertemperament of meantone. The only other index-2 supertemperament of meantone is mohajira (or maybe "mohaha" if you don't bother to map 7).

There are three index-3 supertemperaments: the first is something like this guy (not porcupine):
http://x31eq.com/cgi-bin/rt.cgi?ets=7_43&limit=2_3_5_11
the second is cynder/mothra, and the third is liese.

Keenan

On Wed, Aug 10, 2011 at 6:21 PM, Keenan Pepper <keenanpepper@gmail.com> wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> > OK, and are we going to say that machine[11] is an index-2 subMOS of superpyth?
>
> Makes perfect sense to me.

OK, so coming back to this, let's tighten up the terminology a bit and
say that machine's MOS's are 2nd index sub-MOS's of superpyth. We can
keep calling it "machine" informally, but perhaps something like
2-superpyth[11] would be better. Should be decently extensible, yes?

Again, my goal wasn't to think in terms of these 2nd-index
temperamental subgroups as being new "subtemperaments" per se, nor
different JI subgroup versions of the original temperament, but just
to define a new class of scales that each temperament produces.
2-porcupine produces something that looks like but is not the MOS
series of Hanson, for example, and 2-meantone produces a series of
MOS's of which one is the linearized version of the whole tone scale.

The obvious question, then, is whether or not 2-semaphore and
2-mohajira in some sense "are" meantone. This is more or less a
question of semantics, equally as semantically useless as arguing over
whether or not the perceptual octave or its obvious mathematical
correlate is the "true octave." Let's say that meantone is a 2nd index
"subtemperament" of semaphore, but within the semaphore universe there
also exists a class of scales called 2-semaphore[7], and that these
scales happen also to be called meantone[7] if you're in the 2.3.5
subgroup. The fact that the 2nd-index subtemperament of semaphore
switches you neatly down from 2.3.5.7 to 2.3.5 is just a nice
coincidence is the exception rather than the rule, and to define a new
subgroup temperament name for every index subtemperament of every
temperament out there wasn't my goal anyway, although you're all
welcome to do it if you want.

-Mike