The seven 19=et seven-tone CS scales

Inspired by reading Balzano, I computed these. The rotation chosen is
random, I fear. If there is interest I may make up Scala files and try
to choose it according to some idea or other.

[4, 6, 9, 11, 14, 17, 19]
[3, 4, 7, 10, 13, 16, 19]
[3, 6, 8, 11, 13, 16, 19]
[2, 5, 8, 10, 14, 15, 19]
[2, 5, 8, 10, 13, 16, 19]
[2, 5, 8, 10, 13, 15, 19]
[3, 6, 9, 12, 15, 17, 19]

I should be able to do a nice article on Balzano, because if you get
past the bias, there are things to say about it. For one thing, he
apparently came up with Constant Structure independently. At least I
think it was independently, but maybe he got it from Chalmers.

Wherever he got it from, he had the idea, which lead to a funny
situation with his 12-et obsession; he applied it to 7-note 12-et
scales, and found there aren't any 7-note CS scales in 12-et. This is
actually a characterizing feature of 12-et; among all diatonic
et-tunings in the sense of images of the Pythagorean scale, *only* 12
tempers out the Pythagorean comma, and hence has ambiguous tritone
intervals, which is *not* a characteristic of diatonic scales
generally. From a 5-limit point of view, thinking of diatonic as a
meantone scale, only 12-et tempers out both 2048/2025 and 81/80. From
Balzano's pro-CS point of view you might think this makes 12-et
uniquely bad, but of course he doesn't draw this conclusion. Instead
he concocts a notion of "CS, except for tritones" and is happy with
the conclusion that in 12-et, only the diatonic scale satisfies this
weak CS definition.

Gene Ward Smith wrote:
> Inspired by reading Balzano, I computed these. The rotation chosen is
> random, I fear. If there is interest I may make up Scala files and try
> to choose it according to some idea or other.
> > [4, 6, 9, 11, 14, 17, 19]
> [3, 4, 7, 10, 13, 16, 19]
> [3, 6, 8, 11, 13, 16, 19]
> [2, 5, 8, 10, 14, 15, 19]
> [2, 5, 8, 10, 13, 16, 19]
> [2, 5, 8, 10, 13, 15, 19]
> [3, 6, 9, 12, 15, 17, 19]

So are they all rotations of the same scale?

> I should be able to do a nice article on Balzano, because if you get
> past the bias, there are things to say about it. For one thing, he
> apparently came up with Constant Structure independently. At least I
> think it was independently, but maybe he got it from Chalmers. He did? Do you have a quote? What's a constant structure?

Graham

On 5/26/06, Graham Breed <gbreed@gmail.com> wrote:
> Gene Ward Smith wrote:
> > Inspired by reading Balzano, I computed these. The rotation chosen is
> > random, I fear. If there is interest I may make up Scala files and try
> > to choose it according to some idea or other.
> >
> > [4, 6, 9, 11, 14, 17, 19]
> > [3, 4, 7, 10, 13, 16, 19]
> > [3, 6, 8, 11, 13, 16, 19]
> > [2, 5, 8, 10, 14, 15, 19]
> > [2, 5, 8, 10, 13, 16, 19]
> > [2, 5, 8, 10, 13, 15, 19]
> > [3, 6, 9, 12, 15, 17, 19]
>
> So are they all rotations of the same scale?

No, they're all different scales. One is the diatonic and two more are
permutations of it (5 3s and 2 2s). One is the "harmonic minor" scale
from music theory class and another is its inversion. One is the
"whole tone" scale with an extra small interval (3,3,3,3,3,3,1). The
remaining one is the rather interesting scale (2,4,1,4,2,3,3) which
could be represented (using meantone notation) as E F G# Ab B C D E.

[...]

Keenan

>What's a constant structure?

Haven't you been party to years of discussion
about this?

http://tonalsoft.com/enc/c/constant-structure.aspx

-C.

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

> > I should be able to do a nice article on Balzano, because if you get
> > past the bias, there are things to say about it. For one thing, he
> > apparently came up with Constant Structure independently. At least I
> > think it was independently, but maybe he got it from Chalmers.
>
> He did? Do you have a quote? What's a constant structure?

"Symbolically, we can say a set satisfies *coherence* if for any pair
of scale elements i and i',

j < k ==> v[i,j] < v[i',k] (5)

where j and k are scalestep-counting indicies, and take on values from
0 to m-1 [the size of the scale] inclusive. Equation (5) says that
larger numbers of scale steps are always associated to with larger
numbers of semitones in a coherent scale."

Here Balzano has defined v[i,j] = s[i+j]-s[i], where the index set to
the scale steps s[i] is taken modulo m, the number of elements of s.
Since "<" is a linear order and cyclic groups are not linearly
ordered, that has a problem. One way around it is to make v[i,j] take
values mod n, where n is the et, and this is what Balzano does. Better
might be to define s[i] as a function from the integers (in the form
of the index set) to the integers, which is increasing and
quasiperiodic with quasiperiod m, it seems to me. That is,
s[i+km]-s[i] = kn, where n is the number of the et. Then define v[i,j]
using this s.

>>> I should be able to do a nice article on Balzano, because if you get
>>> past the bias, there are things to say about it. For one thing, he
>>> apparently came up with Constant Structure independently. At least I
>>> think it was independently, but maybe he got it from Chalmers.
>>
>> He did? Do you have a quote? What's a constant structure?
>
>"Symbolically, we can say a set satisfies *coherence* if for any pair
>of scale elements i and i',
>
>j < k ==> v[i,j] < v[i',k] (5)
>
>where j and k are scalestep-counting indicies, and take on values from
>0 to m-1 [the size of the scale] inclusive. Equation (5) says that
>larger numbers of scale steps are always associated to with larger
>numbers of semitones in a coherent scale."

That's strict propriety, not CS. My recollection is
that SP -> CS but not the other way around.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:

> That's strict propriety, not CS. My recollection is
> that SP -> CS but not the other way around.

I told you you were the one who's studied this stuff. However, this is
giving me an excuse to dig into it, and liven things up around here.

It seems we have epimorphic ==> strictly proper, and strictly proper
==> CS, but now I don't know what CS is since I thought it meant
strictly proper. I think Balzano should have cited Rothenberg for
this, BTW, rather than just sticking him in the bibliography.

>> That's strict propriety, not CS. My recollection is
>> that SP -> CS but not the other way around.
>
>I told you you were the one who's studied this stuff. However, this is
>giving me an excuse to dig into it, and liven things up around here.
>
>It seems we have epimorphic ==> strictly proper, and strictly proper
>==> CS, but now I don't know what CS is since I thought it meant
>strictly proper.

Here's offlist communication to me from John Chalmers, circa '99...

""
The only definition of constant structure (CS) that I know is that
every instance of a particular interval subtends the same number of
smaller scalar intervals. I don't think Erv required the subtended
intervals to be in the same order and I don't recall now if they even
have to be the same intervals.

CSness is much weaker than Propriety as improper scales such as the
idealized Aristoxenian enharmonic are CS, but IP.

50 50 400 200 50 50 400
100 450 600 250 100 450 450
500 650 650 300 500 500 500
700 700 700 700 550 550 900
etc.

Every interval here is in one and only one melodic class, but the
scale is very improper.""

Here's something I wrote the next year...

""
CS and propriety both assume that when we hear music, we attempt to
assign scale degree numbers to melodic pitches. I won't argue about
that here; if you don't think it happens, then propriety and CS
aren't for you. But if you're interested in getting it to happen
or not, then they are for you.

Propriety and CS both measure how easy it is for the listener to
un-ambiguously assign scale degree numbers to pitches as he hears them.
CS assumes that a listener can recognize intervals by their _specific_
size -- a 3:2 is distinct from a 7:5, and so on. So CS asks, once you
recognize the interval, will there be any doubt as to what scale degree
it is? If you can answer "no", your scale is CS. In the diatonic scale
in 12-tet, you can answer no for all the intervals but the tritone -- it
can be a fourth or a fifth; you need other notes to clarify its
position in the scale.

Propriety does the same thing, except it assumes that listeners
recognize intervals by their _relative_ size -- that the listener ranks
the intervals he hears by how big they are. The actual tuning of the
intervals doesn't matter, so long as their ranks are preserved. This
means the theory can be tested to see if people perceive a similarity
between different tunings of a scale with the same "rank-order", and if
they perceive a difference between two tunings of a scale with a
different rank-order. Rothenberg has actually suggested some very cool
experiments to test many aspects of his theory, which are yet to be
performed.
""

-Carl

Gene Ward Smith wrote:

> It seems we have epimorphic ==> strictly proper, and strictly proper
> ==> CS, but now I don't know what CS is since I thought it meant
> strictly proper. I think Balzano should have cited Rothenberg for
> this, BTW, rather than just sticking him in the bibliography.

I thought empimorphic meant a scale from JI with the same mapping as an equal temperament. That's what the Tonalsoft definition looks like, anyway. Strict propriety is a purely melodic property. So how is there a relationship between them?

Graham

>> It seems we have epimorphic ==> strictly proper, and strictly proper
>> ==> CS, but now I don't know what CS is since I thought it meant
>> strictly proper. I think Balzano should have cited Rothenberg for
>> this, BTW, rather than just sticking him in the bibliography.
>
>I thought empimorphic meant a scale from JI with the same mapping as an
>equal temperament. That's what the Tonalsoft definition looks like,
>anyway. Strict propriety is a purely melodic property. So how is there
>a relationship between them?

This has always confused me. I could see epimorphic ==> CS, but
not much else. I always assumed I didn't understand epimorphic.

-Carl

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

> I thought empimorphic meant a scale from JI with the same mapping as an
> equal temperament. That's what the Tonalsoft definition looks like,
> anyway. Strict propriety is a purely melodic property. So how is
there
> a relationship between them?

Epimorphic is in good measure a melodic property, because it maps to a
quaisiperiodic scale, and a quasiperioidic scale has a group structure
of a rank one group: adding n to the index i for s[i] leads to s[i+n].
It also has an order structure: i<j iff s[i]<s[j]. The epimorphic
property says that there is a val v such that v(s[i])=i. Hence,
v(s[i]*s[j]) = v(s[i])+v(s[j]) = i+j.

Gene Ward Smith wrote:
> --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
> > >>I thought empimorphic meant a scale from JI with the same mapping as an >>equal temperament. That's what the Tonalsoft definition looks like, >>anyway. Strict propriety is a purely melodic property. So how is
> there >>a relationship between them?
> > > Epimorphic is in good measure a melodic property, because it maps to a
> quaisiperiodic scale, and a quasiperioidic scale has a group structure
> of a rank one group: adding n to the index i for s[i] leads to s[i+n].
> It also has an order structure: i<j iff s[i]<s[j]. The epimorphic
> property says that there is a val v such that v(s[i])=i. Hence, > v(s[i]*s[j]) = v(s[i])+v(s[j]) = i+j.

What's a quasiperiodic scale? What's s[i]?

http://tonalsoft.com/enc/e/epimorphic.aspx

says

"A scale has the epimorphic property, or is epimorphic, if there is a val h such that if qn is the nth scale degree, then h(qn) = n. The val h is the characterizing val of the scale."

So the diatonic scale has a mapping (val) <7 11 16]. You can write it in 12-equal with values in cents

0 200 400 500 700 900 1100 1200
C D E F G A B C

and it's only proper, not strictly proper. Then you can arbitrarily change an interval

0 399 400 500 700 900 1100 1200
C D E F G A B C

It's now horribly improper, and not CS by the melodic interpretation, but the mapping still holds. What am I missing?

Graham

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

> > Epimorphic is in good measure a melodic property, because it maps to a
> > quaisiperiodic scale, and a quasiperioidic scale has a group structure
> > of a rank one group: adding n to the index i for s[i] leads to s[i+n].
> > It also has an order structure: i<j iff s[i]<s[j]. The epimorphic
> > property says that there is a val v such that v(s[i])=i. Hence,
> > v(s[i]*s[j]) = v(s[i])+v(s[j]) = i+j.
>
> What's a quasiperiodic scale? What's s[i]?

See http://www.xenharmony.org/quasi.htm

> http://tonalsoft.com/enc/e/epimorphic.aspx
>
> says
>
> "A scale has the epimorphic property, or is epimorphic, if there is a
> val h such that if qn is the nth scale degree, then h(qn) = n. The
val h
> is the characterizing val of the scale."

> It's now horribly improper, and not CS by the melodic interpretation,
> but the mapping still holds. What am I missing?

Mostly, your missing that the mapping is supposed to be from a scale
in 5-limit JI in this case.

Gene Ward Smith wrote:
> --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
> > >>>Epimorphic is in good measure a melodic property, because it maps to a
>>>quaisiperiodic scale, and a quasiperioidic scale has a group structure
>>>of a rank one group: adding n to the index i for s[i] leads to s[i+n].
>>>It also has an order structure: i<j iff s[i]<s[j]. The epimorphic
>>>property says that there is a val v such that v(s[i])=i. Hence, >>>v(s[i]*s[j]) = v(s[i])+v(s[j]) = i+j.
>>
>>What's a quasiperiodic scale? What's s[i]?
> > See http://www.xenharmony.org/quasi.htm

I can't reach xenharmony.org and I have said so before.

>>http://tonalsoft.com/enc/e/epimorphic.aspx
>>
>>says
>>
>>"A scale has the epimorphic property, or is epimorphic, if there is a >>val h such that if qn is the nth scale degree, then h(qn) = n. The
> val h >>is the characterizing val of the scale."
> >>It's now horribly improper, and not CS by the melodic interpretation, >>but the mapping still holds. What am I missing?
> > Mostly, your missing that the mapping is supposed to be from a scale
> in 5-limit JI in this case.

Then you're missing that in your definition. But no matter, the diatonic scale is a mapping from

1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1

and it can still be tuned improperly.

Graham

On 5/29/06, Graham Breed <gbreed@gmail.com> wrote:
> I can't reach xenharmony.org and I have said so before.

Yeah, Gene, you should take better care of your domain name.

In the meantime just replace "xenharmony.org" with "66.98.148.43/~xenharmo".

At 09:31 PM 5/29/2006, you wrote:
>On 5/29/06, Graham Breed <gbreed@gmail.com> wrote:
>> I can't reach xenharmony.org and I have said so before.
>
>Yeah, Gene, you should take better care of your domain name.
>
>In the meantime just replace "xenharmony.org"

Oh jeez, not again. I just updated all my links.

>with "66.98.148.43/~xenharmo".

Maybe Graham meant it's behind the Great Wall?

-Carl

--- In tuning-math@yahoogroups.com, "Keenan Pepper" <keenanpepper@...>
wrote:
>
> On 5/29/06, Graham Breed <gbreed@...> wrote:
> > I can't reach xenharmony.org and I have said so before.
>
> Yeah, Gene, you should take better care of your domain name.

It works fine for me. However, I have a weird web hosting and domain
serving bunch. On the one hand, they charge basically nothing. On the
other hand, they won't even answer email.

> In the meantime just replace "xenharmony.org" with
"66.98.148.43/~xenharmo".

I'd like to hear if that works out. Carl was harassing me for giving
links like that before for some reason, so I haven't been doing it.

Carl Lumma wrote:

>>with "66.98.148.43/~xenharmo".
> > Maybe Graham meant it's behind the Great Wall?

I thought it might be (or the server was blocking Chinese connections, which can also happen) but the numeric address works fine. I've got a foreign DNS host in my list as well.

Hmm, "host" says www.xenharmony.org has address 216.52.184.240

If anybody wants to give me the numeric address for www.anaphoria.com, I could try that as well.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> If anybody wants to give me the numeric address for
www.anaphoria.com, I
> could try that as well.

Keep this up and soon you'll be able to reach your own web site.

Gene Ward Smith wrote:
> --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
> > >>>Epimorphic is in good measure a melodic property, because it maps to a
>>>quaisiperiodic scale, and a quasiperioidic scale has a group structure
>>>of a rank one group: adding n to the index i for s[i] leads to s[i+n].
>>>It also has an order structure: i<j iff s[i]<s[j]. The epimorphic
>>>property says that there is a val v such that v(s[i])=i. Hence, >>>v(s[i]*s[j]) = v(s[i])+v(s[j]) = i+j.
>>
>>What's a quasiperiodic scale? What's s[i]?
> > > See http://www.xenharmony.org/quasi.htm

Being quasiperiodic doesn't affect much. But s[i] means a mapping to JI. So this is a JI scale that maps *to itself* according to a given val?

>>http://tonalsoft.com/enc/e/epimorphic.aspx
>>
>>says
>>
>>"A scale has the epimorphic property, or is epimorphic, if there is a >>val h such that if qn is the nth scale degree, then h(qn) = n. The
> > val h > >>is the characterizing val of the scale."
> > >>It's now horribly improper, and not CS by the melodic interpretation, >>but the mapping still holds. What am I missing?
> > Mostly, your missing that the mapping is supposed to be from a scale
> in 5-limit JI in this case.

I talked about this with Carl. He says Scala says the Pythagorean diatonic is epimorphic but improper.

Is there a term, then, for a scale that is mapped to from JI in a consistent way? That is, a given JI interval means a given number of scale steps, and the mapping's a homomorphism, no other restrictions. It would cover equal temperaments, MOS subsets of linear temperaments, well temperaments and periodicity blocks with the right scale order.

Graham

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

> Being quasiperiodic doesn't affect much. But s[i] means a mapping to
> JI. So this is a JI scale that maps *to itself* according to a
given val?

No, it means it maps to the index set. For example, the quasiperiodic
scale defined by {9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2}. If h7 is the
7-val, then h7(s[i])=i; for instance, h7(1)=0, h7(2)=7, h7(3/2)=4,
h7(3)=11, and so forth.

> I talked about this with Carl. He says Scala says the Pythagorean
> diatonic is epimorphic but improper.

Ouch! Yes, past 12-et, the augmented fourth becomes larger than the
diminished fifth, and the scales become improper. But above any
interval, you never get both an augmented fourth and a diminished
fifth, just one or the other, if either. That is enough to make them
epimorphic; the ordering is consistently implied by the mapping. A
fourth, augmented or not, is always below a fifth, diminised or not,
because you never get an augmented fourth and a diminised fifth together.

> Is there a term, then, for a scale that is mapped to from JI in a
> consistent way?

What I need to do is figure out what implies what, but I think
epimorphic does exactly what you ask, and proper is stronger than
needed to get that.

That is, a given JI interval means a given number of
> scale steps, and the mapping's a homomorphism, no other restrictions.

Epimorphic does that. Plus, you can define it for a given regular
tuning, in case you need that.

Gene Ward Smith wrote:
> --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
> > >>Being quasiperiodic doesn't affect much. But s[i] means a mapping to >>JI. So this is a JI scale that maps *to itself* according to a
> > given val?
> > No, it means it maps to the index set. For example, the quasiperiodic
> scale defined by {9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2}. If h7 is the
> 7-val, then h7(s[i])=i; for instance, h7(1)=0, h7(2)=7, h7(3/2)=4,
> h7(3)=11, and so forth.

The what? Each interval is a ratio, which maps to that number of scale steps. So it maps to itself. How is this wrong?

>>Is there a term, then, for a scale that is mapped to from JI in a >>consistent way?
> > What I need to do is figure out what implies what, but I think
> epimorphic does exactly what you ask, and proper is stronger than
> needed to get that.

No, epimorphic doesn't do what I want because the scale has to be in JI. Propriety has nothing to do with this.

> That is, a given JI interval means a given number of > >>scale steps, and the mapping's a homomorphism, no other restrictions. > > > Epimorphic does that. Plus, you can define it for a given regular
> tuning, in case you need that.

Yes, but what about an irregular temperament? Or an adaptive tuning? That's something else I need to think about.

Graham