Yahoo Tuning Groups Ultimate Backup tuning open problem

Carl wrote,

>I'm posting what may be the most important open problem in tuning I know
of:

>"How can one choose unison vectors in linear space so he gets only the
>MOS's, and all the MOS's of the generator of the linear space and a given
>interval of equivalence?"

>I've already shown to my satisfaction that 1D periodicity blocks of the
>type Paul Erlich is looking for do not fit the bill.

The type I was looking for would use a unison vector smallest (in cents) for
its level of complexity (measured in distance on the chain of generators).
Would this solve your problem: Use a unison vector smallest for its level of
complexity _and sign_? In other words, a 17-tone MOS exists for a chain of
3:2s despite the fact that its unison vector is -67 cents while that for the
12-tone MOS is 23 cents, because there is no smaller unison vector than the
17-tone one that is negative when moving in a positive direction along the
chain of fifths.

🔗Carl Lumma <clumma@xxx.xxxx>

>>I've already shown to my satisfaction that 1D periodicity blocks of the
>>type Paul Erlich is looking for do not fit the bill.
>
>The type I was looking for would use a unison vector smallest (in cents) for
>its level of complexity (measured in distance on the chain of generators).

Really? I thought it was Kees who included complexity, that you were
simply looking for blocks with no scale steps smaller than their vectors.

Wait- measured in distance on the generator chain? What kind of complexity
is that? Has to do with how much error each generator will bear in ET?

>Would this solve your problem: Use a unison vector smallest for its level of
>complexity _and sign_? In other words, a 17-tone MOS exists for a chain of
>3:2s despite the fact that its unison vector is -67 cents while that for the
>12-tone MOS is 23 cents, because there is no smaller unison vector than the
>17-tone one that is negative when moving in a positive direction along the
>chain of fifths.

This sounds right; like what happens with the zig-zag pattern. But why do
you say "complexity" and sign when all ya need is size and sign? Positive
direction? I would simply say: "You get MOS's by finding successive
reducitions in the size of the unison vector, per sign."

I was hoping for something a little more self-contained, something you
could tell just from the given scale, without appealing to the series...

-C.

Here's 24/41...

MOS pos neg NOT
------------------
2 205
3 293
410 4
5 88
585 6
7 117
380 8
322 9
176 10
527 11
12 29
468 13
234 14
263 15
439 16
17 59
556 18
146 19
351 20
-------------------
351 21
146 22

...etc.

🔗Paul H. Erlich <PErlich@xxxxxxxxxxxxx.xxxx>

>>>I've already shown to my satisfaction that 1D periodicity blocks of the
>>>type Paul Erlich is looking for do not fit the bill.
>
>>The type I was looking for would use a unison vector smallest (in cents)
for
>>its level of complexity (measured in distance on the chain of generators).

>Really? I thought it was Kees who included complexity, that you were
>simply looking for blocks with no scale steps smaller than their vectors.

The two are equivalent in the 1-D case (think about it), which Kees didn't
deal with.

>Wait- measured in distance on the generator chain? What kind of complexity
>is that?

Complexity as measured by distance in the lattice (sorry to be circular, but
what else can I say?)

>Has to do with how much error each generator will bear in ET?

That would be the size of the unison vector (in cents) divided by its
complexity (in steps along the chain of generators).

>This sounds right; like what happens with the zig-zag pattern. But why do
>you say "complexity" and sign when all ya need is size and sign? Positive
>direction? I would simply say: "You get MOS's by finding successive
>reducitions in the size of the unison vector, per sign."

Yes, but by "successive" you're tacitly implying a process of moving one end
of the unison vector farther and farther along the chain of generators while
keeping the other end fixed. If one were simply looking at a big list of
unison vectors, one couldn't apply your instructions to get the correct
result.

>I was hoping for something a little more self-contained, something you
>could tell just from the given scale, without appealing to the series...

Are you allowed to appeal to continued-fraction representations?

🔗Carl Lumma <clumma@xxx.xxxx>

>The two are equivalent in the 1-D case (think about it), which Kees didn't
>deal with.

Right.

>That would be the size of the unison vector (in cents) divided by its
>complexity (in steps along the chain of generators).

Right.

>Yes, but by "successive" you're tacitly implying a process of moving one end
>of the unison vector farther and farther along the chain of generators while
>keeping the other end fixed. If one were simply looking at a big list of
>unison vectors, one couldn't apply your instructions to get the correct
>result.

Right!

>>I was hoping for something a little more self-contained, something you
>>could tell just from the given scale, without appealing to the series...
>
>Are you allowed to appeal to continued-fraction representations?

No. I'd like, first off, to fix that chart I posted a ways back. Does
anybody know how to transpose scales in Scala? I tried move, permute,
key... all with confusing results...

-C.

🔗manuel.op.de.coul@xxx.xxx

Carl Lumma wrote:
> Does anybody know how to transpose scales in Scala? I tried move, permute,
> key... all with confusing results...

I got the same complaint from Brian McLaren once. Then I rewrote the
help text for the KEY command. If it's still unclear I'd be happy to
rewrite it again. The answer is, if the scale has an octave (interval
of equivalence) then use KEY, if not then use MOVE/KEY or MOVE. In the
latter case you can then do NORMALIZE to add an octave afterwards.
Unless your definition of transpose is different from mine of course.

On another subject:
With all the correspondence about Fokker periodicity blocks I guess I
should mention that they can be easily created in Scala with the
PIPEDUM command.
Recently I discovered a bug in it that in some cases can cause the scale
to have a few extra tones. The cause is a numerical comparison after the
coordinate transformation of the vectors where I didn't take numerical precision
into account. It's easily seen when this caused a wrong result because then the
scale will contain some intervals with the same size as
one of the defining intervals (= unison vectors, homophonic intervals).
It will be fixed in the next version.

Manuel Op de Coul coul@ezh.nl

🔗Carl Lumma <clumma@xxx.xxxx>

>>Does anybody know how to transpose scales in Scala? I tried move, permute,
>>key... all with confusing results...
>
>I got the same complaint from Brian McLaren once. Then I rewrote the
>help text for the KEY command. If it's still unclear I'd be happy to
>rewrite it again. The answer is, if the scale has an octave (interval
>of equivalence) then use KEY, if not then use MOVE/KEY or MOVE. In the
>latter case you can then do NORMALIZE to add an octave afterwards.
>Unless your definition of transpose is different from mine of course.

The problem, at least for me, stems from the fact that there are two things
you can do which may be called transposing.

<1.> You multiply each note in the scale by some factor and express the new
notes in terms of the 1/1 of the original scale. This is like what happens
when you make a cross-set of a scale and itself. It is like going from
Cmaj to Gmaj on a piano. The scales are the same, but the second one is
written in terms of the first; you say it starts on "G".

<2.> You keep the tonic fixed, and rotate the 2nds of the scale. This is
like what happens in a tonality diamond.

What I'm getting is that KEY is #2, or is it? And what's #1? Multiply?
Then there's SHOW TRANSPOSE. What does "resp." mean?

PIPEDUM is awesome. Have you thought about its inverse -- finding unison
vectors for a given scale?

Thanks,

-Carl

🔗patrick pagano <ppagano@xxxxxxxxx.xxxx>

hi carl it probbably won help much but when i want to transpose into another key
for JI with scala i set freq to say 384 to trans to G

Carl Lumma wrote:

> From: Carl Lumma <clumma@nni.com>
>
> >>Does anybody know how to transpose scales in Scala? I tried move, permute,
> >>key... all with confusing results...
> >
> >I got the same complaint from Brian McLaren once. Then I rewrote the
> >help text for the KEY command. If it's still unclear I'd be happy to
> >rewrite it again. The answer is, if the scale has an octave (interval
> >of equivalence) then use KEY, if not then use MOVE/KEY or MOVE. In the
> >latter case you can then do NORMALIZE to add an octave afterwards.
> >Unless your definition of transpose is different from mine of course.
>
> The problem, at least for me, stems from the fact that there are two things
> you can do which may be called transposing.
>
> <1.> You multiply each note in the scale by some factor and express the new
> notes in terms of the 1/1 of the original scale. This is like what happens
> when you make a cross-set of a scale and itself. It is like going from
> Cmaj to Gmaj on a piano. The scales are the same, but the second one is
> written in terms of the first; you say it starts on "G".
>
> <2.> You keep the tonic fixed, and rotate the 2nds of the scale. This is
> like what happens in a tonality diamond.
>
> What I'm getting is that KEY is #2, or is it? And what's #1? Multiply?
> Then there's SHOW TRANSPOSE. What does "resp." mean?
>
> PIPEDUM is awesome. Have you thought about its inverse -- finding unison
> vectors for a given scale?
>
> Thanks,
>
> -Carl
>
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🔗manuel.op.de.coul@xxx.xxx

> What I'm getting is that KEY is #2, or is it? And what's #1? Multiply?

Correct. Not multiply. If you want to change the frequencies of all notes,
including the 1/1 then it's what Patrick says: SET FREQUENCY. It won't
change the values in cents, since they are relative to the 1/1.
If you want to change the frequencies of all notes _except_ the 1/1, then
use MOVE. And if you want to do that _and_ change the 1/1 to another degree,
use MOVE/KEY. Any possibilities I forgot?

> Then there's SHOW TRANSPOSE. What does "resp." mean?

respectively. I'm not sure if this word is double Dutch or not.

> PIPEDUM is awesome. Have you thought about its inverse -- finding unison
> vectors for a given scale?

Only a little bit after Paul began about it. It's more difficult.
I also wondered whether the Bohlen-Pierce scale is a periodicity
block or not. I think it is. This is the lattice, ignoring factors
of 3:

hor > 5/1 - ver ^ 7/1

*
* * *
* * * * *
* * *
*

Heinz Bohlen's "commas" are these:
1) 245/243 minor BP diesis (1 2)
2) 3125/3087 major BP diesis (5 -3)
3) 15625/15309 great BP diesis (6 -1)
4) 16875/16807 small BP diesis (4 -5)

Trying all six combinations of two out of these 4, five have
determinant 13 and one is 26.

1)+2)

* *
* * * * *
* * * *
* *

Here we see we can get the Bohlen-Pierce scale if the rightmost two
points are shifted an inverse 3125/3087. If I'm not mistaken this is
like shifting a surrounding parallelogram a bit.
The other combinations give:

1)+3)

* * * * * *
* * * * * * *

1)+4)

*
* * *
* * *
* * *
* *
*

2)+3)

*
* * * *
* * * *
* * * *

2)+4)

*
*
* *
* *
* *
* *
* *
*

3)+4)

*
* * * * *
* * * * *
* * * * *
* * * * *
* * * * *

Manuel Op de Coul coul@ezh.nl

🔗Carl Lumma <clumma@xxx.xxxx>

>>What I'm getting is that KEY is #2, or is it? And what's #1? Multiply?
>
>Correct. Not multiply. If you want to change the frequencies of all notes,
>including the 1/1 then it's what Patrick says: SET FREQUENCY. It won't
>change the values in cents, since they are relative to the 1/1.
>If you want to change the frequencies of all notes _except_ the 1/1, then
>use MOVE. And if you want to do that _and_ change the 1/1 to another degree,
>use MOVE/KEY. Any possibilities I forgot?

None that I can think of. Thanks!

>>Then there's SHOW TRANSPOSE. What does "resp." mean?
>
>respectively. I'm not sure if this word is double Dutch or not.

Double Dutch? I've heard of "going double" or "dutch", but never both at
once... :)

Resp. seems to make sense in the help for PIPEDUM, ironically. But what
does "different resp. identical" (from the SHOW TRANSPOSE help) mean?

>> PIPEDUM is awesome. Have you thought about its inverse -- finding unison
>> vectors for a given scale?
>
>Only a little bit after Paul began about it. It's more difficult.

Best of luck. BTW, I just got R's algorithm for efficiency in the mail today!

-Carl

P.S. Scala is sooo amazing!

🔗manuel.op.de.coul@xxx.xxx

> Double Dutch? I've heard of "going double" or "dutch", but never both at
> once... :)

It's the common term for Dutch idiom translated literally into English,
making bad or funny English.

> Resp. seems to make sense in the help for PIPEDUM, ironically. But what
> does "different resp. identical" (from the SHOW TRANSPOSE help) mean?

It just means the number of identical pitches follows the number of
different pitches in the output.

Paul asked if Scala can shift parallelograms. No, I'll see about adding it.

Manuel Op de Coul coul@ezh.nl

🔗Carl Lumma <clumma@xxx.xxxx>

[me, Carl Lumma]
>You bet! Of course any chain-of-fifths tuning can be viewed as a
>periodicity block whose unison vector is the chromatic interval, not just
>the pentatonic scale, and not just the MOS's.

[Paul Erlich]
>Hmm . . . a 6-note chain of fifths has the tritone as unison vector, and I
>would hardly call a tritone a single chromatic step . . .

[me]
>Sheesh! My bad. I was confusing the note it changed _to_ with the _amount_
>it changed. Let's see...

Actually, if you define the chromatic interval as the amount of change in
the _note that changes_ when the scale is transposed, then I was correct --
that amount is always the unison vector. This is how I would like to
define chromatic interval. I had posted a chart showing the chromatic
interval as the distance between the new note and the nearest unchanged
note...

[me]
>However, the nearest one isn't always the one that changes, at least I
don't >think it is. For example, in the 4-tone case, the change actually
jumps a >scale degree (the chromatic step should be 81/64, or 410 cents in
41tET).
>Can anybody fix the chart? Or better yet, explain how to transpose a
scale >in Scala without confusing the hell out of yourself?

So I've just fixed the chart- the chromatic interval is the unison vector.
Manuel and Pat came to the rescue with the Scala question...

[me]
>Resp. seems to make sense in the help for PIPEDUM, ironically. But what
>does "different resp. identical" (from the SHOW TRANSPOSE help) mean?

[Manuel Op de Coul]
>It just means the number of identical pitches follows the number of
>different pitches in the output.

That solved, I've moved on to being confused by the wording "degree order"
vs. "any order". I have convinced myself by playing with it that "degree
order" enforces my preferred definition of chromatic [it does show more
than one note changing when certain connected cyclic scales are transposed
by their generator, which is impossible with the other definition of
chromatic. Indeed, the "any order" column always shows 1 note change when
transposing by the generator. Where do the >1's in the "degree order"
column actually come from -- Manuel?].

Anyway, I've also convinced myself that MOS's are the scales that have a
"1" in the "degree order" column for a given generator and IE --- that
MOS's are scales whose chromatic changes don't jump scale degrees. Can
anybody prove it? Are MOS's the only such scales?

[me]
>I'm not looking for a shortcut for finding MOS's, I'm hoping to find
>higher-D analogs to MOS's by applying the rule to bigger periodicity
>blocks.

[Paul Erlich]
>Aha! How would such analogs be defined?

When they are transposed by any of their unit vectors, no changing notes
hop over unchanging notes. Currently looking to define this in terms of
size of the unison vectors and 2nds of the scale. Paul, your search rules
out many of these (only the 2nds built from either end of the chain are
important?), as I noted...

>I've already shown to my satisfaction that 1D periodicity blocks of the
type >Paul Erlich is looking for do not fit the bill.

What would a search for these 2-D MOS analogs be like?

-Carl

🔗Paul H. Erlich <PErlich@xxxxxxxxxxxxx.xxxx>

Carl Lumma wrote,

>Actually, if you define the chromatic interval as the amount of change in
>the _note that changes_ when the scale is transposed, then I was correct --
>that amount is always the unison vector.

Since our context here is 1-D chains of notes, this is so obvious as to be
trivial.

>This is how I would like to
>define chromatic interval.

That really only makes sense in the MOS case, but since that's what you're
interested in, OK!

>Anyway, I've also convinced myself that MOS's are the scales that have a
>"1" in the "degree order" column for a given generator and IE --- that
>MOS's are scales whose chromatic changes don't jump scale degrees. Can
>anybody prove it? Are MOS's the only such scales?

I think so. If the scale is not a single connected chain of generators, then
transposition by a generator will produce more than one pitch change. If it
is, but it's not MOS, the one pitch change will cause a reordering of some
of the unchanging scale degrees. This can be seen in terms of the test I
proposed to check single connected chains for MOS -- if and only if there is
no interval in the chain which, when only one direction in the chain is
allowed for measurement, is closer to zero than, and the same sign as, the
unison vector, then the scale is MOS.

>Paul, your search rules
>out many of these

Yes, but with a very minor modification, it becomes equivalent.

>(only the 2nds built from either end of the chain are
>important?), as I noted...

Eh?

>I've already shown to my satisfaction that 1D periodicity blocks of the
type
>Paul Erlich is looking for do not fit the bill.

Again, only a very minor modification is necessary.

>What would a search for these 2-D MOS analogs be like?

Fascinating. As we have seen, even a given pair of unison vectors implies a
very wide variety of 2-D periodicity blocks, in contradistinction to the 1-D
case.

🔗Paul H. Erlich <PErlich@xxxxxxxxxxxxx.xxxx>

🔗Carl Lumma <clumma@xxx.xxxx>

>>This is how I would like to define chromatic interval.
>
>That really only makes sense in the MOS case, but since that's what you're
>interested in, OK!

How does it not make sense with more dimensions?

>>Paul, your search rules out many of these
>
>Yes, but with a very minor modification, it becomes equivalent.

Let me see if I have this right- the set of 1-D scales meeting your
condition is a subset of the scales meeting the MOS condition.

>>(only the 2nds built from either end of the chain are
>>important?), as I noted...
>
>Eh?

Since the endpoints of the chain are the only notes that can change when
the scale is transposed by the generator, only those seconds adjacent to
them matter. And of these, only the ones that are in the same direction as
the unison vector matter. And of these, you only need to check one, since
the scale is symmetrical.

>>When they are transposed by any of their unit vectors, no changing notes
>>hop over unchanging notes.
>
>Well, in the 1-D case, if that's your only criterion, then there are non-MOS
>scales that qualify.

Yes, and I wonder if there's as simple a test to find them as your MOS test.

>So to make any progress I need to know how many pitches you, Carl, would
>allow to change in the 2-D case.

Let's simply insist that the scale is connected and that no changing notes
jump over unchanging notes when the scale is transposed by one step on the
triangular lattice. This is sufficient to get the normal 1-D MOS's, in
case the non-connected 1-D MOS's mentioned above don't work out.

-Carl

🔗Paul H. Erlich <PErlich@Acadian-Asset.com>

>>That really only makes sense in the MOS case, but since that's what you're
>>interested in, OK!

>How does it not make sense with more dimensions?

I meant in the 1-D case.

>>>Paul, your search rules out many of these
>
>>Yes, but with a very minor modification, it becomes equivalent.

>Let me see if I have this right- the set of 1-D scales meeting your
>condition is a subset of the scales meeting the MOS condition.

Right. If you allow for larger steps of the opposite sign from the unison
vector, but not the same sign, then you get MOS.

>Since the endpoints of the chain are the only notes that can change when
>the scale is transposed by the generator, only those seconds adjacent to
>them matter. And of these, only the ones that are in the same direction as
>the unison vector matter. And of these, you only need to check one, since
>the scale is symmetrical.

Sure -- in the n-tone chain, the interval spanning m tones in the chain
occurs n-m+1 times.

>>>When they are transposed by any of their unit vectors, no changing notes
>>>hop over unchanging notes.
>
>>Well, in the 1-D case, if that's your only criterion, then there are
non-MOS
>>scales that qualify.

>Yes, and I wonder if there's as simple a test to find them as your MOS
test.

Hmm . . . It's sort of like propriety but weaker. What should we name it?

>>So to make any progress I need to know how many pitches you, Carl, would
>>allow to change in the 2-D case.

>Let's simply insist that the scale is connected and that no changing notes
>jump over unchanging notes when the scale is transposed by one step on the
>triangular lattice. This is sufficient

but not a necessary condition

>to get the normal 1-D MOS's, in
>case the non-connected 1-D MOS's mentioned above don't work out.

Huh? What non-connected 1-D MOS's? I thought I showed that there are none.

🔗Carl Lumma <clumma@xxx.xxxx>

>>>That really only makes sense in the MOS case, but since that's what you're
>>>interested in, OK!
>>
>>How does it not make sense with more dimensions?
>
>I meant in the 1-D case.

One of us has missed something.

>>Since the endpoints of the chain are the only notes that can change when
>>the scale is transposed by the generator, only those seconds adjacent to
>>them matter. And of these, only the ones that are in the same direction as
>>the unison vector matter. And of these, you only need to check one, since
>>the scale is symmetrical.
>
>Sure -- in the n-tone chain, the interval spanning m tones in the chain
>occurs n-m+1 times.

I don't get it.

>>>>When they are transposed by any of their unit vectors, no changing notes
>>>>hop over unchanging notes.
>>>
>>>Well, in the 1-D case, if that's your only criterion, then there are
>>>non-MOS scales that qualify.
>>
>>Yes, and I wonder if there's as simple a test to find them as your MOS
>>test.
>
>Hmm . . . It's sort of like propriety but weaker. What should we name it?

Hopefully something less tacky than transpositional coherence... ..does
that mean you've got a test for it (maybe Scala's SHOW TRANSPOSE does it
already- of course it would be nice to have a rule that lets us find them)?

>>>So to make any progress I need to know how many pitches you, Carl, would
>>>allow to change in the 2-D case.
>
>>Let's simply insist that the scale is connected and that no changing notes
>>jump over unchanging notes when the scale is transposed by one step on the
>>triangular lattice. This is sufficient
>
>but not a necessary condition

Obviously, since there's an extra dimension. My point was just that it was
backward-compatible with 1-D MOS; an MOS of 3/2's will change all of it's
notes when transposed by a 5/4, and this doesn't violate the condition.

>>to get the normal 1-D MOS's, in case the non-connected 1-D MOS's mentioned
>>above don't work out.
>
>Huh? What non-connected 1-D MOS's? I thought I showed that there are none.

I shouldn't have called them MOS. Howabout "the non-connected 1-D
transpositionally coherent scales" mentioned above? If they do work out
(sound good), then maybe we can throw out the "connected" in the higher-D
definition as well.

-Carl

🔗manuel.op.de.coul@xxx.xxx

[Carl Lumma about SHOW TRANSPOSE]
>I've moved on to being confused by the wording "degree order"
>vs. "any order". I have convinced myself by playing with it that "degree
>order" enforces my preferred definition of chromatic [it does show more
>than one note changing when certain connected cyclic scales are transposed
>by their generator, which is impossible with the other definition of
>chromatic. Indeed, the "any order" column always shows 1 note change when
>transposing by the generator.

With the "degree order" count, each pitch is compared to the pitch with the
same degree number in the scale with the key changed.
With the "any order" count, each pitch is looked up in the scale with the
key changed to see if it's present. The degree numbers don't have to be
the same.
This distinction is the same as with the difference counts shown by
SHOW DIFFERENCE and SHOW/NEAREST DIFFERENCE.

>Where do the >1's in the "degree order"
>column actually come from -- Manuel?].

More than one different, perhaps your MOS doesn't have Myhill's property
then. I can't comment on your criterion, I find your discussion with Paul
confusing.

Manuel Op de Coul coul@ezh.nl

🔗Paul H. Erlich <PErlich@xxxxxxxxxxxxx.xxxx>

>>Sure -- in the n-tone chain, the interval spanning m tones in the chain
>>occurs n-m+1 times.

>I don't get it.

For example, in the chain of fifths F-C-G-D-A-E-B, the major third, which
spans 5 tones in the chain, occurs 7-5+1 = 3 times (C-E, F-A, G-B).

>>>Let's simply insist that the scale is connected and that no changing
notes
>>>jump over unchanging notes when the scale is transposed by one step on
the
>>>triangular lattice. This is sufficient
>
>>but not a necessary condition

>Obviously, since there's an extra dimension.

I was saying that even in 1-D it's not a necessary condition. But I missed
the "connected" part. So now I think you're right, it is a necessary
condition.

>I shouldn't have called them MOS. Howabout "the non-connected 1-D
>transpositionally coherent scales" mentioned above? If they do work out
>(sound good), then maybe we can throw out the "connected" in the higher-D
>definition as well.

Since any proper scale is transpositionally coherent, there are an infinite
number of 7-note TC scales to be found in the infinite chain of 3:2s (or any
other interval logarithmically incommeasurable with the octave). Many of
them will be totally disconnected, while others may consist of two connected
pieces far apart from one another. Etc. So maybe we should keep our lives
simple and stick with TC scales that are fully connected in the triangular
lattice, and start looking for some in two and three dimensions.

🔗Carl Lumma <clumma@xxx.xxxx>

>With the "degree order" count, each pitch is compared to the pitch with the
>same degree number in the scale with the key changed. With the "any order"
>count, each pitch is looked up in the scale with the key changed to see if
>it's present. The degree numbers don't have to be the same. This
>distinction is the same as with the difference counts shown by SHOW
>DIFFERENCE and SHOW/NEAREST DIFFERENCE.

Thanks!

>>Where do the >1's in the "degree order" column actually come from --
>>Manuel?].
>
>More than one different, perhaps your MOS doesn't have Myhill's property
>then.

It was only non-MOS connected cyclic scales that gave me the >1's. So all
good. Incidentally, according to Clampitt Myhill's property is equivalent
to MOS.

>I can't comment on your criterion, I find your discussion with Paul
>confusing.

Me too! What else is new? It's mostly my fault, for asking so many
questions at once.

This thread started by me asking if there were any properties of MOS that
could be applied to higher-dimensional (i.e. 5-limit, 7-limit, etc.)
scales. If Clampitt is right, then Myhill's Property cannot. Paul and I
seem to have settled on something we're calling transpositional coherence:

"A scale is transpositionally coherent iff it is connected across some set
of basic intervals (as on a 5-limit or 7-limit lattice), and transposition
by any of those intervals gives a new scale in which any notes shared with
the original scale occupy the same scale-degree positions they did in the
original scale."

In other words, a scale is transpositionally coherent iff the number in the
degree order column is the same as the number in the any order column.

There's still a few details to work out. Like what exactly "connected" and
"any" mean... Then there's that name. Maybe transpropriety would be better.

-Carl

🔗Carl Lumma <clumma@xxx.xxxx>

>>>Sure -- in the n-tone chain, the interval spanning m tones in the chain
>>>occurs n-m+1 times.
>
>>I don't get it.
>
>For example, in the chain of fifths F-C-G-D-A-E-B, the major third, which
>spans 5 tones in the chain, occurs 7-5+1 = 3 times (C-E, F-A, G-B).

No, I mean I don't get what that has to do with this statement:

>>I shouldn't have called them MOS. Howabout "the non-connected 1-D
>>transpositionally coherent scales" mentioned above? If they do work out
>>(sound good), then maybe we can throw out the "connected" in the higher-D
>>definition as well.
>
>Since any proper scale is transpositionally coherent,

Wow. Really? Having trouble seeing that. The inverse isn't true.

>So maybe we should keep our lives simple and stick with TC scales that are
>fully connected in the triangular lattice, and start looking for some in
two >and three dimensions.

I agree to keeping the connected part. But what is "fully" connected? By
connected, I usually just mean you could draw it on the lattice without
picking up your pen or passing through any un-used lattice points.

Say, there's two more things to iron out:

1) Should we require TC by all of the intervals needed to connect the
scale, or simply by any one of them?

2) What would be the preferred lattice type here, triangular or
rectangular? I would think that connectedness should always be defined on
a triangular lattice. My guess for defining TC would be to use rect. if we
answer "all" to #1, and triangular if we answer "any".

-Carl

🔗Carl Lumma <clumma@xxx.xxxx>

🔗Paul H. Erlich <PErlich@xxxxxxxxxxxxx.xxxx>

>>For example, in the chain of fifths F-C-G-D-A-E-B, the major third, which
>>spans 5 tones in the chain, occurs 7-5+1 = 3 times (C-E, F-A, G-B).

>No, I mean I don't get what that has to do with this statement:

>>>Since the endpoints of the chain are the only notes that can change when
>>>the scale is transposed by the generator, only those seconds adjacent to
>>>them matter. And of these, only the ones that are in the same direction
as
>>>the unison vector matter. And of these, you only need to check one,
since
>>>the scale is symmetrical.

It has to do with it because in this case you're saying you only have to
look at one major third, and in fact only one of each interval class, so you
might as well look at the ones built on F or on B.

>>Since any proper scale is transpositionally coherent,

>Wow. Really? Having trouble seeing that.

Isn't it obvious that any incoherence would have to result from an
impropriety?

>The inverse isn't true.

You mean the converse, and you're correct.

>I agree to keeping the connected part. But what is "fully" connected? By
>connected, I usually just mean you could draw it on the lattice without
>picking up your pen or passing through any un-used lattice points.

Right. That's fully connected. If the scale was broken up into two pieces in
the lattice, each piece could be connected on its own but not to the other
piece. That scale would not be fully connected. If the scale consisted of
notes in the lattice so far apart that no two were connected, that would be
fully disconnected.

>1) Should we require TC by all of the intervals needed to connect the
>scale, or simply by any one of them?

>2) What would be the preferred lattice type here, triangular or
>rectangular? I would think that connectedness should always be defined on
>a triangular lattice. My guess for defining TC would be to use rect. if we
>answer "all" to #1, and triangular if we answer "any".

I would suggest the opposite.

🔗Paul H. Erlich <PErlich@xxxxxxxxxxxxx.xxxx>

Carl wrote,

>Here's a connected, 1-D, non-MOS, proper but not strictly so, scale from a
>few months back...

>...which is not TC.

Comparing the scale with its transposition by 6/5:

0 0
63 63
316 316
379 379
632 632
694 694
947 884
1011 947

So it's coherent but not "strictly coherent". OK?

I see I should amend my statement from before. Connectedness plus propriety
implies coherence (if the propriety is strict, so is the coherence). I was
tacitly assuming connectedness.

🔗Paul H. Erlich <PErlich@xxxxxxxxxxxxx.xxxx>

🔗Carl Lumma <clumma@xxx.xxxx>

; "square set" - cross set of a scale with itself
; "diamond" - cross set of a scale and its inversion

[me]
>1) Should we require TC by all of the intervals needed to connect the scale,
>or simply by any one of them?

What if we throw out the connected idea, and make it so it has to behave
under transpositions by any of it's "visible" intervals (measured up from
the root of the current mode)? Then it winds up being CS, except defined
on the scale's square set rather than on its diamond. That much is right.

What I don't know: is this new TC then a property of modes or of scales?
The square sets of two different modes of the same scale can obviously be
different. But can they differ with respect to this property?

The scale I am most familiar with -- the diatonic scale in 12tET -- is
remarkable in that a mode change and its corresponding key change differ
the same number of notes, in opposite directions. Example: the dorian mode
of Cmaj, re-rooted on C as is done in the diamond, has 2 flats (3rd and
7th). Transposing Cmaj by its second degree gives Dmaj, which has two
sharps (also 3rd and 7th).

-C.