Deep Scale Property?

Hi all,

I ran into my friend today at UM's campus - he was in the same major
as me, so also spent quite a bit of time studying DSP and
psychoacoustics, etc.

He said he was in the process of writing a paper about the "deep scale
property." I didn't quite understand all of his explanation, but it
seems to be this:

http://en.wikipedia.org/wiki/Deep_scale_property

This is also a pretty vague and ambiguous explanation and I'm not sure
exactly what it means. I told him to swing by here so maybe he can
expound on it more, but can anyone comment on the cognitive
significance of scales like these? Are they somehow related to MOS?
He's about to write a MATLAB program to do a systematic search for all
scales with the deep scale property from ET's 7 up to 31. His
motivation is that since a lot of the scales we use in music tend to
have this mathematical property, that it must somehow be cognitively
relevant.

The whole thing to me sounds like something related to MOS, and I'm
curious to see if he keeps barking up this tree long enough if he'll
discover that what he's really after is Fokker. Can anyone comment on
these scales?

-Mike

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

> The whole thing to me sounds like something related to MOS, and I'm
> curious to see if he keeps barking up this tree long enough if he'll
> discover that what he's really after is Fokker. Can anyone comment on
> these scales?

Not me, other than it does seem related to MOS: a MOS with octave period would have the property if I am construing things correctly.

On Fri, Feb 11, 2011 at 12:21 AM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > The whole thing to me sounds like something related to MOS, and I'm
> > curious to see if he keeps barking up this tree long enough if he'll
> > discover that what he's really after is Fokker. Can anyone comment on
> > these scales?
>
> Not me, other than it does seem related to MOS: a MOS with octave period would have the property if I am construing things correctly.

That's what it seemed like to me as well, although I believe he said
that the pentatonic scale didn't have the deep scale property...?

-Mike

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

> > > The whole thing to me sounds like something related to MOS,
> > > and I'm curious to see if he keeps barking up this tree long
> > > enough if he'll discover that what he's really after is Fokker.
> > > Can anyone comment on these scales?
> >
> > Not me, other than it does seem related to MOS: a MOS with
> > octave period would have the property if I am construing things
> > correctly.
>
> That's what it seemed like to me as well, although I believe he
> said that the pentatonic scale didn't have the deep scale
> property...?

Deep scales are discussed in the all-powerful Clough et al.
See my folder in the files section. -C.

"Carl Lumma" <carl@...> wrote:

> Deep scales are discussed in the all-powerful Clough et

"Every interval class has unique multiplicity" is what it
says. I can understand that. What I can't understand is
why it should be important.

Graham

On Fri, Feb 11, 2011 at 2:40 AM, Carl Lumma <carl@...> wrote:
>
> Deep scales are discussed in the all-powerful Clough et al.
> See my folder in the files section. -C.

That's quite a reference. I read some of it now, will read the rest
later. So it looks like it's not MOS after all. It looks also like
most deep scales are generated, except for some weird exception that's
a subset of the whole tone scale. Very interesting. Do you have any
thoughts as to its relevance?

-Mike

On Fri, Feb 11, 2011 at 3:39 AM, Graham Breed <gbreed@...> wrote:
>
> "Carl Lumma" <carl@...> wrote:
>
> > Deep scales are discussed in the all-powerful Clough et
>
> "Every interval class has unique multiplicity" is what it
> says. I can understand that. What I can't understand is
> why it should be important.

Then I don't see how it applies to the diatonic scale or how the
diatonic scale is deep.

I'm going to assume that "interval class" here means specific interval
class, because if it means generic interval class then it really makes
no sense at all. So here it is, worked out for some common 12-tet
scales:

Diatonic:
1\12 - 2
2\12 - 5
3\12 - 4
4\12 - 3
5\12 - 6
6\12 - 2

Looks like we just hit 2 twice, so it doesn't look like the 12-tet
diatonic scale is "deep." I suppose deepness requires constant
structure. Clough doesn't seem to address this, at not least in the
part I read. It's 31 pages long and the text isn't searchable so that
makes things tricky.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Fri, Feb 11, 2011 at 3:39 AM, Graham Breed <gbreed@...> wrote:
> >
> > "Carl Lumma" <carl@...> wrote:
> >
> > > Deep scales are discussed in the all-powerful Clough et
> >
> > "Every interval class has unique multiplicity" is what it
> > says. I can understand that. What I can't understand is
> > why it should be important.
>
> Then I don't see how it applies to the diatonic scale or how the
> diatonic scale is deep.
>
> I'm going to assume that "interval class" here means specific interval
> class, because if it means generic interval class then it really makes
> no sense at all. So here it is, worked out for some common 12-tet
> scales:
>
> Diatonic:
> 1\12 - 2
> 2\12 - 5
> 3\12 - 4
> 4\12 - 3
> 5\12 - 6
> 6\12 - 2
>
> Looks like we just hit 2 twice, so it doesn't look like the 12-tet
> diatonic scale is "deep." I suppose deepness requires constant
> structure. Clough doesn't seem to address this, at not least in the
> part I read. It's 31 pages long and the text isn't searchable so that
> makes things tricky.
>
> -Mike

I don't think we should be counting inversions here, and then the 6/12 would only occur once right?

On Fri, Feb 11, 2011 at 4:24 AM, John Moriarty <JlMoriart@...> wrote:
>
> I don't think we should be counting inversions here, and then the 6/12 would only occur once right?

Oh, that's a good point. I thought they were going to do something
dumb with it and say that it was supposed to be an "augmented fourth"
in one case and a "diminished fifth" in the other, and insist that
they aren't the same because of some kind of universal diatonic
hearing, but this makes more sense. So you would view 6/12 as its own
inversion then, and not count it twice.

So here's the pentatonic scale then:
1\12: 0
2\12: 3
3\12: 2
4\12: 1
5\12: 4
6\12: 0

So I guess this isn't deep then, because 1 and 6 both appear 0 times.
So it seems to be some kind of weird variant of MOS that applies only
for equal temperaments.

I wonder though, if you were to vary this property a bit to make it so
that each specific interval has its own multiplicity - but only when
it does appear at all, meaning we just ignore 1\12 and 6\12 and
discount them from the equation, since they appear 0 times - if that
means that you end up with something equivalent to MOS, but only for
equal temperaments.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Fri, Feb 11, 2011 at 2:40 AM, Carl Lumma <carl@...> wrote:
> >
> > Deep scales are discussed in the all-powerful Clough et al.
> > See my folder in the files section. -C.
>
> That's quite a reference. I read some of it now, will read the rest
> later. So it looks like it's not MOS after all.

Note also that DE is not the same as MOS, and should not be used as a synonym.

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> "Carl Lumma" <carl@...> wrote:
>
> > Deep scales are discussed in the all-powerful Clough et
>
> "Every interval class has unique multiplicity" is what it
> says. I can understand that. What I can't understand is
> why it should be important.
>
> Graham

Indeed. -C.

On Fri, Feb 11, 2011 at 11:34 AM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> >
> > On Fri, Feb 11, 2011 at 2:40 AM, Carl Lumma <carl@...> wrote:
> > >
> > > Deep scales are discussed in the all-powerful Clough et al.
> > > See my folder in the files section. -C.
> >
> > That's quite a reference. I read some of it now, will read the rest
> > later. So it looks like it's not MOS after all.
>
> Note also that DE is not the same as MOS, and should not be used as a synonym.

Last I asked on here, people told me that DE was same as MOS, and
should be used as a synonym. What's the difference? That MOS doesn't
apply for fractional octave periods?

-Mike

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

> > Note also that DE is not the same as MOS, and should not be used as a synonym.
>
> Last I asked on here, people told me that DE was same as MOS, and
> should be used as a synonym. What's the difference? That MOS doesn't
> apply for fractional octave periods?

DE does not need to have Myhill's property; but now that I think on it, maybe that just means it's a MOS for a period which is a fraction of an octave. How else does it get either one or two specific intervals per generic interval?

On Fri, Feb 11, 2011 at 2:49 PM, genewardsmith
<genewardsmith@...> wrote:
>
> > Last I asked on here, people told me that DE was same as MOS, and
> > should be used as a synonym. What's the difference? That MOS doesn't
> > apply for fractional octave periods?
>
> DE does not need to have Myhill's property; but now that I think on it, maybe that just means it's a MOS for a period which is a fraction of an octave. How else does it get either one or two specific intervals per generic interval?

I thought that MOS technically referred to Myhill's property, but that
nowadays people were using it in a way that's synonymous with DE. And
then I thought that the difference between DE and Myhill's property
was that DE can handle fractional-octave periods as well, and it makes
a lot more sense if MOS's can refer to things like 4L4s as well.

I guess one abstract way to make it so that you end up with a DE scale
that doesn't have Myhill's property and doesn't have a fractional
period would be to use a quasi-periodic scale; one that has 2/1 as the
interval of equivalence, but 3/2 as the actual period of repetition or
something like that. This is, of course, dirty rotten cheating.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Fri, Feb 11, 2011 at 4:24 AM, John Moriarty <JlMoriart@...> wrote:
> >
> > I don't think we should be counting inversions here, and then the 6/12 would only occur once right?
>
> Oh, that's a good point. I thought they were going to do something
> dumb with it and say that it was supposed to be an "augmented fourth"
> in one case and a "diminished fifth" in the other, and insist that
> they aren't the same because of some kind of universal diatonic
> hearing, but this makes more sense. So you would view 6/12 as its own
> inversion then, and not count it twice.

Well, the augmented fourth and diminished fifth are inversions, so the same way you wouldn't say that the diatonic scale has the same number of minor thirds and major sixths, you wouldn't say that it has the same number of augmented fourths and diminished fifths.

>
> So here's the pentatonic scale then:
> 1\12: 0
> 2\12: 3
> 3\12: 2
> 4\12: 1
> 5\12: 4
> 6\12: 0
>
> So I guess this isn't deep then, because 1 and 6 both appear 0 times.
> So it seems to be some kind of weird variant of MOS that applies only
> for equal temperaments.
>
> I wonder though, if you were to vary this property a bit to make it so
> that each specific interval has its own multiplicity - but only when
> it does appear at all, meaning we just ignore 1\12 and 6\12 and
> discount them from the equation, since they appear 0 times - if that
> means that you end up with something equivalent to MOS, but only for
> equal temperaments.

I think that would be the correct interpretation. "Each interval that *actually appears in the scale* has a unique multiplicity". Otherwise, you could say that your deep 12-edo diatonic scale wasn't deep anymore if you consider it to be a part of 24-edo where both 1/24 and 3/24 appear 0 times. Or as an extreme, the 200-edo diatonic scale wouldn't be deep because there are TONS of intervals that are used 0 times.

John

Mike wrote:

> I thought that MOS technically referred to Myhill's property,
> but that nowadays people were using it in a way that's
> synonymous with DE. And then I thought that the difference
> between DE and Myhill's property was that DE can handle
> fractional-octave periods as well, and it makes a lot more
> sense if MOS's can refer to things like 4L4s as well.

Correct. And a bit depressing this is being discussed less
than 3 weeks after... it was all discussed
/tuning/topicId_95922.html#95949

Including by Gene himself!
/tuning/topicId_95922.html#95923

> I guess one abstract way to make it so that you end up with
> a DE scale that doesn't have Myhill's property and doesn't
> have a fractional period would be to use a quasi-periodic
> scale; one that has 2/1 as the interval of equivalence, but
> 3/2 as the actual period of repetition or something like that.
> This is, of course, dirty rotten cheating.

It sure is: it would be a rank 3 scale, since the period no
longer generates the IoE. The key thing of importance is
rank 2 scales, and the key term to remember is "MOS". There
is no term for (or good understanding of) rank 3 scales.

-Carl

On Fri, Feb 11, 2011 at 4:27 PM, Carl Lumma <carl@...> wrote:
>
> Correct. And a bit depressing this is being discussed less
> than 3 weeks after... it was all discussed
> /tuning/topicId_95922.html#95949
>
> Including by Gene himself!
> /tuning/topicId_95922.html#95923

What's depressing is the constant flip-flopping about it. I have a
crystal clear understanding of it in my head, which is

- MOS used to be synonymous with Myhill's property
- DE is a variant of Myhill's property that allows for fractional periods
- MOS nowadays is used to refer to DE, such that 3L3s is called an
"MOS" of augmented temperament

That's what I said here. You guys then said that DE and MOS aren't
synonymous, which I assume is somehow relevant to my question about
the deep scale property. But then, in the message Gene posted that you
just linked to, he's saying that MOS's do allow for fractional
periods, and that Paul is being too much of a stickler, which would
seem to indicate that MOS and DE always were the same thing anyway.
Except that now they aren't again.

> > I guess one abstract way to make it so that you end up with
> > a DE scale that doesn't have Myhill's property and doesn't
> > have a fractional period would be to use a quasi-periodic
> > scale; one that has 2/1 as the interval of equivalence, but
> > 3/2 as the actual period of repetition or something like that.
> > This is, of course, dirty rotten cheating.
>
> It sure is: it would be a rank 3 scale, since the period no
> longer generates the IoE. The key thing of importance is
> rank 2 scales, and the key term to remember is "MOS". There
> is no term for (or good understanding of) rank 3 scales.

I don't see how it would be rank 3. There's a scale called the
"hyperlydian" progression, which in meantone notation looks something
like this:

C D E F# G A B C# D E F# G# A B C# D# E - etc

It's basically the C-D-E-F#-G pentachord tiled forever, with 3/2 as
the period. 2/1 is represented in this scale, but it just isn't the
period. The generator is 9/8 and the period is 3/2, so it's rank 2.

-Mike

On Fri, Feb 11, 2011 at 4:23 PM, John Moriarty <JlMoriart@...> wrote:
>
> Well, the augmented fourth and diminished fifth are inversions, so the same way you wouldn't say that the diatonic scale has the same number of minor thirds and major sixths, you wouldn't say that it has the same number of augmented fourths and diminished fifths.

? The diatonic scale does have the same number of minor thirds and major sixths.

> > I wonder though, if you were to vary this property a bit to make it so
> > that each specific interval has its own multiplicity - but only when
> > it does appear at all, meaning we just ignore 1\12 and 6\12 and
> > discount them from the equation, since they appear 0 times - if that
> > means that you end up with something equivalent to MOS, but only for
> > equal temperaments.
>
> I think that would be the correct interpretation. "Each interval that *actually appears in the scale* has a unique multiplicity". Otherwise, you could say that your deep 12-edo diatonic scale wasn't deep anymore if you consider it to be a part of 24-edo where both 1/24 and 3/24 appear 0 times. Or as an extreme, the 200-edo diatonic scale wouldn't be deep because there are TONS of intervals that are used 0 times.

I'm not sure that Clough's interpretation supports this one, though. I
specifically asked my friend at school this question, and he said that
the pentatonic scale wouldn't count because those two intervals both
appear twice. If you use this alternate interpretation, then I think
it really is related to MOS after all, although I'm not sure how to
prove that.

-Mike

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

> Except that now they aren't again.

Sorry for whatever contribution I made to the confusion, which I got into after looking at the Clough paper.

> It's basically the C-D-E-F#-G pentachord tiled forever, with 3/2 as
> the period. 2/1 is represented in this scale, but it just isn't the
> period. The generator is 9/8 and the period is 3/2, so it's rank 2.

And I don't know what Clough, et al, would call it. If anything.

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

> > Correct. And a bit depressing this is being discussed less
> > than 3 weeks after... it was all discussed
> > /tuning/topicId_95922.html#95949
> > Including by Gene himself!
> > /tuning/topicId_95922.html#95923
>
> What's depressing is the constant flip-flopping about it. I have
> a crystal clear understanding of it in my head, which is
> - MOS used to be synonymous with Myhill's property
> - DE is a variant of Myhill's property that allows for fractional
> periods
> - MOS nowadays is used to refer to DE, such that 3L3s is called
> an "MOS" of augmented temperament

Correct. Well, I don't know about nowadays. I do, and I've
presented what I consider to be unassailable arguments why
others should too. Shall I retype them in the 2nd week of
March? Probably.

> That's what I said here. You guys then said that DE and MOS
> aren't synonymous,

Only Gene said that, and it's odd since on Jan 29 he was
critical of Paul doing it.

> > It sure is: it would be a rank 3 scale, since the period no
> > longer generates the IoE. The key thing of importance is
> > rank 2 scales, and the key term to remember is "MOS". There
> > is no term for (or good understanding of) rank 3 scales.
>
> I don't see how it would be rank 3. There's a scale called the
> "hyperlydian" progression, which in meantone notation looks
> something like this:
> C D E F# G A B C# D E F# G# A B C# D# E - etc
> It's basically the C-D-E-F#-G pentachord tiled forever,
> with 3/2 as the period. 2/1 is represented in this scale,
> but it just isn't the period. The generator is 9/8 and the
> period is 3/2, so it's rank 2.

There are no octaves in this scale in JI. If there are, the
generator and/or period have been tempered to divide it and
it's rank 2.

-Carl

On Fri, Feb 11, 2011 at 5:19 PM, Carl Lumma <carl@...> wrote:
>
> Only Gene said that, and it's odd since on Jan 29 he was
> critical of Paul doing it.

What I wrote above is what I think that MOS and DE mean. After using
the words that way in my initial post, Gene said that my
interpretation was wrong, and you said "Indeed." After inquiring
further, you seemed to get frustrated about how this was already dealt
with three weeks ago and how we already went over it. To illuminate on
where I'm going wrong, you linked to a post from yourself in which you
seem to agree after all with my current interpretation, and then one
also from Gene in which he also seems to agree. Then, after laying my
interpretation out again in plain English, it turns out that you
really did agree with it after all, despite our earlier disagreement.
However, you aren't going to say why you agree with it until March. I
think I'm following this now.

> > C D E F# G A B C# D E F# G# A B C# D# E - etc
> > It's basically the C-D-E-F#-G pentachord tiled forever,
> > with 3/2 as the period. 2/1 is represented in this scale,
> > but it just isn't the period. The generator is 9/8 and the
> > period is 3/2, so it's rank 2.
>
> There are no octaves in this scale in JI. If there are, the
> generator and/or period have been tempered to divide it and
> it's rank 2.

It's rank 2 in JI. The generator is 3/2, and the period is 9/8. These
are all 3-limit intervals. If p is the period, and g is the generator,
and g is 9/8 and p is 3/2, then g^2 brings you to 81/64. p^2 * g
brings you to 9/4 * 9/8 = 81/32. The difference between 81/64 and
81/32 is 2/1.

-Mike

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:

> > That's what I said here. You guys then said that DE and MOS
> > aren't synonymous,
>
> Only Gene said that, and it's odd since on Jan 29 he was
> critical of Paul doing it.

It's not weird, just a mistake. I looked at Clough again and got the impression that DE was something other than what I had thought it was.

Mike wrote:

> > Only Gene said that, and it's odd since on Jan 29 he was
> > critical of Paul doing it.
>
> What I wrote above is what I think that MOS and DE mean. After using
> the words that way in my initial post, Gene said that my
> interpretation was wrong, and you said "Indeed."

check again

/tuning/topicId_96189.html#96204

> > There are no octaves in this scale in JI. If there are, the
> > generator and/or period have been tempered to divide it and
> > it's rank 2.
>
> It's rank 2 in JI. The generator is 3/2, and the period is 9/8.
> These are all 3-limit intervals. If p is the period, and g is
> the generator, and g is 9/8 and p is 3/2, then g^2 brings you
> to 81/64. p^2 * g brings you to 9/4 * 9/8 = 81/32. The
> difference between 81/64 and 81/32 is 2/1.

Sorry, I meant to say, it's not periodic at the octave. So
the octave can't be the IoE, as not every pitch has an octave
duplicate. To do that the period must either divide the
octave or the octave must be introduced independently (making
the scale rank 3).

-Carl

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

> It's not weird, just a mistake. I looked at Clough again and got
> the impression that DE was something other than what I had thought
> it was.

Ah. You do that less often than I typically expect so it's easy
for me to forget you're human sometimes. I'm aware of the issue
and how it's not fair to you, so, sorry. -Carl

On Fri, Feb 11, 2011 at 7:37 PM, Carl Lumma <carl@...> wrote:
>
> Mike wrote:
>
> > > Only Gene said that, and it's odd since on Jan 29 he was
> > > critical of Paul doing it.
> >
> > What I wrote above is what I think that MOS and DE mean. After using
> > the words that way in my initial post, Gene said that my
> > interpretation was wrong, and you said "Indeed."
>
> check again
>
> /tuning/topicId_96189.html#96204

Ah. It all makes sense now. Mea culpa.

-Mike

On Fri, Feb 11, 2011 at 5:34 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> > > That's what I said here. You guys then said that DE and MOS
> > > aren't synonymous,
> >
> > Only Gene said that, and it's odd since on Jan 29 he was
> > critical of Paul doing it.
>
> It's not weird, just a mistake. I looked at Clough again and got the impression that DE was something other than what I had thought it was.

So then I ask this again: I proposed an alternative interpretation of
the "deep scale" property that would make something like pentatonic
scales deep. I don't think that right now they qualify, because

Pentatonic scale:
1\12: 0
2\12: 3
3\12: 2
4\12: 1
5\12: 4
6\12: 0

1 and 6 both appear 0 times, so they don't appear a "unique" number of
times. If, however, you alter the property slightly such that an
interval appears a unique number of times when it appears at all, then
this would be said to be "deep" after all.

I think, based on my intuition and some prior exploration on various
MOS scales, that this is the same thing as MOS. If not, it certainly
makes more sense than the deep scale property the way it was first
described. Can anyone shed more insight on if these two properties are
the same thing? I'm not quite sure how to start tackling this one.

-Mike

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

> So then I ask this again: I proposed an alternative
> interpretation of the "deep scale" property

I think I speak for many here when I say, I don't care
one cigar about the "deep scale" property. -Carl

On 12 February 2011 01:35, Mike Battaglia <battaglia01@...> wrote:

> What's depressing is the constant flip-flopping about it. I have a
> crystal clear understanding of it in my head, which is
>
> - MOS used to be synonymous with Myhill's property
> - DE is a variant of Myhill's property that allows for fractional periods
> - MOS nowadays is used to refer to DE, such that 3L3s is called an
> "MOS" of augmented temperament
>
> That's what I said here. You guys then said that DE and MOS aren't
> synonymous, which I assume is somehow relevant to my question about
> the deep scale property. But then, in the message Gene posted that you
> just linked to, he's saying that MOS's do allow for fractional
> periods, and that Paul is being too much of a stickler, which would
> seem to indicate that MOS and DE always were the same thing anyway.
> Except that now they aren't again.

MOS and DE are the same thing. They have been for the past 10 years.
I searched the archives the last time this came up, and after Kraig
corrected me privately about a caveat I expressed then.

There may be a message somewhere buried in the archives that says
otherwise. Maybe somebody can find it and we can see how the myth
originated. But from Erv's documents that have been released since
whenever that was, it looks like Erv has been thinking of non-octave
periods for much longer than 10 years. Whatever message that said the
octave must be the period was a miscommunication. I don't think there
was ever a time when MOS was really different to DE. There was a time
when most of us weren't clear about it.

Graham

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

> MOS and DE are the same thing. They have been for the past 10 years.
> I searched the archives the last time this came up, and after Kraig
> corrected me privately about a caveat I expressed then.
>
> There may be a message somewhere buried in the archives that says
> otherwise. Maybe somebody can find it and we can see how the myth
> originated. But from Erv's documents that have been released since
> whenever that was, it looks like Erv has been thinking of non-octave
> periods for much longer than 10 years. Whatever message that said the
> octave must be the period was a miscommunication. I don't think there
> was ever a time when MOS was really different to DE. There was a time
> when most of us weren't clear about it.
>
> Graham

This makes things simpler. However an important point I raised on 6th - 8th - 10th of march 2010 is left unanswered :

/tuning/topicId_86834.html#86974

Which said in short :
If this is the common use of these terms (= Distributionally even) then some more general term has to be found to refer to octave divisions produced by 3D temperaments.
I suggest "3 sizes-DE" (or 3D-DE or whatever) : this would keep the idea of even distribution, while it precises the number of interval sizes it refers to.

Example of a fractal "3D-DE" easy to understand :
(infinite word defined by the combinatory endless transformation a - > bc, b -> a, c-> b)
octave = 37, generators = 9 and 16 steps :

16 12 9
9 7 12 9
9 7 5 7 9
4 5 7 5 7 4 5
4 5 3 4 5 3 4 4 5

n-D temperaments can't be "DE" otherwise than with n sizes step intervals, it means that 3D-DE and next ones are an essential extension, not only for scales but also for rhythmns and all patterns in general.
- - - - - - -
Jacques

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

> There may be a message somewhere buried in the archives that says
> otherwise. Maybe somebody can find it and we can see how the myth
> originated. But from Erv's documents that have been released since
> whenever that was, it looks like Erv has been thinking of non-octave
> periods for much longer than 10 years.

Thanks Graham. Can you link us to any Wilson PDF(s) showing
nonoctave periods? I don't think this is necessary to support
DE = MOS, I'm just interested to see them.

> Whatever message that said the
> octave must be the period was a miscommunication.

IIRC, it took place on MMM between Paul and Kraig.

-Carl

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Fri, Feb 11, 2011 at 4:23 PM, John Moriarty <JlMoriart@...> wrote:
> >
> > Well, the augmented fourth and diminished fifth are inversions, so the same way you wouldn't say that the diatonic scale has the same number of minor thirds and major sixths, you wouldn't say that it has the same number of augmented fourths and diminished fifths.
>
> ? The diatonic scale does have the same number of minor thirds and major sixths.

Took me a while but I see what you thought I meant.
What I meant by saying "you wouldn't say the diatonic scale has the same number of minor thirds and major sixths" was that you wouldn't *point this out as a reason* to not count the scale as deep, because they are the same interval.

Sorry about that!

John