Re: that scale

>
> For a more *interesting* scale, how about 17 notes, which is an MOS.
>
> Bb--D\--F---A\--C---E\--G---B\
> \
> C/--E---G/--B---D/--F# \
> \ / \ / \ / \ / \ / \ \
> / / / / / \ Bb
> / \ / \ / \ / \ / \ \
> C---E\--G---B\--D---F/--A---C/--E
> \ \ / \ / \ / \ / \ /
> F# \ / / / / /
> \ \ / \ / \ / \ / \ / \
> \ Bb--D\--F---A\--C---E\
> \
> F/--A---C/--E---G/--B---D/--F#
>
>

Thank you for these diagrams. I messed around with two "scales
through lattices" last weekend. One of ithese was "septimal
minor thirds and good fifths". I got the very nice 12-out-of-22

131311313131

which is what I would have gotten if I had said "meantone in 22"

4 4 14 4 4 1
313113131311

(no I don't want to open up what is meantone aggain).

But I got stuck when trying to do a neutral third scale. It kept
on wrapping around and leaving holes in itself. I'll take these home
and see what happens.

Bob Valentine

Robert C Valentine wrote,

>which is what I would have gotten if I had said "meantone in 22"

> 4 4 14 4 4 1
> 313113131311

>(no I don't want to open up what is meantone aggain).

It would have been more correct (but still wrong) to call this "Pythagorean
in 22"

>
> >which is what I would have gotten if I had said "meantone in 22"
>
> > 4 4 14 4 4 1
> > 313113131311
>
> It would have been more correct (but still wrong) to call this "Pythagorean
> in 22"
>

Can I call it a "diatonic in 22?" When I say "meantone",
it is mainly a shorthand for "four equal sized fifths
are a major third", from a scale structure point of view,
rather than how nice it sounds. I will refrain from this
terminology in the future...

If I call LLsLLLs diatonic, what is LMsLMLs?

19 : 3323332 22: 4324342
31 : 5535553 34: 6536563
50 : 8858885 53: 9859895

Regarding propriety. The scale in 22 above is not proper. But it
has some neat features.

I think of a 12-out-of-N as having a set of 12 tone scales via
rotation, like modes of a normal scale. If we look at the sets
of intervals per scale degree, it has

scale step
degree sizes
1 [1,3] so even though the step sizes fo degree
2 [2,4] '1' cross degree '2', the are only
3 [5,7] equal for the one interval, '11' which
4 [6,8] is the augmented-fourth/diminished-fifth
5 [9,11]
6 [10,12] The appearance of interval 7 as a minor
7 [11,13] third is... a treasure akin to schismatic
8 [14,16] tuning...
9 [15,17]
10 [18,20] I dunno... it was one of the more
11 [19,21] inspiring things I tuned up...

Bob Valentine

[Robert C Valentine:]
> If I call LLsLLLs diatonic, what is LMsLMLs?

I'd say they're both diatonic. But I'd call the second LLsLLLs (where
in 22e L=4,3 & s=2) instead of "LMsLMLs." If you map the syntonic
diatonic ((LOG(N)-LOG(D))*(n/LOG(2)), you'll see that the permutations
of the second step structure you outlined (22, 34 and 53e) mimic that
JI step structure (i.e. L=9/8, 10/9 & s=16/15). 22e is the first
temperament (22, 29, 34, 41, 46 and 53e are the others equal to or
less than 53) that both mimics this step structure and is also
consistent... 7, 12, 19, 24 and 31e are the temperaments (equal to or
less than 53) that are also consistent with the 1/1, 9/8, 5/4, 4/3,
3/2, 5/3, 15/8, 2/1, but do not mimic the JI step structure.

Dan

Robert C Valentine wrote:

>Can I call it a "diatonic in 22?"
>If I call LLsLLLs diatonic, what is LMsLMLs?

It depends who you ask or what book your refernce is. That's the problem,
there's no standard terminology for these things. Calling the first
pseudo-Pythagorean major, and the second pseudo-just major, should be clear
enough to get your meaning across to speakers of various dialects

I love the 4 4 1 4 4 4 1 scale and devoted a paragraph to it in my paper
(http://www-math.cudenver.edu/~jstarret/22ALL.pdf):

"The hexachordal dodecatonic scale contains within it seven consecutive
notes in a cycle of Qs. This forms a diatonic scale where the three "minor"
triads are tuned 9:7:6 and the three "major" triads are tuned 1/6:1/7:1/9 -
a reversal of the usual harmo nic/subharmonic distinctions. For example, the
"minor" mode of this scale, which could be termed "sub-minor" due to its
small minor thirds, would be 0 4 5 9 13 14 18. This scale can be mapped to
the white notes of a keyboard, with the remainder of the hexachordal
dodecatonic scale mapped to the black keys. Then the white-note scale
satisfies all the properties of the ordinary diatonic scale except that the
triads are not complete sonorities (although they can all be partially
completed since the 8 or 1/8 is always a scale tone, the required pattern of
scale steps is not the same for the minor and major cases), and the dorian
and mixolydian modes can no longer support a static tonality."

BTW, the hexachordal dodecatonic scale is 2 2 2 2 1 2 2 2 2 2 2 1.

Dan Stearns wrote,

>But I'd call the second LLsLLLs (where
>in 22e L=4,3 & s=2) instead of "LMsLMLs."
<[ . . .]
>22e is the first
>temperament (22, 29, 34, 41, 46 and 53e are the others equal to or
>less than 53) that both mimics this step structure and is also
>consistent...

What about 15e and 27e?

>7, 12, 19, 24 and 31e are the temperaments (equal to or
>less than 53) that are also consistent with the 1/1, 9/8, 5/4, 4/3,
>3/2, 5/3, 15/8, 2/1, but do not mimic the JI step structure.

What about 26e and 43e?

By "mimic the step structure" you mean exactly what Robert meant by
"LMsLMLs," and by "not mimic the step structure" you mean exactly what
Robert meant by "LLsLLLs", so your first statement above is really confusing
the issue!

>> 22e is the first temperament (22, 29, 34, 41, 46 and 53e are the
others equal to or less than 53) that both mimics this step structure
and is also consistent...

[Paul H. Erlich:]
> What about 15e and 27e?

Both mimic the step structure but neither are consistent...

> > 7, 12, 19, 24 and 31e are the temperaments (equal to or less than
53) that are also consistent with the 1/1, 9/8, 5/4, 4/3, 3/2, 5/3,
15/8, 2/1, but do not mimic the JI step structure.

[Paul Erlich:]
> What about 26e and 43e?

Again, neither are consistent...

[Paul Erlich:]
>so your first statement above is really confusing the issue!

Hmm... I though it was all pretty clear!

Dan

My last two posts came up in reverse order for some reason.

Anyway, Dan, I see you're continuing to use a notion of consistency relative
to a JI scale rather then a harmonic limit. I find that illogical. The only
reason the JI ratios come about in the first place is in order to make
certain intervals as consonant as possible. If an ET makes those intervals
consonant, then why worry about the dissonant intervals (like 32/27, 40/27,
etc.)? If those intervals are not represented by their best approximations
to JI, who cares? Approximating JI only matters when it gives you a sense of
how consonant your intervals are going to be.

On the other hand, if you're trying to find out whether the ET has a
_melodic_ representation of the diatonic scale, I would ask you to consider
this: Would anyone say that the pseudo-Pythagorean diatonic scale in
53-equal fails melodically? I don't think so. In that case, it's clearly not
consistency with the 1/1, 9/8, 5/4, 4/3, 3/2, 5/3,
15/8, 2/1 that determines melodic viability.

As I see it, the diatonic scale is generally (in common practice music)
associated with a harmonic limit of 5. 15-, 26-, 27-, and 43-equal are
consistent within that harmonic limit. But 15 and 27 (and 22, 29, 34, 41)
have a LMsLMLs structure, while 26 and 43 (and 12, 19, 31) have a LLsLLLs
structure. Only the latter are appropriate for common-practice music since
the former totally screw up a I-vi-ii-V-I or I-IV-ii-V-I progression (i.e.,
trying to use all the common tones leads to a pitch drift downward by one
degree of the tuning). So I would really deem 15, 22, 27, 29, 34, and maybe
41 (though there you might not notice the drift) to be truly inconsistent
with the diatonic scale in common practice, while 12, 19, 26, 31, and 43
would be truly consistent with it. Of course this is a different type of
consistency that the type used in the context of harmonic limits, but it is
a more appropriate way of judging consistency with a certain usage of a
scale, rather than divorcing the scale from its usage and pretending that it
was one big chord, which is essentially what you're doing, as Carl has been
trying to tell you.

Please excuse my unfriendly tone, Dan, I'm very stressed-out at the moment
and I want you to know that I greatly appreciate your coming to my defense
in recent weeks. Hopefully you'll understand that and purely consider that
content of this message and not its tone.

[Paul H. Erlich:]
>Please excuse my unfriendly tone, Dan, I'm very stressed-out at the
moment and I want you to know that I greatly appreciate your coming to
my defense in recent weeks. Hopefully you'll understand that and
purely consider that content of this message and not its tone.

I really don't see anything unfriendly about it. I see it as you
believing in what your saying and therefore saying it when you think
it needs saying... seems like things pretty much the way they should
be... But let me try and back things up for a moment.

The main point in the original post to which I was responding was that
the three step size 22, 34, and 53e scales were in fact the syntonic
diatonic (call it major and not diatonic if you must!). And that they
consist of two step sizes: whole and half... As far as mapping whole
scales for consistency - you think it's illogical (and as such you
certainly should say so), but me, I'm still unsure... so consider all
the rest of that post as perhaps useless (to which you've stated your
case), perhaps interesting, or perhaps useful... but all of it as
pretty much besides the main point, which was that the "LMsLMLs" that
were posted, were in fact the 1/1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2/1.

Dan

Dan Stearns wrote,

>The main point in the original post to which I was responding was that
>the three step size 22, 34, and 53e scales were in fact the syntonic
>diatonic (call it major and not diatonic if you must!).

Well, if pressed to come up with a major scale using 7 JI ratios, I would
just use the major triads and say 1/1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8 (2/1);
but if pressed to come up with a minor scale using 7 JI ratios, I would use
the minor triads and say 1/1, 9/8, 6/5, 4/3, 3/2, 8/5, 9/5 (2/1), whose
third mode is 1/1, 10/9, 5/4, 4/3, 3/2, 5/3, 15/8 (2/1). Since the term
"diatonic" today so often refers to a pitch set which can be used as either
a major scale or a minor scale (not to mention other modes), I would argue
that no set of 7 JI pitches can alone really capture what is today meant by
"diatonic".

>And that they
>consist of two step sizes: whole and half...

Dan, what could be gained, other that confusion, by saying that a scale with
three step sizes consists of two step sizes?

[Paul H. Erlich:]
Dan, what could be gained, other that confusion, by saying that a
scale with three step sizes consists of two step sizes?

Well not too long ago I posted this (ear derived)

0 7 12
8 15 20
[11] 18 3

+3 +4 +1 +4 +3 +3 +2 scale in 20e... and while it has four different
step sizes, I still see it as (some variety of) a "two step size" 5L &
2s "diatonic" scale... and I guess I really don't see why a:

L(3)L(4)s(1)L(4)L(3)L(3)s(2)

would be quite so objectionable (or only confusing)... I mean I really
don't think that anyone would want to call the recently discussed:

((LOG(1)-LOG(1))*(n/LOG(2))
((LOG(12)-LOG(11))*(n/LOG(2))
((LOG(9)-LOG(8))*(n/LOG(2))
((LOG(11)-LOG(9))*(n/LOG(2))
((LOG(4)-LOG(3))*(n/LOG(2))
((LOG(16)-LOG(11))*(n/LOG(2))
((LOG(3)-LOG(2))*(n/LOG(2))
((LOG(18)-LOG(11))*(n/LOG(2))
((LOG(16)-LOG(9))*(n/LOG(2))
((LOG(11)-LOG(6))*(n/LOG(2))
((LOG(2)-LOG(1))*(n/LOG(2))

a "LsMLLsLMsL."

But while I've used (and posted about) quite a few scales that are
constructed of more than two distinct step sizes as well as quite a
few that are constructed of a larger number of (more or less) easily
reducible step sizes (of which 1/1, 7/6, 11/9, 23/18, 4/3, 3/2, 7/4,
11/6, 23/12, 2/1 would be a good example of say a 5-into-3), and
therefore think that it's worth my while to attempt to make some
general demarcations between a distinct step size and a not distinct
step size - that's not to say that this sort of a thing isn't going to
cause some confusion! For example, what would you call a scale like
1/1, 13/12, 7/6, 5/4, 4/3, 3/2, 13/8, 7/4, 15/8, 2/1? A scale with
five step sizes... a scale with five step sizes falling into three
general size categories (though this too could be interpreted in a
number of different ways, see for example the difference between how
it could sit in 15 and 16e)... a scale with five step sizes falling
into two general size categories...

Dan

Earlier I wrote:

"a scale with five step sizes falling into three general size
categories (though this too could be interpreted in a
number of different ways, see for example the difference between how
it could sit in 15 and 16e)..."

However the parenthetical reference references a different scale (one
that I had originally typed in then changed) than the 1/1, 13/12, 7/6,
5/4, 4/3, 3/2, 13/8, 7/4, 15/8, 2/1, and should have read:

(though this too could be interpreted in a number of different ways,
see for example the difference between how it could sit in 31, 36 and
43e)...

Dan

I wrote,

>>Dan, what could be gained, other that confusion, by saying that a scale
with
>>three step sizes consists of two step sizes? >>

Daniel Wolf wrote,

>Pardon me for stepping in here, but a lot can be gained from a coarser
>definition of step sizes. For one, it's a central feature in the the
>tradition of defining tetrachordal genera.

Yes, yes. I was just pointing out how confusing Dan Stearns's statement
sounded:

"The main point in the original post to which I was responding was that
the three step size 22, 34, and 53e scales were in fact the syntonic
diatonic (call it major and not diatonic if you must!). And that they
consist of two step sizes: whole and half..."

Doesn't that sound like he's saying 3=2?

>For another, it is very often more
>useful to define a scale as a collection where a given scale step is
>represented by more than one ratio -- e.g. the sixth degree in a major
scale
>might be represented by either 27/16 or 5/3, depending on the context, and
>that difference is important but also important is the fact that both
>rational values share qualities that can be group under the term "Major
>sixth".

Totally agree -- which is why I have a problem with Dan Stearns's rendering
of the major scale in 22-, 34-, 41-, and 53-equal, where the 5/3 but not the
27/16 are represented. By contrast, in the major scale in meantone tunings
like 19-, 31-, 43-, and 55-equal, the sixth can act as both 5/3 and 27/16.

Dan Stearns wrote,

>However the parenthetical reference references a different scale (one
>that I had originally typed in then changed) than the 1/1, 13/12, 7/6,
>5/4, 4/3, 3/2, 13/8, 7/4, 15/8, 2/1, and should have read:

>(though this too could be interpreted in a number of different ways,
>see for example the difference between how it could sit in 31, 36 and
>43e)...

Although this is tangential to your point, I'd say that none of these ETs
can realistically represent that JI scale, in part because none of them are
consistent in the 13-limit.

[Paul H. Erlich:]
>Although this is tangential to your point,

But what about the point... You did ask "what could be gained, other
that confusion, by saying that a scale with three step sizes consists
of two step sizes?" And I tried to offer some (concrete) examples of
something other than "confusion."

>I'd say that none of these ETs can realistically represent that JI
scale, in part because none of them are consistent in the 13-limit.

I offered those as hypothetical examples in a hypothetical argument -
I'd personally gravitate to the mappings like 19 & 29e where the scale
would be a 1L & 9s. And though the 1/1, 13/12, 7/6, 5/4, 4/3, 3/2,
13/8, 7/4, 15/8, 2/1 scale would also be consistent in these
temperaments, that is not why I'd gravitate towards them... as here it
would be the reduction to two step sizes (but while you think
otherwise, I'm not so sure that their consistency and reduction would
be a purely trivial coincidence here).

Dan

>>Although this is tangential to your point,

>But what about the point... You did ask "what could be gained, other
>that confusion, by saying that a scale with three step sizes consists
>of two step sizes?" And I tried to offer some (concrete) examples of
>something other than "confusion."

I have no problem with your examples; as I said before, my problem was with
your language in this statement:

"The main point in the original post to which I was responding was that
the three step size 22, 34, and 53e scales were in fact the syntonic
diatonic (call it major and not diatonic if you must!). And that they
consist of two step sizes: whole and half..."

. . . where it really sounds like you're saying 3=2 without any
qualification.

>I offered those as hypothetical examples in a hypothetical argument -
>I'd personally gravitate to the mappings like 19 & 29e where the scale
>would be a 1L & 9s. And though the 1/1, 13/12, 7/6, 5/4, 4/3, 3/2,
>13/8, 7/4, 15/8, 2/1 scale would also be consistent in these
>temperaments,

Again I beseech you not to think of consistency in this way

>that is not why I'd gravitate towards them... as here it
>would be the reduction to two step sizes

Well, remember, I was the one who liked the diatonic scale only in those ETs
in which it has two, rather than three step sizes -- so I'd certainly be in
favor of such a criterion in this case too. Why do we agree here and not in
the diatonic case?

>(but while you think
>otherwise, I'm not so sure that their consistency and reduction would
>be a purely trivial coincidence here).

On the contrary, I do not think anything of the sort.

[Paul H. Erlich:]
>as I said before, my problem was with your language in this
statement: "The main point in the original post to which I was
responding was that the three step size 22, 34, and 53e scales were in
fact the syntonic diatonic (call it major and not diatonic if you
must!). And that they consist of two step sizes: whole and half..."

> . . . where it really sounds like you're saying 3=2

Yikes... The whole AND ONLY REAL point was that I objected to calling
them "LMsLMLs..."

> without any qualification.

...Which I really think I did "qualified" (and many times over now!)

[me:]
>I offered those as hypothetical examples in a hypothetical argument -
I'd personally gravitate to the mappings like 19 & 29e where the scale
would be a 1L & 9s. And though the 1/1, 13/12, 7/6, 5/4, 4/3, 3/2,
13/8, 7/4, 15/8, 2/1 scale would also be consistent in these
temperaments,

[Paul:]
> Again I beseech you not to think of consistency in this way

I know, and I certainly do take your opinions here in high regard...
But I'm curious as to why you later say "On the contrary, I do not
think anything of the sort," to my: "but while you think otherwise,
I'm not so sure that their consistency and reduction would be a purely
trivial coincidence here..." When 19e would only be consistent through
the 13-limit in the 1/1, 13/12, 7/6, 5/4, 4/3, 3/2, 13/8, 7/4, 15/8,
2/1 scale mapping...

[Paul:]
> Well, remember, I was the one who liked the diatonic scale only in
those ETs in which it has two, rather than three step sizes -- so I'd
certainly be in favor of such a criterion in this case too. Why do we
agree here and not in the diatonic case?

Again the ONLY thing I was really favoring there was NOT calling the
22, 34, and 53e syntonic diatonic "LMsLMLs."

Dan

>I know, and I certainly do take your opinions here in high regard...
>But I'm curious as to why you later say "On the contrary, I do not
>think anything of the sort," to my: "but while you think otherwise,
>I'm not so sure that their consistency and reduction would be a purely
>trivial coincidence here..." When 19e would only be consistent through
>the 13-limit in the 1/1, 13/12, 7/6, 5/4, 4/3, 3/2, 13/8, 7/4, 15/8,
>2/1 scale mapping...

I don't get your last statement. I was saying that I certainly think
consistency as you are evaluating (with respect to the scale) it is not in
any way equivalent to a reduction in the number of step sizes.

I wrote,

>>Well, remember, I was the one who liked the diatonic scale only in
>>those ETs in which it has two, rather than three step sizes -- so I'd
>>certainly be in favor of such a criterion in this case too. Why do we
>>agree here and not in the diatonic case?

Dan wrote,

>Again the ONLY thing I was really favoring there was NOT calling the
>22, 34, and 53e syntonic diatonic "LMsLMLs."

#1 -- your formulation of consistency and your fleshing out of its
consequences _did_ seem to amount to a favoring of certain ETs over others
for the diatonic scale, and in _that_ we disagreed -- remember?

#2 -- on this other point, how on earth could you justify calling your 22e
scale, 4 3 2 4 3 4 2, LLsLLLs and not LMsLMLS, on purely acoustical grounds
(rather than appealing to perception in terms of experientially engrained
categories)?

[Paul Erlich:]
> Yes, yes. I was just pointing out how confusing Dan Stearns's
statement sounded: "The main point in the original post to which I was
responding was that the three step size 22, 34, and 53e scales were in
fact the syntonic diatonic (call it major and not diatonic if you
must!). And that they consist of two step sizes: whole and half..."
>
> Doesn't that sound like he's saying 3=2?

Well, to the extent that they both =L, I certainly was.

> Totally agree -- which is why I have a problem with Dan Stearns's
rendering of the major scale in 22-, 34-, 41-, and 53-equal, where the
5/3 but not the 27/16 are represented.

You must be misunderstanding me, because in each of these the "27/16"
would just be the "5/3" +1 (17/22, 26/34, 31/41, 40/53).

Dan

I wrote,

>>Totally agree -- which is why I have a problem with Dan Stearns's
>>rendering of the major scale in 22-, 34-, 41-, and 53-equal, where the
>>5/3 but not the 27/16 are represented.

>You must be misunderstanding me, because in each of these the "27/16"
>would just be the "5/3" +1 (17/22, 26/34, 31/41, 40/53).

Right, but you're not including those pitches in your version of the major
scale in those tunings, correct? For example, in 22-equal you have the 16
but not the 17 -- true?

[Paul H. Erlich:]
>I was saying that I certainly think consistency as you are evaluating
(with respect to the scale) it is not in any way equivalent to a
reduction in the number of step sizes.

Right... which is what I thought... and implied that you thought. (And
just for the record; I never said "equivalent," I said: "I'm not so
sure that their consistency and reduction would be a purely trivial
coincidence here..." They may be - but as I said, I'm not yet sure.)

>#1 -- your formulation of consistency and your fleshing out of its
consequences _did_ seem to amount to a favoring of certain ETs over
others for the diatonic scale, and in _that_ we disagreed -- remember?

Yes... Though as I've said (many times), unlike you, I'm not yet sure
one way or the other. And while I very much respect, and want to know,
your (and others) take on it (or to speak up if something I'm saying,
or appear to be saying on it seems wrong and whatnot), I've just began
to look at it (consistency) and won't abandon my inclinations and ways
of working things out until I think I've run into a wall, or better
understand it to my satisfaction.

>#2 -- on this other point, how on earth could you justify calling
your 22e scale, 4 3 2 4 3 4 2, LLsLLLs and not LMsLMLS, on purely
acoustical grounds (rather than appealing to perception in terms of
experientially engrained categories)?

Hmm, I'm not sure I understand... When did I ever say anything about
acoustics (never mind "on purely acoustical grounds")?

Dan

I wrote,

>>#2 -- on this other point, how on earth could you justify calling
>>your 22e scale, 4 3 2 4 3 4 2, LLsLLLs and not LMsLMLS, on purely
>>acoustical grounds (rather than appealing to perception in terms of
>>experientially engrained categories)?

Dan Stearns wrote,

>Hmm, I'm not sure I understand... When did I ever say anything about
>acoustics (never mind "on purely acoustical grounds")?

OK, then, on what grounds _do_ you justify it? The only rationale I could
see would be environmental conditioning, but I guess I'm assuming that our
theoretical considerations assume that such conditioning could be undone by
enough exposure to microtonal music.

[me:]
> You must be misunderstanding me, because in each of these the
"27/16" would just be the "5/3" +1 (17/22, 26/34, 31/41, 40/53).

[Paul Erlich:]
> Right, but you're not including those pitches in your version of the
major scale in those tunings, correct? For example, in 22-equal you
have the 16 but not the 17 -- true?

Well yes, but this is no different than the (1/1, 9/8, 5/4, 4/3, 3/2,
5/3, 15/8, 2/1) syntonic major... and the point of Daniel Wolf's post
to which you were agreeing... no? What am I missing here!

Dan

[Paul H. Erlich:]
>OK, then, on what grounds _do_ you justify it?

Sure, "environmental conditioning" sounds good enough for me too...
but I'm still not even sure what your trying to say in all this -
Would you rather call the 4, 3, 2, 4, 3, 4, 2, 22e scale a LMsLMLS as
opposed to a L(4)L(3)s(2)L(4)L(3)L(4)s(2) "syntonic diatonic?"

>The only rationale I could see would be environmental conditioning,
but I guess I'm assuming that our theoretical considerations assume
that such conditioning could be undone by enough exposure to
microtonal music.

Well, I'm not so sure that any amount of exposure to microtonal music
is going to turn a

((LOG(1)-LOG(1))*(n/LOG(2))
((LOG(12)-LOG(11))*(n/LOG(2))
((LOG(9)-LOG(8))*(n/LOG(2))
((LOG(11)-LOG(9))*(n/LOG(2))
((LOG(4)-LOG(3))*(n/LOG(2))
((LOG(16)-LOG(11))*(n/LOG(2))
((LOG(3)-LOG(2))*(n/LOG(2))
((LOG(18)-LOG(11))*(n/LOG(2))
((LOG(16)-LOG(9))*(n/LOG(2))
((LOG(11)-LOG(6))*(n/LOG(2))
((LOG(2)-LOG(1))*(n/LOG(2))

LsLLLsLLsL into a "LsMLLsLMsL."

Dan

Dan Stearns wrote,

>Well yes, but this is no different than the (1/1, 9/8, 5/4, 4/3, 3/2,
>5/3, 15/8, 2/1) syntonic major... and the point of Daniel Wolf's post
>to which you were agreeing... no? What am I missing here!

I was agreeing with Daniel Wolf that the sixth degree of the major scale
should be able to act as _either_ 5/3 _or_ 27/16. Only in the meantone-type
ETs do the major scales you came up with have this property.

Dan Stearns wrote,

>Sure, "environmental conditioning" sounds good enough for me too...
>but I'm still not even sure what your trying to say in all this -
>Would you rather call the 4, 3, 2, 4, 3, 4, 2, 22e scale a LMsLMLS as
>opposed to a L(4)L(3)s(2)L(4)L(3)L(4)s(2) "syntonic diatonic?"

>Well, I'm not so sure that any amount of exposure to microtonal music
>is going to turn a

>((LOG(1)-LOG(1))*(n/LOG(2))
>((LOG(12)-LOG(11))*(n/LOG(2))
>((LOG(9)-LOG(8))*(n/LOG(2))
>((LOG(11)-LOG(9))*(n/LOG(2))
>((LOG(4)-LOG(3))*(n/LOG(2))
>((LOG(16)-LOG(11))*(n/LOG(2))
>((LOG(3)-LOG(2))*(n/LOG(2))
>((LOG(18)-LOG(11))*(n/LOG(2))
>((LOG(16)-LOG(9))*(n/LOG(2))
>((LOG(11)-LOG(6))*(n/LOG(2))
>((LOG(2)-LOG(1))*(n/LOG(2))

>LsLLLsLLsL into a "LsMLLsLMsL."

Dan, there is a major quantitative difference here. In the first example, M
(163.6�) is exactly halfway between L (218.2�) and s (109.1�) in size.
Therefore using M for the 163.6� step seems totally appropriate, and calling
it "L" would make as little sense as calling it "s". In the second example,
M (143.5�) is 13 times closer to L (150.6�) than to s (53.3�). In that case,
it seems clear that M and L would fall into one category once the scale has
been truly absorbed on its own terms, so calling the 143.5� step "L" is
perfectly justified.

[Paul H. Erlich:]
> Dan, there is a major quantitative difference here.

Yes, I was exaggerating... But where exactly do you draw these lines
(or the line)? The whole thrust of my carrying on here has been about
making some useful distinctions between reducible, or collapsible and
distinct step size categories.

Dan

>But where exactly do you draw these lines
>(or the line)?

I don't know _exactly_ where, or if such a question can or even needs to be
answered. But certainly these last two examples were about as clear-cut as
they come.

[Paul H. Erlich:]
>But certainly these last two examples were about as clear-cut as they
come.

Ah ha. Well then it's done - we just disagree: You don't see the
syntonic diatonic (JI major scale) as a 5L & 2s scale and I do.

Dan

[Paul Erlich:]
> I was agreeing with Daniel Wolf that the sixth degree of the major
scale should be able to act as _either_ 5/3 _or_ 27/16. Only in the
meantone-type ETs do the major scales you came up with have this
property.

Maybe I'm reading it wrong (and I'm sure he'll let me know if I am),
but that isn't what I took this statement to mean:

"For another, it is very often more useful to define a scale as a
collection where a given scale step is represented by more than one
ratio -- e.g. the sixth degree in a major scale might be represented
by either 27/16 or 5/3, depending on the context, and that difference
is important but also important is the fact that both rational values
share qualities that can be group under the term "Major sixth".

Dan

Daniel Wolf wrote,

>If you want to define "major" by
>the presence of both ratios (as either exact ratios or tempered
>representations), you will then be limiting the definition to a rather
narrow
>repertoire.

I disagree completely. The entire common-practice/tonal use of major is tied
to a meantone conception, where both ratios are approximated by the very
same sixth degree of the major scale. On the other hand, if you are forced
to choose only one of the two ratios, as Dan's major scales in 22, 34, 41,
and 53 do, you will be limiting yourself to a rather narrow repertoire.

>Ah ha. Well then it's done - we just disagree: You don't see the
>syntonic diatonic (JI major scale) as a 5L & 2s scale and I do.

Far from it -- 10:9 is over 3 times closer to 9:8 than to 16:15. So it is
likely to be heard as 5L & 2s.

[me:]
>Ah ha. Well then it's done -
[Paul H. Erlich:]
Far from it --

Ahhhhhhhhhh!