Yahoo Tuning Groups Ultimate Backup tuning Octatonic Scale

Now to make this a tuning issue. One can see here:

that the 'diminished' temperament, also known as the 'octatonic'
temperament, is based on tempering out the interval 648:625, and
yields scales like the octatonic most naturally, since it has a
period of 300 cents and an optimal generator of about 94 cents. While
12-equal represents the latter by 100 cents, other ETs fall along
the 'diminished' temperament line and yield octatonic scales with
slightly different values for the two steps sizes (though they always
sum to 300 cents). For example, you can see that 28-equal also falls
along the 'diminished' line, and that the generator is 2 degrees of
28-equal, or about 86 cents. You can also see that the major and
minor triads won't be tuned too much worse than in 12-equal, but
they're quite different in sound. So take the example below, set h=86
cents and w=214 cents, and you have 'xenharmonic Messaien'. Or use
the 'optimal' values, h=94 cents and w=206 cents, and the progression
won't sound much different than normal, though you might notice some
improvement in the concordance of the triads.

--- In tuning@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> Not really a tuning issue, but want to point out a cool use of
scales
> in "Vingt Regards sur L'Enfant Jesus," Mvmt X. by Messaien. He has
> about 6 chord passages in this movement which are comprised of 6
> layers of alternating octatonic scales, like thus:
>
> whwhwhwhwhwhwhwh
> hwhwhwhwhwhwhwhw
> whwhwhwhwhwhwhwh
> hwhwhwhwhwhwhwhw
> whwhwhwhwhwhwhwh
> hwhwhwhwhwhwhwhw
>
> etc. in the horizontal direction. The intervals between each line
> are
>
> P4-M3-P4-M3-P4-M3-P4 etc
> M3-P4-M3-P4-M3-P4-M3
> P4-M3-P4-M3-P4-M3-P4
> M3-P4-M3-P4-M3-P4-M3
> P4-M3-P4-M3-P4-M3-P4
>
> in the vertical direction. It also creates a nice crunchy
combination
> of alternating Minor Triad (6-3) over Major Triad (6-4) alternating
> with Major Triad (6-4) over Minor Triad (6-3)
>
> What a wonderful use of a mathematical idea, which is also very
> effective musically as well. These cascading chord passages occur
at
> the end of melodic lines ("fill") in the middle of this piece
(Regard
> for the Spirit of Joy) Worth checking out, especially around the
> Holidays
>
> Merry Christmas
>
> Paul

🔗Kurt Bigler <kkb@breathsense.com>

on 12/2/03 10:10 AM, Paul Erlich <paul@stretch-music.com> wrote:

> Now to make this a tuning issue. One can see here:
>
> http://sonic-arts.org/dict/eqtemp.htm
>
> that the 'diminished' temperament, also known as the 'octatonic'
> temperament, is based on tempering out the interval 648:625, and
> yields scales like the octatonic most naturally, since it has a
> period of 300 cents and an optimal generator of about 94 cents. While
> 12-equal represents the latter by 100 cents, other ETs fall along
> the 'diminished' temperament line and yield octatonic scales with
> slightly different values for the two steps sizes (though they always
> sum to 300 cents). For example, you can see that 28-equal also falls
> along the 'diminished' line, and that the generator is 2 degrees of
> 28-equal, or about 86 cents. You can also see that the major and
> minor triads won't be tuned too much worse than in 12-equal, but
> they're quite different in sound. So take the example below, set h=86
> cents and w=214 cents, and you have 'xenharmonic Messaien'. Or use
> the 'optimal' values, h=94 cents and w=206 cents, and the progression
> won't sound much different than normal, though you might notice some
> improvement in the concordance of the triads.

And what happens if you allow the octave to be inexact? Does this allow
other improvements to be made? This is actually two questions...

One is when you are restricted to 12et (the term et still applies with
inexact octaves, right?). What is the best octave size for a 12et-based
octatonic?

The other is when you are not so rectricted - even if it would create
asymmetries in relation to the layered octatonic scales that Paul H. was
referring to, and even if it would make one such scale preferable over the
other two possibilities. Actually I see his layers use only 2 of the 3
possible octatonies and I mean to include all 3 in the question. So what is
the best octatonic that can be created in this circumstance, even if only 1
of the 3 possibilities is optimized?

However, even in the latter case I am assuming a symmetrical 4et diminished
chord. But if it is worth relaxing this restriction for some reason, please
say so.

-Kurt

>
>
> --- In tuning@yahoogroups.com, "Paul G Hjelmstad"
> <paul.hjelmstad@u...> wrote:
>> Not really a tuning issue, but want to point out a cool use of
> scales
>> in "Vingt Regards sur L'Enfant Jesus," Mvmt X. by Messaien. He has
>> about 6 chord passages in this movement which are comprised of 6
>> layers of alternating octatonic scales, like thus:
>>
>> whwhwhwhwhwhwhwh
>> hwhwhwhwhwhwhwhw
>> whwhwhwhwhwhwhwh
>> hwhwhwhwhwhwhwhw
>> whwhwhwhwhwhwhwh
>> hwhwhwhwhwhwhwhw
>>
>> etc. in the horizontal direction. The intervals between each line
>> are
>>
>> P4-M3-P4-M3-P4-M3-P4 etc
>> M3-P4-M3-P4-M3-P4-M3
>> P4-M3-P4-M3-P4-M3-P4
>> M3-P4-M3-P4-M3-P4-M3
>> P4-M3-P4-M3-P4-M3-P4
>>
>> in the vertical direction. It also creates a nice crunchy
> combination
>> of alternating Minor Triad (6-3) over Major Triad (6-4) alternating
>> with Major Triad (6-4) over Minor Triad (6-3)
>>
>> What a wonderful use of a mathematical idea, which is also very
>> effective musically as well. These cascading chord passages occur
> at
>> the end of melodic lines ("fill") in the middle of this piece
> (Regard
>> for the Spirit of Joy) Worth checking out, especially around the
>> Holidays
>>
>> Merry Christmas
>>
>> Paul
>
>
>
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🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> on 12/2/03 10:10 AM, Paul Erlich <paul@s...> wrote:
>
> > Now to make this a tuning issue. One can see here:
> >
> > http://sonic-arts.org/dict/eqtemp.htm
> >
> > that the 'diminished' temperament, also known as the 'octatonic'
> > temperament, is based on tempering out the interval 648:625, and
> > yields scales like the octatonic most naturally, since it has a
> > period of 300 cents and an optimal generator of about 94 cents.
While
> > 12-equal represents the latter by 100 cents, other ETs fall along
> > the 'diminished' temperament line and yield octatonic scales with
> > slightly different values for the two steps sizes (though they
always
> > sum to 300 cents). For example, you can see that 28-equal also
falls
> > along the 'diminished' line, and that the generator is 2 degrees
of
> > 28-equal, or about 86 cents. You can also see that the major and
> > minor triads won't be tuned too much worse than in 12-equal, but
> > they're quite different in sound. So take the example below, set
h=86
> > cents and w=214 cents, and you have 'xenharmonic Messaien'. Or use
> > the 'optimal' values, h=94 cents and w=206 cents, and the
progression
> > won't sound much different than normal, though you might notice
some
> > improvement in the concordance of the triads.
>
> And what happens if you allow the octave to be inexact? Does this
allow
> other improvements to be made?

Yes, though it becomes harder to specify your set of 'target
intervals' -- do you use an 'integer limit'?

> This is actually two questions...
>
> One is when you are restricted to 12et (the term et still applies
with
> inexact octaves, right?).

Umm . . . I'll follow you on that, but it's best to be specific.

> What is the best octave size for a 12et-based
> octatonic?

It depends what your set of 'target intervals' is and how you want to
weight them -- for example, you might not want to include 8:5 at all,
but then again, you might . . .

> The other is when you are not so rectricted - even if it would
create
> asymmetries in relation to the layered octatonic scales that Paul
H. was
> referring to,

It wouldn't, unless you weighted certain *occurrences* of some
intervals more than other occurrences of the same intervals -- which
seems to be what you're suggesting below.

> and even if it would make one such scale preferable over the
> other two possibilities. Actually I see his layers use only 2 of
the 3
> possible octatonies and I mean to include all 3 in the question.

What are octatonies? The octatonic scale has 4 major triads and 4
minor triads . . .

> So what is
> the best octatonic that can be created in this circumstance, even
if only 1
> of the 3 possibilities is optimized?
>
> However, even in the latter case I am assuming a symmetrical 4et
diminished
> chord.

You are? Then I don't know what you could have possibly meant
by "even if it would create asymmetries" above. Can you elaborate the
above with more specifics?

> But if it is worth relaxing this restriction for some reason, please
> say so.

As I was thinking above, if you weight some of the positions of a
given interval more than others, you'll tend to break the 4-equal
symmetry. One terrific approach would be to create a MIDI file of the
chord progression in question -- I could send it to John deLaubenfels
and see what he gets for a COFT:

http://sonic-arts.org/dict/coft.htm

🔗Kurt Bigler <kkb@breathsense.com>

on 12/3/03 12:50 PM, Paul Erlich <paul@stretch-music.com> wrote:

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>> on 12/2/03 10:10 AM, Paul Erlich <paul@s...> wrote:
>>
>>> Now to make this a tuning issue. One can see here:
>>>
>>> http://sonic-arts.org/dict/eqtemp.htm
>>>
>>> that the 'diminished' temperament, also known as the 'octatonic'
>>> temperament, is based on tempering out the interval 648:625, and
>>> yields scales like the octatonic most naturally, since it has a
>>> period of 300 cents and an optimal generator of about 94 cents.
> While
>>> 12-equal represents the latter by 100 cents, other ETs fall along
>>> the 'diminished' temperament line and yield octatonic scales with
>>> slightly different values for the two steps sizes (though they
> always
>>> sum to 300 cents). For example, you can see that 28-equal also
> falls
>>> along the 'diminished' line, and that the generator is 2 degrees
> of
>>> 28-equal, or about 86 cents. You can also see that the major and
>>> minor triads won't be tuned too much worse than in 12-equal, but
>>> they're quite different in sound. So take the example below, set
> h=86
>>> cents and w=214 cents, and you have 'xenharmonic Messaien'. Or use
>>> the 'optimal' values, h=94 cents and w=206 cents, and the
> progression
>>> won't sound much different than normal, though you might notice
> some
>>> improvement in the concordance of the triads.
>>
>> And what happens if you allow the octave to be inexact? Does this
> allow
>> other improvements to be made?
>
> Yes, though it becomes harder to specify your set of 'target
> intervals' -- do you use an 'integer limit'?

Why does it become harder to specify the target because a restriction is
removed? What were the assumptions that led to 94 cents as the optimal
generator of 94 cents in the 2:1 octave cast?

>> This is actually two questions...
>>
>> One is when you are restricted to 12et (the term et still applies
> with
>> inexact octaves, right?).
>
> Umm . . . I'll follow you on that, but it's best to be specific.

I was trying to refer to 12 equally spaced intervals within a possibly
non-2:1 octave. Do you call this 12et or not? I was calling it 12et,
tentatively, in the following...

>> What is the best octave size for a 12et-based
>> octatonic?
>
> It depends what your set of 'target intervals' is and how you want to
> weight them -- for example, you might not want to include 8:5 at all,
> but then again, you might . . .

Again, what were the assumed weights for your original statement? My
intention is to ask the most general possible question. Your original
statement was pretty general, and did not appear to depend on stated
targets. What if I were to weight equally all intervals less than an octave
wide? What if I were to say "for playing french romantic octatonic music"?

>> The other is when you are not so rectricted - even if it would
> create
>> asymmetries in relation to the layered octatonic scales that Paul
> H. was
>> referring to,

> It wouldn't, unless you weighted certain *occurrences* of some
> intervals more than other occurrences of the same intervals -- which
> seems to be what you're suggesting below.

Wait on this point until more ambiguities are clarified.

>> and even if it would make one such scale preferable over the
>> other two possibilities. Actually I see his layers use only 2 of
> the 3
>> possible octatonies and I mean to include all 3 in the question.
>
> What are octatonies? The octatonic scale has 4 major triads and 4
> minor triads . . .

Typo - meant to be "octatonics" meaning "octatonic scales". If there is a
12-note underlying base scale, you can create 3-different octatonic scales
on that base:

0 1 3 4 6 7 9 10
0 2 3 5 6 8 9 11
1 2 4 5 7 8 10 11

These are the "3 octatonics" I was trying to refer to, of which Paul H's
example used 2, if I understood it correctly. But I based this on his
"whwhwhwhwhwhwhwh" style spelling, and may have drawn the wrong conclusions
since I did not follow the rest of the analysis.

>> So what is
>> the best octatonic that can be created in this circumstance, even
> if only 1
>> of the 3 possibilities is optimized?
>>
>> However, even in the latter case I am assuming a symmetrical 4et
> diminished
>> chord.
>
> You are? Then I don't know what you could have possibly meant
> by "even if it would create asymmetries" above. Can you elaborate the
> above with more specifics?

The 4-note diminished chord appears twice in the octatonic scale. I was
originally thinking that I did not want to disturb the equal-temperedness of
that chord so I would not disturb the 4-way symmetry within the "octave".
But at the end I opened that assumption to questioning.

But back to the original assumption: I thought there might still be some
value in freeing up the relative position fo the 2 diminished chords that
appear in the octatonic scale. Or in the more extended question of creating
a 12-tone scale out of which 3 different octatonics can be extracted, I was
wanting to free up the relative positions of the 3 diminished chords that
appear in the 12-tone scale.

So let's focus on the 12-tone super-octatonic scale, whatever it should be
called. I was first asking the question what can happen if you add 1 degree
of freedom: the width of the octave which is divided into 12 equal
intervals.

Then the second quesition was what can happen if I add 2 more degrees of
freedom by allowing the 3 diminished chords to shift. Perhaps it was nieve
to ask this question without saying that one of the 3 octatonic scales is
more important than the other (never mind target intervals). But that was
what I meant to ask, anyway.

The 3rd question then was to allow 11 additional degrees of freedom for the
super-scale (7 for the octatonic). This question was an afterthought in my
original email because I *thought* I for sure wanted to preserve 4-way
symmetry within the octave.

>
>> But if it is worth relaxing this restriction for some reason, please
>> say so.
>
> As I was thinking above, if you weight some of the positions of a
> given interval more than others, you'll tend to break the 4-equal
> symmetry. One terrific approach would be to create a MIDI file of the
> chord progression in question -- I could send it to John deLaubenfels
> and see what he gets for a COFT:
>
> http://sonic-arts.org/dict/coft.htm

Well, again the motivation for the question is to find a scale for playing a
range of already composed music. I could easily narrow that to french
romantic, but not with particular progressions in mind. So perhaps the
question has no good answer then? Perhaps the question stated that way
would steer things toward an equal-tempered result? If so, the octave size
is still a question.

-Kurt

🔗Paul Erlich <paul@stretch-music.com>

Hey Kurt, look at this:

http://www.io.com/~hmiller/music/temp-diminished.html

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> > Yes, though it becomes harder to specify your set of 'target
> > intervals' -- do you use an 'integer limit'?
>
> Why does it become harder to specify the target because a
restriction is
> removed? What were the assumptions that led to 94 cents as the
optimal
> generator of 94 cents in the 2:1 octave cast?

Hi Kurt, I'm glad you're giving me an excuse to stay up all night. We
weighted all 5-(odd)-limit intervals (a.k.a. 5-limit consonances)
equally, and minimized the sum of their squared errors.

> >> This is actually two questions...
> >>
> >> One is when you are restricted to 12et (the term et still applies
> > with
> >> inexact octaves, right?).
> >
> > Umm . . . I'll follow you on that, but it's best to be specific.
>
> I was trying to refer to 12 equally spaced intervals within a
possibly
> non-2:1 octave. Do you call this 12et or not? I was calling it
12et,
> tentatively, in the following...

Yes, I was tentatively following this usage . . . though some people
around here say 12ED2, 12ED2.1, etc.

> What if I were to weight equally all intervals less than an octave
> wide?

*all* intervals? What would the targets for the small and large steps
themselves be? Or do you really just mean all 5-limit consonances?

> >> The other is when you are not so rectricted - even if it would
> > create
> >> asymmetries in relation to the layered octatonic scales that Paul
> > H. was
> >> referring to,
>
> > It wouldn't, unless you weighted certain *occurrences* of some
> > intervals more than other occurrences of the same intervals --
which
> > seems to be what you're suggesting below.
>
> Wait on this point until more ambiguities are clarified.
>
> >> and even if it would make one such scale preferable over the
> >> other two possibilities. Actually I see his layers use only 2 of
> > the 3
> >> possible octatonies and I mean to include all 3 in the question.
> >
> > What are octatonies? The octatonic scale has 4 major triads and 4
> > minor triads . . .
>
> Typo - meant to be "octatonics" meaning "octatonic scales". If
there is a
> 12-note underlying base scale,

Oh! I wasn't assuming that at all. Why would you?

> These are the "3 octatonics" I was trying to refer to, of which
Paul H's
> example used 2, if I understood it correctly.

Oh, I thought it was just 1!

> But back to the original assumption: I thought there might still
be some
> value in freeing up the relative position fo the 2 diminished
chords that
> appear in the octatonic scale.

Yes, absolutely. Again, under one set of assumptions, the 2 diminised
chords should be 94 cents, instead of 100 cents, apart.

> Or in the more extended question of creating
> a 12-tone scale out of which 3 different octatonics can be
extracted, I was
> wanting to free up the relative positions of the 3 diminished
chords that
> appear in the 12-tone scale.

That's harder to come by because then you have to tune the octatonic
scales differently from one another, which would require some
assumptions. The simplest is that you're only going to use 2 of the
octatonic scales, in which case you space the three diminished chords
so that one lies 94 cents below, one lies 94 cents above a central
one.

> So let's focus on the 12-tone super-octatonic scale, whatever it
should be
> called.

Hmm . . . it depends what harmonies you assume.

> I was first asking the question what can happen if you add 1 degree
> of freedom: the width of the octave which is divided into 12 equal
> intervals.

OK, what criterion do you want to use?

> The 3rd question then was to allow 11 additional degrees of freedom
for the
> super-scale (7 for the octatonic). This question was an
afterthought in my
> original email because I *thought* I for sure wanted to preserve 4-
way
> symmetry within the octave.

Well, if we calculated a COFT for an octatonic music, we might find
interesting beasts, but since the example in question was
so 'symmetrical', we're likely to see the 4-way symmetry almost
perfectly preserved.

> I could easily narrow that to french
> romantic,

Which composers do you consider french romantic? I didn't know any of
them used the octatonic scale at all . . .

🔗Kurt Bigler <kkb@breathsense.com>

on 12/3/03 11:45 PM, Paul Erlich <paul@stretch-music.com> wrote:

> Hey Kurt, look at this:
>
> http://www.io.com/~hmiller/music/temp-diminished.html

Yikes, you know, the terminology in your messages goes into some territory
that I have not entirely learned, but unfortunately the link above stays
almost entirely in territory I have not yet learned. And please: do not
try to tempt me into learning it right now. But I will *definitely* learn
this material sooner or later. (I will try to let my health determine the
best time for all of that. I am probably older than you (and so can be
trusted even less?), and probably took worse care of myself for a longer
amount of that time.)

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>
>>> Yes, though it becomes harder to specify your set of 'target
>>> intervals' -- do you use an 'integer limit'?
>>
>> Why does it become harder to specify the target because a
> restriction is
>> removed? What were the assumptions that led to 94 cents as the
> optimal
>> generator of 94 cents in the 2:1 octave cast?
>
> Hi Kurt, I'm glad you're giving me an excuse to stay up all night.

Well I just hope you're staying healthy, and not involved in substance
abuse. For me the long day yesterday was abusive to my own substance (me),
and I intend not to repeat it too often.

> We
> weighted all 5-(odd)-limit intervals (a.k.a. 5-limit consonances)
> equally, and minimized the sum of their squared errors.
>
>>>> This is actually two questions...
>>>>
>>>> One is when you are restricted to 12et (the term et still applies
>>> with
>>>> inexact octaves, right?).
>>>
>>> Umm . . . I'll follow you on that, but it's best to be specific.
>>
>> I was trying to refer to 12 equally spaced intervals within a
> possibly
>> non-2:1 octave. Do you call this 12et or not? I was calling it
> 12et,
>> tentatively, in the following...
>
> Yes, I was tentatively following this usage . . . though some people
> around here say 12ED2, 12ED2.1, etc.
>
>> What if I were to weight equally all intervals less than an octave
>> wide?
>
> *all* intervals? What would the targets for the small and large steps
> themselves be? Or do you really just mean all 5-limit consonances?

I was making a perhaps very nieve assumption - addmittely this was
unconscious at the time I wrote the above, but I'll state it anyway, and see
if it is not too embarassing. I was thinking of the french romantic
octatonic literature, combined with the use of 12-et-based octatonic, as a
"defacto" choice of important intervals. Therefore I was thinking of every
"important" interval that *occurs* in the actual 12et2.0 octatonic as being
implicitly targeting some interval. I did not bother to figure out what
those intervals were. So then, is there any way to redeem this original
assumption of a defacto choice beyind the octatonic 12et literature and ask
can we do better with a non-2:1 octave?

>>>> The other is when you are not so rectricted - even if it would
>>> create
>>>> asymmetries in relation to the layered octatonic scales that Paul
>>> H. was
>>>> referring to,
>>
>>> It wouldn't, unless you weighted certain *occurrences* of some
>>> intervals more than other occurrences of the same intervals --
> which
>>> seems to be what you're suggesting below.
>>
>> Wait on this point until more ambiguities are clarified.
>>
>>>> and even if it would make one such scale preferable over the
>>>> other two possibilities. Actually I see his layers use only 2 of
>>> the 3
>>>> possible octatonies and I mean to include all 3 in the question.
>>>
>>> What are octatonies? The octatonic scale has 4 major triads and 4
>>> minor triads . . .
>>
>> Typo - meant to be "octatonics" meaning "octatonic scales". If
> there is a
>> 12-note underlying base scale,
>
> Oh! I wasn't assuming that at all. Why would you?

Only a tentative assumption actually. It would be interesting if creating
one optimal octatonic turned out to create 2 other "interesting" octatonics.
Either that or I have to have 3 scales available on my instrument, to assure
constant optimality.

>> These are the "3 octatonics" I was trying to refer to, of which
> Paul H's
>> example used 2, if I understood it correctly.
>
> Oh, I thought it was just 1!

Well indeed, with "xmw" in place modulating to another octatonic could well
preserve optimality. But I wasn't thinking that way, because indeed I am
thinking of already having both hands and both feet busy with the piece I am
playing.

ASIDE: Incidentally, for some reason I am just not interesting in automated
retuning. It goes against the grain. (I exclude returning that is
deterministically and *directly* based on explicit cues from the "automatic"
category.) Not that things wouldn't sound good, but with compositional
desires I also going on in my, I am thinking towards having total choice,
and would rather create an awkward protocol for 2 hands and 2 feet than have
something automatic happen. I can't help that I think this way. Maybe I
will change. Automatic retuning is of academic interest to me. I think I
would learn from it, but I would never try to depend on it, and expect I
would never use it for anything important.

But let's assume anyway that I am interested in that "1" octatonic as a
primary goal, but I am curious about the other 2, and I definitely want to
tie down that remaining degree of freedom because in fact I *will* be using
a 12-tone instrument and the other notes will not be muted, so I want to
give them a pitch!

>> But back to the original assumption: I thought there might still
> be some
>> value in freeing up the relative position fo the 2 diminished
> chords that
>> appear in the octatonic scale.
>
> Yes, absolutely. Again, under one set of assumptions, the 2 diminised
> chords should be 94 cents, instead of 100 cents, apart.

Ok, so with 3 of them then you have 94+94+112, and I could try to place the
112 wisely for the music at hand, in case it happens to have "layers" like
in Paul H.'s example.

>> Or in the more extended question of creating
>> a 12-tone scale out of which 3 different octatonics can be
> extracted, I was
>> wanting to free up the relative positions of the 3 diminished
> chords that
>> appear in the 12-tone scale.
>
> That's harder to come by because then you have to tune the octatonic
> scales differently from one another, which would require some
> assumptions. The simplest is that you're only going to use 2 of the
> octatonic scales, in which case you space the three diminished chords
> so that one lies 94 cents below, one lies 94 cents above a central
> one.

Exactly.

>> So let's focus on the 12-tone super-octatonic scale, whatever it
> should be
>> called.
>
> Hmm . . . it depends what harmonies you assume.

>> I was first asking the question what can happen if you add 1 degree
>> of freedom: the width of the octave which is divided into 12 equal
>> intervals.
>
> OK, what criterion do you want to use?

Let's start with the "defacto" idea above and see if you can come up with an
objective interpretation of that.

>> The 3rd question then was to allow 11 additional degrees of freedom
> for the
>> super-scale (7 for the octatonic). This question was an
> afterthought in my
>> original email because I *thought* I for sure wanted to preserve 4-
> way
>> symmetry within the octave.
>
> Well, if we calculated a COFT for an octatonic music, we might find
> interesting beasts, but since the example in question was
> so 'symmetrical', we're likely to see the 4-way symmetry almost
> perfectly preserved.
>
>> I could easily narrow that to french
>> romantic,
>
> Which composers do you consider french romantic? I didn't know any of
> them used the octatonic scale at all . . .

Yes, well admittely Messiaen organ music tends to "fall out" of my usual
sense of what "french romantic" means. But Messiaen piano music for some
reason does not. (Thanks, JP for the recommendation of Vingt Regards played
by Loriod.) Perhaps the problem is that pianos end up sounding too much
like pianos. (Hey, I have one. ;)

So let's just forget "french romantic" and focus on Messiaen since that is
my top priority anyway. The issue of "layering" probably will come up, and
with any luck it will stick to 2 layers - and I think that may well be the
case - makes you wonder whether Messiaen was *really* thinking in 12et.

-Kurt

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> on 12/3/03 11:45 PM, Paul Erlich <paul@s...> wrote:
>
> > Hey Kurt, look at this:
> >
> > http://www.io.com/~hmiller/music/temp-diminished.html
>
> Yikes, you know, the terminology in your messages goes into some
territory
> that I have not entirely learned, but unfortunately the link above
stays
> almost entirely in territory I have not yet learned. And please:
do not
> try to tempt me into learning it right now. But I will
*definitely* learn
> this material sooner or later.

Great -- everything on Herman's page is pretty basic.

> Therefore I was thinking of every
> "important" interval that *occurs* in the actual 12et2.0 octatonic
as being
> implicitly targeting some interval.

Well, I think maybe the consonances do, but the steps (and sevenths)
are just 'resultants', if you will.

> Either that or I have to have 3 scales available on my instrument,
to assure
> constant optimality.

If all you want is the optimal octatonic at 3 different
transpositions, then 16 notes would do -- 4-equal repeated at 94-94-
94, so that the last 4-equal set is just 18 cents from the first.

> >> These are the "3 octatonics" I was trying to refer to, of which
> > Paul H's
> >> example used 2, if I understood it correctly.
> >
> > Oh, I thought it was just 1!
>
> Well indeed, with "xmw" in place modulating to another octatonic

I meant I thought it was just 1 octatonic scale . . . what's "xmw"?

> ASIDE: Incidentally, for some reason I am just not interesting in
automated
> retuning. It goes against the grain.

OK. COFT doesn't necessarily imply automated retuning, of course.

> But let's assume anyway that I am interested in that "1" octatonic
as a
> primary goal, but I am curious about the other 2, and I definitely
want to
> tie down that remaining degree of freedom because in fact I *will*
be using
> a 12-tone instrument and the other notes will not be muted, so I
want to
> give them a pitch!

If you go 94-94-112, repeated every 300 cents, you will have
2 "optimal" octatonic scales and one "funky" one. How's that?

> Ok, so with 3 of them then you have 94+94+112,

read my mind.

> >> I was first asking the question what can happen if you add 1
degree
> >> of freedom: the width of the octave which is divided into 12
equal
> >> intervals.
> >
> > OK, what criterion do you want to use?
>
> Let's start with the "defacto" idea above and see if you can come
up with an
> objective interpretation of that.

Dunno. Let me do this: I'll equally weight the ratios within an
integer limit of 6, including unreduced fractions as well as reduced
ones (so, for example, the octave gets counted thrice), and minimize
the sum-squared errors. I get that four times the period is 1198.974
cents, and the generator is 97.852 cents.

Gene would probably avoid unreduced fractions and get that four times
the period is 1198.859 cents, and the generator is 96.515 cents. Not
a big difference.

🔗Kurt Bigler <kkb@breathsense.com>

on 12/4/03 1:10 AM, Paul Erlich <paul@stretch-music.com> wrote:

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>> on 12/3/03 11:45 PM, Paul Erlich <paul@s...> wrote:
>>
>>> Hey Kurt, look at this:
>>>
>>> http://www.io.com/~hmiller/music/temp-diminished.html
>>
>> Yikes, you know, the terminology in your messages goes into some
> territory
>> that I have not entirely learned, but unfortunately the link above
> stays
>> almost entirely in territory I have not yet learned. And please:
> do not
>> try to tempt me into learning it right now. But I will
> *definitely* learn
>> this material sooner or later.
>
> Great -- everything on Herman's page is pretty basic.

Good. But not tonight.

>> Therefore I was thinking of every
>> "important" interval that *occurs* in the actual 12et2.0 octatonic
> as being
>> implicitly targeting some interval.
>
> Well, I think maybe the consonances do, but the steps (and sevenths)
> are just 'resultants', if you will.
>
>> Either that or I have to have 3 scales available on my instrument,
> to assure
>> constant optimality.
>
> If all you want is the optimal octatonic at 3 different
> transpositions, then 16 notes would do -- 4-equal repeated at 94-94-
> 94, so that the last 4-equal set is just 18 cents from the first.
>
>>>> These are the "3 octatonics" I was trying to refer to, of which
>>> Paul H's
>>>> example used 2, if I understood it correctly.
>>>
>>> Oh, I thought it was just 1!
>>
>> Well indeed, with "xmw" in place modulating to another octatonic
>
> I meant I thought it was just 1 octatonic scale . . .

That's what I thought you meant.

> what's "xmw"?

Xenharmonic Moving Windows. I thought that was something Carl had discussed
on this list once. Its a spec for how explicit retuning can be controlled
by assining a midi channel as a control channel - to control modulation.
This would allow moving the same set of octatonic intervals to be rooted at
a different position on the keyboard when modulating to a different
octatonic scale (probably with a diminished chord in common).

>> ASIDE: Incidentally, for some reason I am just not interesting in
> automated
>> retuning. It goes against the grain.
>
> OK. COFT doesn't necessarily imply automated retuning, of course.

Yes, I got that it didn't. I was just answering an antipated suggestion
that software could determine on the fly which octatonic scale was active
and make the necessary adjustments. Actually for such a limited purpose
heuristic retuning might be highly reliable.

>> But let's assume anyway that I am interested in that "1" octatonic
> as a
>> primary goal, but I am curious about the other 2, and I definitely
> want to
>> tie down that remaining degree of freedom because in fact I *will*
> be using
>> a 12-tone instrument and the other notes will not be muted, so I
> want to
>> give them a pitch!
>
> If you go 94-94-112, repeated every 300 cents, you will have
> 2 "optimal" octatonic scales and one "funky" one. How's that?

Yes, that's fine for 1200 octave. So on to the other questions.

>> Ok, so with 3 of them then you have 94+94+112,
>
> read my mind.
>
>>>> I was first asking the question what can happen if you add 1
> degree
>>>> of freedom: the width of the octave which is divided into 12
> equal
>>>> intervals.
>>>
>>> OK, what criterion do you want to use?
>>
>> Let's start with the "defacto" idea above and see if you can come
> up with an
>> objective interpretation of that.
>
> Dunno. Let me do this: I'll equally weight the ratios within an
> integer limit of 6, including unreduced fractions as well as reduced
> ones (so, for example, the octave gets counted thrice)

don't follow that. You're making a list of intervals, not a set, so that
some intervals recur, and if the recur you weight them by # of occurrences?

> , and minimize
> the sum-squared errors. I get that four times the period is 1198.974
> cents, and the generator is 97.852 cents.

Interesting! So this reduces the discrepancy between the 3 octatonics in a
12-tone super-scale. I suppose that might have been expected intuitively.
But I didn't guess the generator would increase from 94 to 98. That's most
of the way home toward an edo.

> Gene would probably avoid unreduced fractions and get that four times
> the period is 1198.859 cents, and the generator is 96.515 cents. Not
> a big difference.

But it closes the gap a lot less (between the 3 octatonics).

And are you including any intervals greater than an octave, out of
curiosity? I'm assuming not.

Still, there is the question of the optimal 12edo-non-2.0 based octatonic
set (all 3 octatonic subsets identical). But I'm guessing the answer would
be near an octave near 1199.

-Kurt

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> > Dunno. Let me do this: I'll equally weight the ratios within an
> > integer limit of 6, including unreduced fractions as well as
reduced
> > ones (so, for example, the octave gets counted thrice)
>
> don't follow that. You're making a list of intervals, not a set,

Why "not a set"?

> so that
> some intervals recur, and if the recur you weight them by # of
>occurrences?

Yes.

> And are you including any intervals greater than an octave, out of
> curiosity? I'm assuming not.

Yes, I was! As large as 6:1. French Romantics use plenty such
intervals :)

> Still, there is the question of the optimal 12edo-non-2.0 based
octatonic
> set (all 3 octatonic subsets identical). But I'm guessing the
answer would
> be near an octave near 1199.

Oh yeah. The results are (again keeping unreduced fractions)

Integer Limit 'Octave'
3................1201.101
4................1200.379
5................1196.273
6................1198.187
7................1193.252
8................1195.371

Now assuming at least 12 notes it makes sense to say

9................1197.273
10...............1196.582

If we the assume the 'tritone' can represent 11:8, etc., we get

11...............1192.580
12...............1194.402

However if we assume instead the 'fourth' can represent 11:8, etc.,
we get

11...............1202.487
12...............1202.370

🔗Kurt Bigler <kkb@breathsense.com>

on 12/4/03 1:55 AM, Paul Erlich <paul@stretch-music.com> wrote:

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>
>>> Dunno. Let me do this: I'll equally weight the ratios within an
>>> integer limit of 6, including unreduced fractions as well as
> reduced
>>> ones (so, for example, the octave gets counted thrice)
>>
>> don't follow that. You're making a list of intervals, not a set,
>
> Why "not a set"?

Technicality in any case, but I thought I was taught once that a "set"
doesn't have the same element twice. But I may be confusing this with
another term.

>
>> so that
>> some intervals recur, and if the recur you weight them by # of
>> occurrences?
>
> Yes.
>
>> And are you including any intervals greater than an octave, out of
>> curiosity? I'm assuming not.
>
> Yes, I was! As large as 6:1. French Romantics use plenty such
> intervals :)

I just thought it would create too much ambiguity. For example if you are
optimizing 3:2, 3:1, and 6:1 you get a kind of over-determining - any octave
size will work well for one of them, so to speak. It would be interesting
to see the same results with the errors also shown, and then repeat with all
the same intervals represented and weighted the same, but folded to less
than one octave. The optimized errors would no doubt be a lot less in the
folded case. If the errors are great enough in the non-folded case, it
makes the aesthetic meaning of the result somewhat suspect, doesn't it?

>> Still, there is the question of the optimal 12edo-non-2.0 based
> octatonic
>> set (all 3 octatonic subsets identical). But I'm guessing the
> answer would
>> be near an octave near 1199.
>
> Oh yeah. The results are (again keeping unreduced fractions)
>
> Integer Limit 'Octave'
> 3................1201.101
> 4................1200.379
> 5................1196.273
> 6................1198.187
> 7................1193.252
> 8................1195.371
>
> Now assuming at least 12 notes it makes sense to say
>
> 9................1197.273
> 10...............1196.582
>
> If we the assume the 'tritone' can represent 11:8, etc., we get
>
> 11...............1192.580
> 12...............1194.402
>
> However if we assume instead the 'fourth' can represent 11:8, etc.,
> we get
>
> 11...............1202.487
> 12...............1202.370

Well overall it does not lead me to a confident single choice. Need to try
a lot of different things.

This is the first time I've looked at results like this though.

It makes me think RMS error may be the wrong criterion. You want to address
the (proposed) fact that some good consonances will make up for some
less-good consonances. This is especially true with non-2:1 octave
involved, I think. Based on having used Alaska 1 a lot with Bach a lot the
flattenned octaves (1197) are easily forgiven, the important consonances
stand out in a piece, the reduced consonances fall into the background of
"normalcy", and the true dissonances remain occasional enough to fall where
they would be expected for a well-temperament. With multiple flat octaves
present, I'm wondering whether the ear won't hear the extremely consonant
ones as increased consonance while the simultaneously slightly off-just ones
just get ignored. I know I'm not being terribly clear here, but maybe you
can get the drift, especially if you get a glimmer of a reason why RMS error
*might* be missing an important piece of the truth.

Consider this hypothesis: a note and its (not exactly 2:1) octave with a
rich timbre spectrum, occurring together and also with a 3rd non-octave note
which is very close to a clear JI consonance with one of the two octaves. I
think the ear will hear the primary consonance and the less strong
consonance won't drown it out. This *might* even depend on the listening
head being moveable and the 3 notes emanating from separate speakers. Mind
you this is the scenario I am most used to hearing. Something to try
(tomorrow) - but I think I've already experienced this.

The tweak that would be needed compared to RMS error might be as simple as
patching together a quadratic curve with a linear one that kicks in for
values under some particular fraction of a cent threshold, or simply
removing a slice from the middle of the parabola. In all cases the sign of
the error is removed before accumulating. Perhaps take the absolute value,
subtract the "threshold" value, and then square. The result will be more
linear for small values and won't flatten out at zero, but will have a
point.

Or else an entirely different kind of analysis might be applied to intervals
involving the factor 2. Just thinking out loud now.

-Kurt

🔗Kurt Bigler <kkb@breathsense.com>

correction to my previous post...

on 12/4/03 2:32 AM, Kurt Bigler <kkb@breathsense.com> wrote:

> on 12/4/03 1:55 AM, Paul Erlich <paul@stretch-music.com> wrote:
>
> It makes me think RMS error may be the wrong criterion. You want to address
> the (proposed) fact that some good consonances will make up for some less-good
> consonances. This is especially true with non-2:1 octave involved, I think.
> Based on having used Alaska 1 a lot with Bach a lot the flattenned octaves
> (1197) are easily forgiven, the important consonances stand out in a piece,
> the reduced consonances fall into the background of "normalcy", and the true
> dissonances remain occasional enough to fall where they would be expected for
> a well-temperament. With multiple flat octaves present, I'm wondering whether
> the ear won't hear the extremely consonant ones as increased consonance while
> the simultaneously slightly off-just ones just get ignored. I know I'm not
> being terribly clear here, but maybe you can get the drift, especially if you
> get a glimmer of a reason why RMS error *might* be missing an important piece
> of the truth.
>
> Consider this hypothesis: a note and its (not exactly 2:1) octave with a rich
> timbre spectrum, occurring together and also with a 3rd non-octave note which
> is very close to a clear JI consonance with one of the two octaves. I think
> the ear will hear the primary consonance and the less strong consonance won't
> drown it out. This *might* even depend on the listening head being moveable
> and the 3 notes emanating from separate speakers. Mind you this is the
> scenario I am most used to hearing. Something to try (tomorrow) - but I think
> I've already experienced this.
>
> The tweak that would be needed compared to RMS error might be as simple as
> patching together a quadratic curve with a linear one that kicks in for values
> under some particular fraction of a cent threshold, or simply removing a slice
> from the middle of the parabola. In all cases the sign of the error is
> removed before accumulating. Perhaps take the absolute value, subtract the
> "threshold" value, and then square.

I should have said "add" not "substract".

(|x| + thresh) ^ 2

and "thresh" is not *exactly* a threshold, but it is a value related to the
point near which linear behavior transitions to quadratic behavior.

> The result will be more linear for small
> values and won't flatten out at zero, but will have a point.

🔗Kurt Bigler <kkb@breathsense.com>

clarification to my previous 2 posts...

on 12/4/03 2:41 AM, Kurt Bigler <kkb@breathsense.com> wrote:

> correction to my previous post...
>
>
> on 12/4/03 2:32 AM, Kurt Bigler <kkb@breathsense.com> wrote:
>
>> The tweak that would be needed compared to RMS error might be as simple as
>> patching together a quadratic curve with a linear one that kicks in for
>> values
>> under some particular fraction of a cent threshold, or simply removing a
>> slice
>> from the middle of the parabola. In all cases the sign of the error is
>> removed before accumulating. Perhaps take the absolute value, subtract the
>> "threshold" value, and then square.
>
> I should have said "add" not "substract".
>
> (|x| + thresh) ^ 2
>
> and "thresh" is not *exactly* a threshold, but it is a value related to the
> point near which linear behavior transitions to quadratic behavior.
>
>> The result will be more linear for small
>> values and won't flatten out at zero, but will have a point.

I could have done a bit better at explaining the point. Whether the formula
I suggested achieves this is less important.

My point was that the normal RMS method flattens out at the bottom of the
curve. In doing so it says that small errors are less important than large
errors, less important even than their "size" would indicate. Thus in
effect, tiny errors disappear, and large errors are magnified. I think this
is good above a certain size error. But it may be that when thinking in
terms of consonance (the reverse criterion of error) that consonances of a
certain goodness (below a certain error) are "extra good" and count for more
consonance. This is the *opposite* of what squaring the error achieves. I
may be wrong, but that was my point.

But for this to account for the behavior I was suggesting regarding non-2:1
octaves, the "threshold" would have to be as large as a cent or two, which
does not seem right. So I'm also looking for ideas about pre-reducing
octave-related intervals prior to computing an RMS error. Again, this may
be bologna, but at least I wanted to make it clearer bologna, to avoid
unnecessary rounds of clarification with Paul.

-Kurt

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Carl Lumma <ekin@lumma.org>

>It makes me think RMS error may be the wrong criterion. You want to
>address the (proposed) fact that some good consonances will make up
>for some less-good consonances.

RMS does that, of course. I haven't been following this thread but
I don't think Paul is allowing circulating temperaments here, which
is maybe what you meant.

>Based on having used Alaska 1 a lot with Bach a lot the flattenned
>octaves (1197) are easily forgiven, the important consonances stand
>out in a piece, the reduced consonances fall into the background of
>"normalcy",

This is often put forth by the well-temperament crowd. However, I
think they're just not playing in the bad keys. Take Werckmeister 3
and play something in F# and get back to me.

>and the true dissonances remain occasional enough to fall where
>they would be expected for a well-temperament.

They're aren't any true discordances in a well-temperament; by
definition all the intervals are supposed to be useable.

I simply can't tolerate anything worse than et, though, which is
why for years I had no interest in wt's (because the music I'm
interested in is all over the place). Alaska 5 is the first wt
I've found that I'm interested in. The good keys aren't clumped
at one end of the circle of fifths, and the bad keys aren't any
worse than 12-tET.

>especially if you get a glimmer of a reason why RMS error
>*might* be missing an important piece of the truth.

I've subjected various error functions to many listening tests
over the years, and RMS always came out on top.

>Consider this hypothesis: a note and its (not exactly 2:1) octave with
>a rich timbre spectrum, occurring together and also with a 3rd non-octave
>note which is very close to a clear JI consonance with one of the two
>octaves. I think the ear will hear the primary consonance and the less
>strong consonance won't drown it out.

Interestingly, this came up re. harmonic entropy. As you mistune one
note of a tetrad, when does the ear give up on the tetrad and hear a
triad with a noise component? On the harmonic_entropy list I've
suggested 'hierarchical harmonic entropy' to address this.

But the present issue has to do finding the optimal tuning for a known
target, not with identifying the target....

-Carl

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Carl Lumma <ekin@lumma.org>

🔗Kurt Bigler <kkb@breathsense.com>

on 12/4/03 10:59 AM, Carl Lumma <ekin@lumma.org> wrote:

>> It makes me think RMS error may be the wrong criterion. You want to
>> address the (proposed) fact that some good consonances will make up
>> for some less-good consonances.
>
> RMS does that, of course.

Yes, that's why I wrote my clarification message which was more explicit
about what RMS does not do.

> I haven't been following this thread but
> I don't think Paul is allowing circulating temperaments here, which
> is maybe what you meant.

No I was just talking about flattenned octaves, and one particular instance
of flattenned octaves. Octatonic is already better than a well-tempered (at
what it does) with 2:1 octaves. Flatting octatonic octaves should be "no
worse" than flattening well-tempered octaves, crudely speaking.

>> Based on having used Alaska 1 a lot with Bach a lot the flattenned
>> octaves (1197) are easily forgiven, the important consonances stand
>> out in a piece, the reduced consonances fall into the background of
>> "normalcy",
>
> This is often put forth by the well-temperament crowd. However, I
> think they're just not playing in the bad keys. Take Werckmeister 3
> and play something in F# and get back to me.

I've done all this. I find the "strain" happenns in the same places in all
well-temperaments, including Alaska 1. I can apparently tolerate things you
can't and vice-versa.

>> and the true dissonances remain occasional enough to fall where
>> they would be expected for a well-temperament.
>
> They're aren't any true discordances in a well-temperament; by
> definition all the intervals are supposed to be useable.

Useable schmoozable. ;)

> I simply can't tolerate anything worse than et,

There you go!

But Alasksa 1 has 8 major thirds that are 33% worse than et, while 4 are
much better.

> though, which is
> why for years I had no interest in wt's
>
>> especially if you get a glimmer of a reason why RMS error
>> *might* be missing an important piece of the truth.
>
> I've subjected various error functions to many listening tests
> over the years, and RMS always came out on top.

Check my clarification message and tell me whether you tried a function that
addresses this. Trying random error functions without specific goals in
mind may not lead anywhere.

>> Consider this hypothesis: a note and its (not exactly 2:1) octave with
>> a rich timbre spectrum, occurring together and also with a 3rd non-octave
>> note which is very close to a clear JI consonance with one of the two
>> octaves. I think the ear will hear the primary consonance and the less
>> strong consonance won't drown it out.
>
> Interestingly, this came up re. harmonic entropy. As you mistune one
> note of a tetrad, when does the ear give up on the tetrad and hear a
> triad with a noise component? On the harmonic_entropy list I've
> suggested 'hierarchical harmonic entropy' to address this.
>
> But the present issue has to do finding the optimal tuning for a known
> target, not with identifying the target....

Paul has made it into that though, in spite of the lack of specific input
from me regarding targets.

-Kurt

>
> -Carl

🔗Carl Lumma <ekin@lumma.org>

>> I haven't been following this thread but
>> I don't think Paul is allowing circulating temperaments here, which
>> is maybe what you meant.
>
>No I was just talking about flattenned octaves, and one particular
>instance of flattenned octaves.

Then I'm confused. What is the case you're thinking of, where some
good consonances make up for other bad ones?

>Octatonic is already better than a well-tempered (at what it does)
>with 2:1 octaves. Flatting octatonic octaves should be "no
>worse" than flattening well-tempered octaves, crudely speaking.

Keep in mind that around here, "well tempered" just means a
circulating temperament. So there is such a thing as well-tempered
octatonic scales.

>>> Based on having used Alaska 1 a lot with Bach a lot the flattenned
>>> octaves (1197) are easily forgiven, the important consonances stand
>>> out in a piece, the reduced consonances fall into the background of
>>> "normalcy",
>>
>> This is often put forth by the well-temperament crowd. However, I
>> think they're just not playing in the bad keys. Take Werckmeister 3
>> and play something in F# and get back to me.
>
>I've done all this. I find the "strain" happenns in the same places
>in all well-temperaments, including Alaska 1. I can apparently
>tolerate things you can't and vice-versa.

The strain?

>> I simply can't tolerate anything worse than et,
>
>There you go!
>
>But Alasksa 1 has 8 major thirds that are 33% worse than et, while
>4 are much better.

And yes, Alaska 1 was designed to be like historical Bach-era
temperaments, which I generally don't like for modern music. But
the 8 bad thirds of Alaska 1 are much better than in those
temperaments.

How are you getting 33%? The bad major thirds are 3.5 cents
sharp of 12-tET, and the bad major 10ths are only .5 cents sharp
of 12-tET.

>Check my clarification message and tell me whether you tried a
>function that addresses this. Trying random error functions without
>specific goals in mind may not lead anywhere.

I did see the other message, but I'm not sure if any of:

sum-of-squares
arithmatic mean
geometric mean
max error

address your conern.

>>> Consider this hypothesis: a note and its (not exactly 2:1) octave with
>>> a rich timbre spectrum, occurring together and also with a 3rd non-octave
>>> note which is very close to a clear JI consonance with one of the two
>>> octaves. I think the ear will hear the primary consonance and the less
>>> strong consonance won't drown it out.
>>
>> Interestingly, this came up re. harmonic entropy. As you mistune one
>> note of a tetrad, when does the ear give up on the tetrad and hear a
>> triad with a noise component? On the harmonic_entropy list I've
>> suggested 'hierarchical harmonic entropy' to address this.
>>
>> But the present issue has to do finding the optimal tuning for a known
>> target, not with identifying the target....
>
>Paul has made it into that though, in spite of the lack of specific input
>from me regarding targets.

Sorry, I lost you here. Maybe it's not important.

-Carl

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> on 12/4/03 1:55 AM, Paul Erlich <paul@s...> wrote:
>
> > --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> >
> >>> Dunno. Let me do this: I'll equally weight the ratios within an
> >>> integer limit of 6, including unreduced fractions as well as
> > reduced
> >>> ones (so, for example, the octave gets counted thrice)
> >>
> >> don't follow that. You're making a list of intervals, not a set,
> >
> > Why "not a set"?
>
> Technicality in any case, but I thought I was taught once that
a "set"
> doesn't have the same element twice. But I may be confusing this
with
> another term.

Oh, I get it. So it's not a set of intervals, but it *is* a set of
pairs, which are then interpreted as intervals, some of which turn
out to be identical. Whew!

> I just thought it would create too much ambiguity. For example if
you are
> optimizing 3:2, 3:1, and 6:1 you get a kind of over-determining -
any octave
> size will work well for one of them, so to speak.

Yes, but you find one that works best overall, which I think is
pretty necessary.

> It would be interesting
> to see the same results with the errors also shown,

For now, I'll leave that as an exercise for the reader.

> and then repeat with all
> the same intervals represented and weighted the same, but folded to
less
> than one octave.

Then the optimum starts to look like the octave-equivalent one.

> The optimized errors would no doubt be a lot less in the
> folded case.

A lot less?

> If the errors are great enough in the non-folded case, it
> makes the aesthetic meaning of the result somewhat suspect, doesn't
>it?

Yes, but folding won't really alleviate that. You have to judge for
yourself if 7-odd-limit harmonies, for example, are being well-
represented or not -- if not, then there's really little that can be
done about that without ruining something else even more.

> >> Still, there is the question of the optimal 12edo-non-2.0 based
> > octatonic
> >> set (all 3 octatonic subsets identical). But I'm guessing the
> > answer would
> >> be near an octave near 1199.
> >
> > Oh yeah. The results are (again keeping unreduced fractions)
> >
> > Integer Limit 'Octave'
> > 3................1201.101
> > 4................1200.379
> > 5................1196.273
> > 6................1198.187
> > 7................1193.252
> > 8................1195.371
> >
> > Now assuming at least 12 notes it makes sense to say
> >
> > 9................1197.273
> > 10...............1196.582
> >
> > If we the assume the 'tritone' can represent 11:8, etc., we get
> >
> > 11...............1192.580
> > 12...............1194.402
> >
> > However if we assume instead the 'fourth' can represent 11:8,
etc.,
> > we get
> >
> > 11...............1202.487
> > 12...............1202.370
>
> Well overall it does not lead me to a confident single choice.
Need to try
> a lot of different things.
>
> This is the first time I've looked at results like this though.
>
> It makes me think RMS error may be the wrong criterion. You want
to address
> the (proposed) fact that some good consonances will make up for some
> less-good consonances.

That's exactly what RMS does, while the most popular method (used for
example by George Secor and Graham Breed), minimax, only looks at the
worst consonances.

> This is especially true with non-2:1 octave
> involved, I think. Based on having used Alaska 1

Do you happen to know how that was derived?

> Consider this hypothesis: a note and its (not exactly 2:1) octave
with a
> rich timbre spectrum, occurring together and also with a 3rd non-
octave note
> which is very close to a clear JI consonance with one of the two
octaves. I
> think the ear will hear the primary consonance and the less strong
> consonance won't drown it out.

Yes, but compromising both consonances equally sounds even better,
doesn't it? If not, you might be suggesting a MAD (mean absolute
deviation) criterion, or something even more "pointy" where p<1. If
that's what your ears tell you, I'll be happy to oblige. For some
reason, Gene Ward Smith and Dave Keenan were willing to consider the
RMS (p=2) case, the minimax (p=infinity) case, and everything in-
between, but wouldn't consider p<2. But perhaps even p<1 is important
for some listeners -- I wouldn't rule it out.

What does p mean? The error criterion being minimized is

(sum(error^p))^(1/p).

> The tweak that would be needed compared to RMS error might be as
simple as
> patching together a quadratic curve with a linear one that kicks in
for
> values under some particular fraction of a cent threshold, or simply
> removing a slice from the middle of the parabola.

And pasting in a wedge?

> In all cases the sign of
> the error is removed before accumulating. Perhaps take the
absolute value,
> subtract the "threshold" value, and then square. The result will
be more
> linear for small values and won't flatten out at zero, but will
have a
> point.

Sounds complex. How about just MAD?

🔗Paul Erlich <paul@stretch-music.com>

🔗Carl Lumma <ekin@lumma.org>

🔗Kurt Bigler <kkb@breathsense.com>

on 12/4/03 2:47 PM, Paul Erlich <paul@stretch-music.com> wrote:

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

>> I just thought it would create too much ambiguity. For example if
> you are
>> optimizing 3:2, 3:1, and 6:1 you get a kind of over-determining -
> any octave
>> size will work well for one of them, so to speak.
>
> Yes, but you find one that works best overall, which I think is
> pretty necessary.

I was thinking that in the case of 3:2, 3:1, and 6:1 RMS is just going to
optimize the middle one of the 3, pushed to the middle by the "pressure"
from the outer 2 octaves (3:2 and 6:1). All 3 track together linearly,
locked by the octave ratio.

But maybe you are still right about "best overall".

>> and then repeat with all
>> the same intervals represented and weighted the same, but folded to
> less
>> than one octave.
>
> Then the optimum starts to look like the octave-equivalent one.

It is not only the octave that is changed by the change in the octave.

>> The optimized errors would no doubt be a lot less in the
>> folded case.
>
> A lot less?

Well, I'm probably not going to do the exercise this week to find out. My
thought was that if you triple the number of data points by spanning 3
octaves, and make it impossible for more than 1/3 of the data points to be
"good" relative to any given interval, then I figure that greatly increases
the error. Of course it might also make it possible to "bring in" some
intervals that were out of reach, so maybe I was wrong.

>> If the errors are great enough in the non-folded case, it
>> makes the aesthetic meaning of the result somewhat suspect, doesn't
>> it?
>
> Yes, but folding won't really alleviate that. You have to judge for
> yourself if 7-odd-limit harmonies, for example, are being well-
> represented or not -- if not, then there's really little that can be
> done about that without ruining something else even more.
>
>>>> Still, there is the question of the optimal 12edo-non-2.0 based
>>> octatonic
>>>> set (all 3 octatonic subsets identical). But I'm guessing the
>>> answer would
>>>> be near an octave near 1199.
>>>
>>> Oh yeah. The results are (again keeping unreduced fractions)
>>>
>>> Integer Limit 'Octave'
>>> 3................1201.101
>>> 4................1200.379
>>> 5................1196.273
>>> 6................1198.187
>>> 7................1193.252
>>> 8................1195.371
>>>
>>> Now assuming at least 12 notes it makes sense to say
>>>
>>> 9................1197.273
>>> 10...............1196.582
>>>
>>> If we the assume the 'tritone' can represent 11:8, etc., we get
>>>
>>> 11...............1192.580
>>> 12...............1194.402
>>>
>>> However if we assume instead the 'fourth' can represent 11:8,
> etc.,
>>> we get
>>>
>>> 11...............1202.487
>>> 12...............1202.370
>>
>> Well overall it does not lead me to a confident single choice.
> Need to try
>> a lot of different things.
>>
>> This is the first time I've looked at results like this though.
>>
>> It makes me think RMS error may be the wrong criterion. You want
> to address
>> the (proposed) fact that some good consonances will make up for some
>> less-good consonances.
>
> That's exactly what RMS does, while the most popular method (used for
> example by George Secor and Graham Breed), minimax, only looks at the
> worst consonances.

I'll think more about this before I say more. My suggestion was to reshape
the curve a little from the square, retain square at the extremes. The
reason was that I thought that 2 different phenomena might operate in
different regions of error.

>> This is especially true with non-2:1 octave
>> involved, I think. Based on having used Alaska 1
>
> Do you happen to know how that was derived?

I think Gene did it with input from Carl. It has 2 sizes of 3rds, 4 much
better than et (at C, G, D, and A) and 8 a little worse. The octave is
1197.

>> Consider this hypothesis: a note and its (not exactly 2:1) octave
> with a
>> rich timbre spectrum, occurring together and also with a 3rd non-
> octave note
>> which is very close to a clear JI consonance with one of the two
> octaves. I
>> think the ear will hear the primary consonance and the less strong
>> consonance won't drown it out.
>
> Yes, but compromising both consonances equally sounds even better,
> doesn't it?

Not based on my experience with certain consonances in Alaska 1.

> If not, you might be suggesting a MAD (mean absolute
> deviation) criterion, or something even more "pointy" where p<1. If
> that's what your ears tell you, I'll be happy to oblige. For some
> reason, Gene Ward Smith and Dave Keenan were willing to consider the
> RMS (p=2) case, the minimax (p=infinity) case, and everything in-
> between, but wouldn't consider p<2. But perhaps even p<1 is important
> for some listeners -- I wouldn't rule it out.
>
> What does p mean? The error criterion being minimized is
>
> (sum(error^p))^(1/p).
>
>> The tweak that would be needed compared to RMS error might be as
> simple as
>> patching together a quadratic curve with a linear one that kicks in
> for
>> values under some particular fraction of a cent threshold, or simply
>> removing a slice from the middle of the parabola.
>
> And pasting in a wedge?

No, here I mean removing the slice and slamming the remaining curve
fragments back together to fill the gap.

Basically I want to to use a different p for small errors. Removing the
slice in effect creates this result without a discontinuity except at 0. It
looks like p=1 (MAD) near 0 and like p=2 (RMS) for large values.

>> In all cases the sign of
>> the error is removed before accumulating. Perhaps take the
> absolute value,
>> subtract the "threshold" value, and then square. The result will
> be more
>> linear for small values and won't flatten out at zero, but will
> have a
>> point.
>
> Sounds complex. How about just MAD?

I suspected everybody can not be wrong about RMS, and that RMS probably
treats large errors correctly. However maybe there also comes a point where
larger does not matter because you will just forget that interval. But that
is probably best reflected by simply refining the target. On the other hand
in a sufficiently complex scenario it might be useful to have an algorithm
that drops the worst data points automatically, so that the "optimized"
result yields possibly interesting things to try, for reasons that could not
have been anticipated.

-Kurt

🔗Kurt Bigler <kkb@breathsense.com>

on 12/4/03 2:13 PM, Carl Lumma <ekin@lumma.org> wrote:

>>> I haven't been following this thread but
>>> I don't think Paul is allowing circulating temperaments here, which
>>> is maybe what you meant.
>>
>> No I was just talking about flattenned octaves, and one particular
>> instance of flattenned octaves.
>
> Then I'm confused. What is the case you're thinking of, where some
> good consonances make up for other bad ones?

I described such a case already. A very sweet consonance in Alaska 1 that
is not disturbed by one of the participating notes appearing in several
different octaves. I will get more specific data on this and report back.
>
>> Octatonic is already better than a well-tempered (at what it does)
>> with 2:1 octaves. Flatting octatonic octaves should be "no
>> worse" than flattening well-tempered octaves, crudely speaking.
>
> Keep in mind that around here, "well tempered" just means a
> circulating temperament. So there is such a thing as well-tempered
> octatonic scales.

Yes, it is questionable how much circulating would be necessary. And the
most general case has not been addressed yet, I don't think.

>>>> Based on having used Alaska 1 a lot with Bach a lot the flattenned
>>>> octaves (1197) are easily forgiven, the important consonances stand
>>>> out in a piece, the reduced consonances fall into the background of
>>>> "normalcy",
>>>
>>> This is often put forth by the well-temperament crowd. However, I
>>> think they're just not playing in the bad keys. Take Werckmeister 3
>>> and play something in F# and get back to me.
>>
>> I've done all this. I find the "strain" happenns in the same places
>> in all well-temperaments, including Alaska 1. I can apparently
>> tolerate things you can't and vice-versa.
>
> The strain?

The discomfort of hearing intervals that are a little "bad".

>>> I simply can't tolerate anything worse than et,
>>
>> There you go!
>>
>> But Alasksa 1 has 8 major thirds that are 33% worse than et, while
>> 4 are much better.
>
> And yes, Alaska 1 was designed to be like historical Bach-era
> temperaments, which I generally don't like for modern music. But
> the 8 bad thirds of Alaska 1 are much better than in those
> temperaments.
>
> How are you getting 33%? The bad major thirds are 3.5 cents
> sharp of 12-tET, and the bad major 10ths are only .5 cents sharp
> of 12-tET.

Maybe I remember wrong. The percentage was relative to the amount that 12et
thirds are off from 5:4.

>> Check my clarification message and tell me whether you tried a
>> function that addresses this. Trying random error functions without
>> specific goals in mind may not lead anywhere.
>
> I did see the other message, but I'm not sure if any of:
>
> sum-of-squares
> arithmatic mean
> geometric mean
> max error
>
> address your conern.

Not off-hand. I've already zeroed in a little more on what I think I want,
described elsewhere in some details. But more thought is needed, and I'll
hold off on this for the moment.

-Kurt

🔗Carl Lumma <ekin@lumma.org>

>>> This is especially true with non-2:1 octave
>>> involved, I think. Based on having used Alaska 1
>>
>> Do you happen to know how that was derived?
>
>I think Gene did it with input from Carl.

It's actually the other way round. Gene wrote...

>I was surprised at how much octave flattening one can get away with.
>The above example flattens by 9 cents; in comparison, the Riemann Zeta
>tuning I proposed some time ago flattens by a mere 2.314 cents. Anyway
>it seems clear that all of the people who want to sharpen octaves to
>brighten things up are indeed headed in the wrong direction if we are
>talking about common practice music.
//
>For 7-limit, at least, it appears 12-et wants to have flattened
>octaves. Aside from the zeta function octave I mentioned, we might
>note that diminished, augmented, pajara, injera, tripletone, meantone,
>schismic, diaschismic and duodecimal all are 7-limit temperaments
>covered by [12,19,28,34]. The following tabulates the octave for the
>canonical map of all of these temperaments:
>
>dominant seventh 1190.496
>diminished 1200
>augmented 1200
>pajara 1198.222
>injera 1200
>tripletone 1195.496
>meantone 1200
>schismic 1199.848
>diaschismic 1198.998
>duodecimal 1200
//
>> > I assumed Gene was finding the optimal generators for
>> > linear temperaments compatible with common-practice
>> > music. That is, 'including the 2s', as we discussed
>> > a while back.
>>
>> I wasn't, actually. However, I think I will now do this.
>
>For the dominant seventh temperament, the results are
>disconcertingly varied:
>
>even 7-limit minimax [1190.496, 493.917]
>(This is the canonical map.)
>
>even 7-limit rms [1192.064, 494.731]
>
>even 8-limit rms [1194.349, 496.068]
>
>even 8-limit minimax [1200, 497.085]

I then built some tools in scheme to check out circulating versions
of flattened-octave temperaments, and both Gene and I used Scala to
create audio examples.

With two sizes of fifth and 12 tones, the Alaska series of tunings
pretty much covers all the points you'd be interested in. All of
it was documented here, Paul, this past June.

Finally, Gene contributed two versions of Alaska (baked and fried)
with synchronous beating. Maybe that's what you're remembering,
Kurt?

-Carl

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> > That's exactly what RMS does, while the most popular method (used
for
> > example by George Secor and Graham Breed), minimax, only looks at
the
> > worst consonances.
>
> I'll think more about this before I say more. My suggestion was to
reshape
> the curve a little from the square, retain square at the extremes.

Square? You said parabola, which made sense . . . (?)

> The
> reason was that I thought that 2 different phenomena might operate
in
> different regions of error.

Excellent

> >> Consider this hypothesis: a note and its (not exactly 2:1)
octave
> > with a
> >> rich timbre spectrum, occurring together and also with a 3rd non-
> > octave note
> >> which is very close to a clear JI consonance with one of the two
> > octaves. I
> >> think the ear will hear the primary consonance and the less
strong
> >> consonance won't drown it out.
> >
> > Yes, but compromising both consonances equally sounds even better,
> > doesn't it?
>
> Not based on my experience with certain consonances in Alaska 1.

Well then your formula isn't going to do it. You need p<1,
essentially. For example, exp(-|error|).

> >> In all cases the sign of
> >> the error is removed before accumulating. Perhaps take the
> > absolute value,
> >> subtract the "threshold" value, and then square. The result will
> > be more
> >> linear for small values and won't flatten out at zero, but will
> > have a
> >> point.
> >
> > Sounds complex. How about just MAD?
>
> I suspected everybody can not be wrong about RMS, and that RMS
probably
> treats large errors correctly.

Well, lots of people seem to thing RMS is wrong and minimax
(p=infinity) is right.

> But that
> is probably best reflected by simply refining the target. On the
>other hand
> in a sufficiently complex scenario it might be useful to have an
>algorithm
> that drops the worst data points automatically, so that
>the "optimized"
> result yields possibly interesting things to try, for reasons that
>could not
> have been anticipated.

Sure.

🔗Kurt Bigler <kkb@breathsense.com>

on 12/4/03 9:08 PM, Paul Erlich <paul@stretch-music.com> wrote:

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>
>>> That's exactly what RMS does, while the most popular method (used
> for
>>> example by George Secor and Graham Breed), minimax, only looks at
> the
>>> worst consonances.
>>
>> I'll think more about this before I say more. My suggestion was to
> reshape
>> the curve a little from the square, retain square at the extremes.
>
> Square? You said parabola, which made sense . . . (?)

Square (p = 2). Squared. Parabola.

>> The
>> reason was that I thought that 2 different phenomena might operate
> in
>> different regions of error.
>
> Excellent

Say more. You resonate with this idea? In any case there is plenty of
oppportunity for testing it. I'm looking for a formula with an additional
constant that can be the threshold between two separate zones of shaping.
The outer zone may well still be RMS. There is too much good experience
with RMS to throw it out. It may just be that if an additional phenomenon
was not recognized that there was no motivation to try more complex
functions. Now if we can clarify this, maybe we can come up with something
worth testing.

>>>> Consider this hypothesis: a note and its (not exactly 2:1)
> octave
>>> with a
>>>> rich timbre spectrum, occurring together and also with a 3rd non-
>>> octave note
>>>> which is very close to a clear JI consonance with one of the two
>>> octaves. I
>>>> think the ear will hear the primary consonance and the less
> strong
>>>> consonance won't drown it out.
>>>
>>> Yes, but compromising both consonances equally sounds even better,
>>> doesn't it?
>>
>> Not based on my experience with certain consonances in Alaska 1.
>
> Well then your formula isn't going to do it. You need p<1,
> essentially. For example, exp(-|error|).

Yes, essentially, for that part of the curve. There are a lot of different
conceivable ways to implement that - not sure how to find the "simplest",
for testing purposes. Maybe economy of calculation should be a priority,
lacking any good reason to avoid it - like a clearer underlying theory.

But the other point may well be that if there is a conditional perceptual
dropping-out of non-consonances under some conditions that this can not be
accounted for by changing the functional mapping at this level.

Carl mentioned some issue related to this having come up on the harmonic
entropy list...

on 12/4/03 10:59 AM, Carl Lumma <ekin@lumma.org> wrote (in respone to my
"Consider this hypothesis" paragraph, still quoted above):
> Interestingly, this came up re. harmonic entropy. As you mistune one
> note of a tetrad, when does the ear give up on the tetrad and hear a
> triad with a noise component? On the harmonic_entropy list I've
> suggested 'hierarchical harmonic entropy' to address this.

So that takes the whole thing in another direction entirely.

It might also be that octaves are special? Either that or more generally
there might be a good reason for weighting different intervals differently,
and the octave becomes a special case of that.

>>>> In all cases the sign of
>>>> the error is removed before accumulating. Perhaps take the
>>> absolute value,
>>>> subtract the "threshold" value, and then square. The result will
>>> be more
>>>> linear for small values and won't flatten out at zero, but will
>>> have a
>>>> point.
>>>
>>> Sounds complex. How about just MAD?
>>
>> I suspected everybody can not be wrong about RMS, and that RMS
> probably
>> treats large errors correctly.
>
> Well, lots of people seem to thing RMS is wrong and minimax
> (p=infinity) is right.

And presumably this is on aesthetic grounds, from both sides - otherwise -
who cares? If so even that might support that there are (at least) 2
phenomena involved.

>> But that
>> is probably best reflected by simply refining the target. On the
>> other hand
>> in a sufficiently complex scenario it might be useful to have an
>> algorithm
>> that drops the worst data points automatically, so that
>> the "optimized"
>> result yields possibly interesting things to try, for reasons that
>> could not
>> have been anticipated.
>
> Sure.

Well it might take us several years to figure this out!

-Kurt

Octatonic Scale

🔗Paul G Hjelmstad <paul.hjelmstad@us.ing.com>

🔗Paul Erlich <paul@stretch-music.com>

🔗Kurt Bigler <kkb@breathsense.com>

🔗Paul Erlich <paul@stretch-music.com>

🔗Kurt Bigler <kkb@breathsense.com>

🔗Paul Erlich <paul@stretch-music.com>

🔗Kurt Bigler <kkb@breathsense.com>

🔗Paul Erlich <paul@stretch-music.com>

🔗Kurt Bigler <kkb@breathsense.com>

🔗Paul Erlich <paul@stretch-music.com>

🔗Kurt Bigler <kkb@breathsense.com>

🔗Kurt Bigler <kkb@breathsense.com>

🔗Kurt Bigler <kkb@breathsense.com>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Carl Lumma <ekin@lumma.org>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Carl Lumma <ekin@lumma.org>

🔗Kurt Bigler <kkb@breathsense.com>

🔗Carl Lumma <ekin@lumma.org>

🔗Paul Erlich <paul@stretch-music.com>

🔗Paul Erlich <paul@stretch-music.com>

🔗Paul Erlich <paul@stretch-music.com>

🔗Carl Lumma <ekin@lumma.org>

🔗Kurt Bigler <kkb@breathsense.com>

🔗Kurt Bigler <kkb@breathsense.com>

🔗Carl Lumma <ekin@lumma.org>

🔗Paul Erlich <paul@stretch-music.com>

🔗Kurt Bigler <kkb@breathsense.com>

🔗Kurt Bigler <kkb@breathsense.com>

🔗Carl Lumma <ekin@lumma.org>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Carl Lumma <ekin@lumma.org>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Paul Erlich <paul@stretch-music.com>

🔗Kurt Bigler <kkb@breathsense.com>

🔗Kurt Bigler <kkb@breathsense.com>

🔗Kurt Bigler <kkb@breathsense.com>

🔗Joseph Pehrson <jpehrson@rcn.com>