Erlich's Decatonics in 12-equal?

Hi!

I've been reading Paul Erlich's old paper on 22-equal and decatonics.
Decatonic scales in 22-equal have proportions 8s+2L=22, when s=2 and
L=3. Now if s=1 and L=2, 8s+2L=12. Does this mean that 12-equal
contain these scales and if not, why not?

Kalle

Hi Kalle,

Yes, and the symmetric decatonic is actually an Olivier Messiaen
scale... but as Paul was trying to optimize things in the 7-limit,
12-tet obviously wasn't the best setting for his intentions.

take care,

--Dan Stearns

----- Original Message -----
From: "kalleaho" <kalleaho@mappi.helsinki.fi>
To: <tuning@yahoogroups.com>
Sent: Sunday, June 02, 2002 10:20 AM
Subject: [tuning] Erlich's Decatonics in 12-equal?

> Hi!
>
> I've been reading Paul Erlich's old paper on 22-equal and
decatonics.
> Decatonic scales in 22-equal have proportions 8s+2L=22, when s=2 and
> L=3. Now if s=1 and L=2, 8s+2L=12. Does this mean that 12-equal
> contain these scales and if not, why not?
>
> Kalle
>
>
>
>
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--- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:
> Hi Kalle,
>
> Yes, and the symmetric decatonic is actually an Olivier Messiaen
> scale... but as Paul was trying to optimize things in the 7-limit,
> 12-tet obviously wasn't the best setting for his intentions.

This is an example of a linear temperament, in this case pajera, which has commas of 50/49 and 64/63. It can also be done in the
32 and 54 ets, but really there isn't much point to using anything but
12 or 22. It's essentially a 22 exclusive.

--- In tuning@y..., "kalleaho" <kalleaho@m...> wrote:
> Hi!
>
> I've been reading Paul Erlich's old paper on 22-equal and
decatonics.
> Decatonic scales in 22-equal have proportions 8s+2L=22, when s=2
and
> L=3. Now if s=1 and L=2, 8s+2L=12. Does this mean that 12-equal
> contain these scales and if not, why not?
>
> Kalle

hi kalle,

yes and no.

in addition to what's already been said, i'd like to bring up two
points:

(a) in 12-equal, the characteristic dissonances are no longer
dissonances at all!

(b) perhaps the most important point of all -- the 164-cent scale
steps that aid in the "browneian position finding" so important to my
view of tonality. in 12-equal these steps are 200 cents wide, and no
different in sound from the "thirds(10)" that are found throughout
the scale. at 164 cents, though, these steps really, with a little
exposure, begin to stand out and demand attention, making resolutions
to the tonic all the more satisfying.

furthermore . . . i have some experience with "translating" music in
the symmetrical decatonic scale directly from 22-equal to 12-equal.
this scale, unlike the pentachordal one, doesn't have the 545-cent
interval that i was referring to in point (a) above. what i can say
about the experience is this:

*the consonant chords are a good deal less consonant, particularly
the "major tetrad";

*the scale sounds melodically more 'jagged' in 12-equal, since it
gives the impression of "skipping" notes;

*modulations are a lot less interesting, because the explicit
microtonality of the resulting 1/22-octave steps disappears;

*but overall, the music comes through pretty much intact!

cheers and hope you'll fire back with some more questions,
paul

--- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:

/tuning/topicId_37170.html#37190

> --- In tuning@y..., "kalleaho" <kalleaho@m...> wrote:
> > Hi!
> >
> > I've been reading Paul Erlich's old paper on 22-equal and
> decatonics.
> > Decatonic scales in 22-equal have proportions 8s+2L=22, when s=2
> and
> > L=3. Now if s=1 and L=2, 8s+2L=12. Does this mean that 12-equal
> > contain these scales and if not, why not?
> >
> > Kalle
>
> hi kalle,
>
> yes and no.
>
> in addition to what's already been said, i'd like to bring up two
> points:
>
> (a) in 12-equal, the characteristic dissonances are no longer
> dissonances at all!
>
> (b) perhaps the most important point of all -- the 164-cent scale
> steps that aid in the "browneian position finding" so important to
my
> view of tonality. in 12-equal these steps are 200 cents wide, and
no
> different in sound from the "thirds(10)" that are found throughout
> the scale. at 164 cents, though, these steps really, with a little
> exposure, begin to stand out and demand attention, making
resolutions
> to the tonic all the more satisfying.
>
> furthermore . . . i have some experience with "translating" music
in
> the symmetrical decatonic scale directly from 22-equal to 12-equal.
> this scale, unlike the pentachordal one, doesn't have the 545-cent
> interval that i was referring to in point (a) above. what i can say
> about the experience is this:
>
> *the consonant chords are a good deal less consonant, particularly
> the "major tetrad";
>
> *the scale sounds melodically more 'jagged' in 12-equal, since it
> gives the impression of "skipping" notes;
>
> *modulations are a lot less interesting, because the explicit
> microtonality of the resulting 1/22-octave steps disappears;
>
> *but overall, the music comes through pretty much intact!
>
> cheers and hope you'll fire back with some more questions,
> paul

***Paul, could you please explain to me what's going on here?? Is
Kalle saying that 22-tET pitches can be played by 12-equal??
That seems really peculiar to me since *24-tET* would contain 12 and
that's so close to 22. I wouldn't think that would work out. Could
you please run this by me...

Thanks!

Joseph

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:

> ***Paul, could you please explain to me what's going on here?? Is
> Kalle saying that 22-tET pitches can be played by 12-equal??
> That seems really peculiar to me since *24-tET* would contain 12
and
> that's so close to 22. I wouldn't think that would work out.
Could
> you please run this by me...
>
> Thanks!
>
> Joseph

kalle is talking about the particular 10-tone scales in my paper.

let's look at something analogous: the diatonic scale.

the diatonic scale has 2 small steps and 5 large steps, so the octave
is 2s+5L.

plugging in s=1, L=2 gives you 2s+5L = 12, the familiar 12-equal.

plugging in s=2, L=3 gives you 2s+5L = 19, 19-tone equal temperament.

plugging in s=1, L=3 gives you 2s+5L = 17, 17-tone equal temperament.

all of these are eminently acceptable tunings for the diatonic scale,
though of course the consonance of the harmonies varies drastically.

now kalle is talking about my 10-note scales, which have 2 large and
8 small steps.

so, in a sense, plugging any integers s and L into the formula 8s+2L
will give you an ET where my 10-note scales "work" . . .

. . . though in another sense they "don't" . . .

making sense now?

--- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:

/tuning/topicId_37170.html#37193

> --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
>
> > ***Paul, could you please explain to me what's going on here??
Is
> > Kalle saying that 22-tET pitches can be played by 12-equal??
> > That seems really peculiar to me since *24-tET* would contain 12
> and
> > that's so close to 22. I wouldn't think that would work out.
> Could
> > you please run this by me...
> >
> > Thanks!
> >
> > Joseph
>
> kalle is talking about the particular 10-tone scales in my paper.
>
>
> let's look at something analogous: the diatonic scale.
>
> the diatonic scale has 2 small steps and 5 large steps, so the
octave
> is 2s+5L.
>
> plugging in s=1, L=2 gives you 2s+5L = 12, the familiar 12-equal.
>
> plugging in s=2, L=3 gives you 2s+5L = 19, 19-tone equal
temperament.
>
> plugging in s=1, L=3 gives you 2s+5L = 17, 17-tone equal
temperament.
>
> all of these are eminently acceptable tunings for the diatonic
scale,
> though of course the consonance of the harmonies varies drastically.
>
>
> now kalle is talking about my 10-note scales, which have 2 large
and
> 8 small steps.
>
> so, in a sense, plugging any integers s and L into the formula
8s+2L
> will give you an ET where my 10-note scales "work" . . .
>
> . . . though in another sense they "don't" . . .
>
> making sense now?

***Hi Paul!

Thanks for the help on this!

This is interesting, but it still seems mighty "peculiar" to me...
In other words, as long as you have two small steps and 5 large
steps, it really doesn't matter how large or small the steps really
*are* and it's still considered *diatonic?*

Does that fit into the "classical" definition of the diatonic scale??

Thanks!

Joseph

--- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:
> Hi Kalle,
>
> Yes, and the symmetric decatonic is actually an Olivier Messiaen
> scale... but as Paul was trying to optimize things in the 7-limit,
> 12-tet obviously wasn't the best setting for his intentions.
>
>
> take care,
>
> --Dan Stearns

Hi Dan,

Thanks for your response. Yes, it is a Messiaen scale. Did Messiaen's
harmonic usage of this scale have anything to do with 7-limit harmony?
Sometimes he used to "explain" his modes of limited transposition
with harmonic series up to at least 7th harmonic.

Kalle

Hi Paul!

Thanks for your reply. I find your decatonic scales and the paper
extremely interesting.

> in addition to what's already been said, i'd like to bring up two
> points:
>
> (a) in 12-equal, the characteristic dissonances are no longer
> dissonances at all!

This is interesting! Can't they work as dissonances depending on the
context?

> (b) perhaps the most important point of all -- the 164-cent scale
> steps that aid in the "browneian position finding" so important to
my
> view of tonality. in 12-equal these steps are 200 cents wide, and
no
> different in sound from the "thirds(10)" that are found throughout
> the scale. at 164 cents, though, these steps really, with a little
> exposure, begin to stand out and demand attention, making
resolutions
> to the tonic all the more satisfying.
>
> furthermore . . . i have some experience with "translating" music
in
> the symmetrical decatonic scale directly from 22-equal to 12-equal.
> this scale, unlike the pentachordal one, doesn't have the 545-cent
> interval that i was referring to in point (a) above. what i can say
> about the experience is this:
>
> *the consonant chords are a good deal less consonant, particularly
> the "major tetrad";

True. I think this has something to do with articulation. While 12-
equal is consistent up to 9-limit, major tetrads don't quite have
that 7-limit sound. You kind of can't escape the 5-limit sound in 12-
tet. In 22-equal every interval inside a 7-limit tetrad is
approximated by a different number of 22-equal steps, am I right? I
think this helps to give a clear impression of 7-limit sound and
consonance.

>
> *the scale sounds melodically more 'jagged' in 12-equal, since it
> gives the impression of "skipping" notes;
>
> *modulations are a lot less interesting, because the explicit
> microtonality of the resulting 1/22-octave steps disappears;
>
> *but overall, the music comes through pretty much intact!
>
> cheers and hope you'll fire back with some more questions,
> paul

I'd like to know what others think about your criteria for
generalizing diatonicity?

Do you think it is a good idea to maximize sensory consonance for
timbres used in 22-equal compositions?

I am thinking of detuning harmonics of sounds to the nearest 22-equal
frequency a la Sethares. How does this technique relate to
consistency? We know that waveforms are most periodic and have the
clearest pitch when their partials are as close as possible to the
integer multiples of the fundamental but spectral mapping doesn't
necessarily preserve the intervals between partials in consistent
manner. I suppose this is not a problem but I could be wrong.

Kalle

Joseph wrote:
>This is interesting, but it still seems mighty "peculiar" to me...
>In other words, as long as you have two small steps and 5 large
>steps, it really doesn't matter how large or small the steps really
>*are* and it's still considered *diatonic?*

Yes, this is Blackwood's definition of diatonicity. You can find
out for yourself that in this case the fifth is always recognisable.
Which is exactly Blackwood's definition of recognisable fifths.

>Does that fit into the "classical" definition of the diatonic scale??

If you mean generated by a chain of fifths, yes.

Manuel

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:

> ***Hi Paul!
>
> Thanks for the help on this!
>
> This is interesting, but it still seems mighty "peculiar" to me...
> In other words, as long as you have two small steps and 5 large
> steps, it really doesn't matter how large or small the steps really
> *are* and it's still considered *diatonic?*
>
> Does that fit into the "classical" definition of the diatonic
scale??
>
> Thanks!
>
> Joseph

i don't know what exactly you mean by 'classical'. certainly
blackwood's book on the subject would agree with me here. but try it -
- improvise around with a few of these versions of the diatonic
scale --you'll find that a certain 'diatonicity' is quite evident
even if the large-step-to-small-step-ratio is distorted to values as
extreme as 5-to-1 (as in 27-equal 'hyper-pythagorean') or 4-to-3 (as
in 26-equal 'hypo-meantone'). the 19-equal and 17-equal versions i
mentioned figure quite prominently in blackwood's pieces in said
tunings, so have a listen -- can you deny that 'diatonic' is a fair
descriptor of these scales?

--- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:

/tuning/topicId_37170.html#37211

> --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
>
> > ***Hi Paul!
> >
> > Thanks for the help on this!
> >
> > This is interesting, but it still seems mighty "peculiar" to
me...
> > In other words, as long as you have two small steps and 5 large
> > steps, it really doesn't matter how large or small the steps
really
> > *are* and it's still considered *diatonic?*
> >
> > Does that fit into the "classical" definition of the diatonic
> scale??
> >
> > Thanks!
> >
> > Joseph
>
> i don't know what exactly you mean by 'classical'. certainly
> blackwood's book on the subject would agree with me here. but try
it -
> - improvise around with a few of these versions of the diatonic
> scale --you'll find that a certain 'diatonicity' is quite evident
> even if the large-step-to-small-step-ratio is distorted to values
as
> extreme as 5-to-1 (as in 27-equal 'hyper-pythagorean') or 4-to-3
(as
> in 26-equal 'hypo-meantone'). the 19-equal and 17-equal versions i
> mentioned figure quite prominently in blackwood's pieces in said
> tunings, so have a listen -- can you deny that 'diatonic' is a fair
> descriptor of these scales?

***That's really interesting, Paul! I'm sure it works. I'll try it
out...

JP

--- In tuning@y..., "kalleaho" <kalleaho@m...> wrote:

> Hi Paul!
>
> Thanks for your reply. I find your decatonic scales and the paper
> extremely interesting.

thank you! the latest version is at
http://www-math.cudenver.edu/~jstarret/22ALL.pdf . . .

> This is interesting! Can't they work as dissonances depending on
the
> context?

i don't see how. the basic 'dissonant' tetrads that the 545-cent
intervals belong to . . . they turn into very familiar, relatively
consonant tetrads in 12-equal. check it out!

> > *the consonant chords are a good deal less consonant,
particularly
> > the "major tetrad";
>
> True. I think this has something to do with articulation. While 12-
> equal is consistent up to 9-limit, major tetrads don't quite have
> that 7-limit sound. You kind of can't escape the 5-limit sound in
12-
> In 22-equal every interval inside a 7-limit tetrad is
> approximated by a different number of 22-equal steps, am I right?

not quite -- 10:7 and 7:5 are both 11 steps, so in some voicings,
this may not hold.

> I
> think this helps to give a clear impression of 7-limit sound and
> consonance.

i think it's more just a function of absolute cents deviations from
the just chord.

> I'd like to know what others think about your criteria for
> generalizing diatonicity?

the subject gets a lot of discussion on the tuning-math list . . .

> Do you think it is a good idea to maximize sensory consonance for
> timbres used in 22-equal compositions?

sure . . .

> I am thinking of detuning harmonics of sounds to the nearest 22-
equal
> frequency a la Sethares.

another approach would be to use adaptive retuning a la john
delaubenfels. or you could use a combination of both, such that
neither the retuning nor the "retimbring" would be large enough to be
noticeable, and yet the two together would give you complete sensory
consonance for many chords . . .

> How does this technique relate to
> consistency? We know that waveforms are most periodic and have the
> clearest pitch when their partials are as close as possible to the
> integer multiples of the fundamental but spectral mapping doesn't
> necessarily preserve the intervals between partials in consistent
> manner. I suppose this is not a problem but I could be wrong.

i'm not sure what you mean by consistency here. as far as the clarity
of the pitch . . . bill sethares posted an example where he remapped
the partials of a guitar timbre to 19-equal. this is about as much
retuning as would be involved in what you're proposing. the effect
was a tone which was still perfectly clear in pitch. it has a very
subtle 'phasing' effect, not altogether unpleasant . . . recall that
the hammond organ uses 'additive synthesis' with partials (1, 2, 3,
4, 5, 6, and 8) within 1 cent of 12-equal, rather than harmonic
series, values -- and lots of people love the hammond organ tone
(particularly through a rotating speaker which 'phases' the sound
even more) . . .

--- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:

> i'm not sure what you mean by consistency here.

Well, we can think about a sound as a chord of sine waves. Now you
can't map all the partials to the nearest ET approximation so that
every interval inside this "chord" is approximated with the best
approximation the ET has.

> as far as the clarity
> of the pitch . . . bill sethares posted an example where he
remapped
> the partials of a guitar timbre to 19-equal. this is about as much
> retuning as would be involved in what you're proposing. the effect
> was a tone which was still perfectly clear in pitch. it has a very
> subtle 'phasing' effect, not altogether unpleasant . . . recall
that
> the hammond organ uses 'additive synthesis' with partials (1, 2, 3,
> 4, 5, 6, and 8) within 1 cent of 12-equal, rather than harmonic
> series, values -- and lots of people love the hammond organ tone
> (particularly through a rotating speaker which 'phases' the sound
> even more) . . .

Strange synchronicity going on here! I experimented today with
additive synthesis in Vienna SoundFont Studio and first I did an
organ sound with 7-equal partials. It sounded kind of metallic. Then
I did the same thing for 12-equal. First I included the 7th partial
but it sounded wrong. What I had in the end was an organ sound with
partials 1, 2, 3, 4, 5, 6 and 8!!! Though it's digital it sounds very
hammond-like. I must say this spectral mapping technique works! I
haven't been playing in 12-equal for some time but now I did because
chords sound so smooth with this organ sound.

Kalle

--- In tuning@y..., "kalleaho" <kalleaho@m...> wrote:
> --- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:
>
> > i'm not sure what you mean by consistency here.
>
> Well, we can think about a sound as a chord of sine waves. Now you
> can't map all the partials to the nearest ET approximation so that
> every interval inside this "chord" is approximated with the best
> approximation the ET has.

well, for 22-equal, you can do so through the 12th partial . . . the
rest doesn't much matter for consonance/dissonance anyhow . . .

> Strange synchronicity going on here! I experimented today with
> additive synthesis in Vienna SoundFont Studio and first I did an
> organ sound with 7-equal partials. It sounded kind of metallic.
>Then
> I did the same thing for 12-equal. First I included the 7th partial
> but it sounded wrong. What I had in the end was an organ sound with
> partials 1, 2, 3, 4, 5, 6 and 8!!!

wow, i'm glad i was so explicit in my post! strange synchronicity
indeed!

> Though it's digital it sounds very
> hammond-like.

no surprises there! ;)

> I must say this spectral mapping technique works! I
> haven't been playing in 12-equal for some time but now I did
because
> chords sound so smooth with this organ sound.

now try 22-equal with partials 1-12 of your favorite timbre . . .