Question for Paul: defining a paultone

Hello, there, Paul, and I just want to confirm a point about the
definition of a paultone that seems clear from some previous posts on
this topic that I looked up, but wouldn't hurt to double-check.

In one discussion, if I'm correct, you say that a paultone disperses
the 64:63 and 50:49 unison vectors, but keeps a chromatic vector of
49:48.

That suggests to me that we have chains of two, three, and four fifths
yielding approximations of 7:4, 12:7, and 9:7 in an eventone-like
fashion (64:63 unison vector), and with the 12:7 and 7:4, for example,
distinct, unlike in 29-tET, for example (49:48 chromatic vector).

At first the 50:49 unison vector wasn't so obvious, but I figured out
that it likely represents the distinction between 7:5 and 10:7, and
this brings us to what I take as an important definitional point that
comes up elsewhere.

Specifically, I take a paultone to be tunings with _two_ chains of
fifths at the distance of 600 cents or 1/2 octave, as in 22-tET, also
meeting the other criteria mentioned above.

This means that while paultones have fifths of the "2-3-7-9 eventone"
variety, they additionally have _two_ such chains at 600 cents apart,
a feature distinguishing them from the general category of eventones
say from around 707 cents to maybe something like 711 cents, or about
two cents on either size of 22-tET.

Of course, by this definition, 22-tET would have the status of being at
once a paultone and an eventone, where 11 fifths yield an interval of
precisely 600 cents -- and 56-tET was also mentioned somewhere.

Any confirmation, corrections, or other comments would be most deeply
appreciated; I want to get this right.

Most appreciatively,

Margo Schulter
mschulter@value.net

--- In tuning@y..., mschulter <MSCHULTER@V...> wrote:
> Hello, there, Paul, and I just want to confirm a point about the
> definition of a paultone that seems clear from some previous posts
on
> this topic that I looked up, but wouldn't hurt to double-check.
>
> In one discussion, if I'm correct, you say that a paultone disperses
> the 64:63 and 50:49 unison vectors, but keeps a chromatic vector of
> 49:48.

Yes. 225:224 is dispersed too, and the chromatic vector can be
identified with 25:24 or 28:27 as well as 49:48.

> Specifically, I take a paultone to be tunings with _two_ chains of
> fifths at the distance of 600 cents or 1/2 octave, as in 22-tET,
also
> meeting the other criteria mentioned above.

You betcha!
>
> This means that while paultones have fifths of the "2-3-7-9
eventone"
> variety, they additionally have _two_ such chains at 600 cents
apart,
> a feature distinguishing them from the general category of eventones
> say from around 707 cents to maybe something like 711 cents, or
about
> two cents on either size of 22-tET.

Good point.
>
> Of course, by this definition, 22-tET would have the status of
being at
> once a paultone and an eventone, where 11 fifths yield an interval
of
> precisely 600 cents -- and 56-tET was also mentioned somewhere.

56-tET has better JI approximations when _not_ considered a paultone,
and it has been mentioned in such a non-paultone context. What I
think is interesting, though, is that 76-tET supports meantone,
paultone, and also the "double-diatonic" system (familiar from 26-
tET) in which two meantone diatonic scales, at opposite sides of the
circle of fifths, each complete one another's triads into 7-limit
tetrads. Doubling 76-tET to 152-tET gets you wafso-just 11-limit
approximations as well -- thus I call 152-tET "Universal tuning".

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

> > Specifically, I take a paultone to be tunings with _two_ chains of
> > fifths at the distance of 600 cents or 1/2 octave, as in 22-tET,
> also
> > meeting the other criteria mentioned above.

> You betcha!

I was just calculating a least squares fit of such a system in the
7-limit, since I am producing an example of adaptive tempering for us
to play with. I ended up with very nearly 1/4 septimal comma
tempering, which happens to be very close to the 22-et fifth. Between
the 50-et (good as an approximation to 7/26-comma meantone) and the
22-et (good as an approximation to 1/4-septimal comma paultone) I
really don't need anything else, aside from JI, for the 7-limit.

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> > > Specifically, I take a paultone to be tunings with _two_ chains
of
> > > fifths at the distance of 600 cents or 1/2 octave, as in 22-
tET,
> > also
> > > meeting the other criteria mentioned above.
>
> > You betcha!
>
> I was just calculating a least squares fit of such a system in the
> 7-limit, since I am producing an example of adaptive tempering for
us
> to play with. I ended up with very nearly 1/4 septimal comma
> tempering, which happens to be very close to the 22-et fifth.

In my paper, I got a fifth of

(30.5 + 7*log(3) - 5*log(35))/27

octaves, or 708.8143 cents -- is this what you got?

> Between
> the 50-et (good as an approximation to 7/26-comma meantone) and the
> 22-et (good as an approximation to 1/4-septimal comma paultone) I
> really don't need anything else, aside from JI, for the 7-limit.

Hmm . . . 7/26-comma meantone of course has nothing to do with 7-
limit, being derived from 5-limit . . .

. . . and what about the various systems we discussed at tuning-
math@yahoogroups.com, including your wonderful 15-tone system with
126:125 and 64:63 unison vectors? Or are all these somehow irrelevant
to the "adaptive tempering" you envision?

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

> In my paper, I got a fifth of
>
> (30.5 + 7*log(3) - 5*log(35))/27
>
> octaves, or 708.8143 cents -- is this what you got?

That looks right, and if it's what you get by fitting to 3,5,7,5/3,
and 7/3 with sqrt(2) fixed then it is right.

> > Between
> > the 50-et (good as an approximation to 7/26-comma meantone) and
the
> > 22-et (good as an approximation to 1/4-septimal comma paultone) I
> > really don't need anything else, aside from JI, for the 7-limit.

> Hmm . . . 7/26-comma meantone of course has nothing to do with 7-
> limit, being derived from 5-limit . . .

I wanted the 5-limit, since I don't propose to use it on 7-limit
notes.

> . . . and what about the various systems we discussed at tuning-
> math@y..., including your wonderful 15-tone system with
> 126:125 and 64:63 unison vectors? Or are all these somehow
irrelevant
> to the "adaptive tempering" you envision?

It's irrelevant to this example, which tempers the well-known hymn
tune "Materna", orginally written in 12-et.

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> > In my paper, I got a fifth of
> >
> > (30.5 + 7*log(3) - 5*log(35))/27
> >
> > octaves, or 708.8143 cents -- is this what you got?
>
> That looks right, and if it's what you get by fitting to 3,5,7,5/3,
> and 7/3 with sqrt(2) fixed then it is right.
>
> > > Between
> > > the 50-et (good as an approximation to 7/26-comma meantone) and
> the
> > > 22-et (good as an approximation to 1/4-septimal comma paultone)
I
> > > really don't need anything else, aside from JI, for the 7-limit.
>
> > Hmm . . . 7/26-comma meantone of course has nothing to do with 7-
> > limit, being derived from 5-limit . . .
>
> I wanted the 5-limit, since I don't propose to use it on 7-limit
> notes.

7-limit notes . . . hmm . . . you mean on notes that don't even
participate in any 7-limit harmonies with some preassigned "tonic"?
>
> > . . . and what about the various systems we discussed at tuning-
> > math@y..., including your wonderful 15-tone system with
> > 126:125 and 64:63 unison vectors? Or are all these somehow
> irrelevant
> > to the "adaptive tempering" you envision?
>
> It's irrelevant to this example, which tempers the well-known hymn
> tune "Materna", orginally written in 12-et.

Hmm . . . I don't know it -- who wrote it? Eliminating 64:63 I
understand, given your point of view, while eliminating 50:49 would
show up only in more "modern" music, say since the late 19th century,
I would think . . . 50:49 would be associated with harmonic
techniques such as "tritone substitution" . . .

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

> 7-limit notes . . . hmm . . . you mean on notes that don't even
> participate in any 7-limit harmonies with some preassigned "tonic"?

Eh? You lost me.

> > It's irrelevant to this example, which tempers the well-known
hymn
> > tune "Materna", orginally written in 12-et.

> Hmm . . . I don't know it -- who wrote it?

Samuel Augustus Ward wrote it, and you do know it, since it is the
tune usually used for "America the Beautiful". I did a JI version to
end the piece "September 11, 2001" I just finished, and thought I
would recycle it as a tuning example.

Eliminating 64:63 I
> understand, given your point of view, while eliminating 50:49 would
> show up only in more "modern" music, say since the late 19th
century,
> I would think . . . 50:49 would be associated with harmonic
> techniques such as "tritone substitution" . . .

Elimianting it is a natural thing to do given how I was
conceptualizing the process, though harldy required. However, it
seems I don't gain much by not eliminating it.

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> > 7-limit notes . . . hmm . . . you mean on notes that don't even
> > participate in any 7-limit harmonies with some
preassigned "tonic"?
>
> Eh? You lost me.

How do you define which notes are "7-limit notes" in a piece that was
written in 12-tET?
>
> Eliminating 64:63 I
> > understand, given your point of view, while eliminating 50:49
would
> > show up only in more "modern" music, say since the late 19th
> century,
> > I would think . . . 50:49 would be associated with harmonic
> > techniques such as "tritone substitution" . . .
>
> Elimianting it is a natural thing to do given how I was
> conceptualizing the process,

Which is how? I'm not nagging you, I'm genuinely interested in
anything that goes on inside your brain.

> though harldy required. However, it
> seems I don't gain much by not eliminating it.

Is that because the "7-limit" diminished fifth is all the way on the
other side of the half-octave relative to the meantone diminished
fifth?

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

> How do you define which notes are "7-limit notes" in a piece that
was
> written in 12-tET?

They are all the notes which are 7-units (ie not having a 7 in the
factorization) in the 7-limit JI version I wrote.

> > Elimianting it is a natural thing to do given how I was
> > conceptualizing the process,

> Which is how? I'm not nagging you, I'm genuinely interested in
> anything that goes on inside your brain.

I was starting from a notation for the 7-limit of 12,22,31,55 and
then dropping those ets which seem to be a problem, but this is
really the same as using either the 22-et or the 50-et in problem
patches, and JI the rest of the time. I'll see if it works, and then
try to get John to compare results.

> Is that because the "7-limit" diminished fifth is all the way on
the
> other side of the half-octave relative to the meantone diminished
> fifth?

Not really, it's just that the approximations you get from 64/63~1
alone don't seem significantly better.

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> > How do you define which notes are "7-limit notes" in a piece that
> was
> > written in 12-tET?
>
> They are all the notes which are 7-units (ie not having a 7 in the
> factorization) in the 7-limit JI version I wrote.

_Not_ having a 7 in the factorization? I would have thought the
reverse (perhaps naively). Also, this piece must be an exception from
the norm if it lends itself to a JI version at all.

> > Is that because the "7-limit" diminished fifth is all the way on
> the
> > other side of the half-octave relative to the meantone diminished
> > fifth?
>
> Not really, it's just that the approximations you get from 64/63~1
> alone don't seem significantly better.

I think that might have something to do with what I said . . . ?

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., genewardsmith@j... wrote:
> > --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

> > > How do you define which notes are "7-limit notes" in a piece
that
> > was
> > > written in 12-tET?

> > They are all the notes which are 7-units (ie not having a 7 in
the
> > factorization) in the 7-limit JI version I wrote.

> _Not_ having a 7 in the factorization?

Sorry, I meant to say *not* a 7-unit.

I would have thought the
> reverse (perhaps naively). Also, this piece must be an exception
from
> the norm if it lends itself to a JI version at all.

It depends on your definition of "lends itself", I suppose. You can
always produce a JI version, or versions; the trick is to get a good
one. That's easier than most people seem to think, but perhaps my
standards are low. I like them, at any rate.