RE: Disappearing Commas

>Could anyone tell me - is there a unique theoretical limit at which a
>comma (of different kinds?) may be said to disappear?

I don't understand this--can you rephrase?

>Are there
>strict and not so strict definitions of which systems (or musical
>sequences?) of which this can be said to pertain/occur? Is there a
>single accepted definitive reference which defines the 'disappearance
>of a comma'? (Helmholtz?) These are things I missed in my education.

With regard to the syntonic comma and some of the "better" ETs, you
probably can get quite a lot from references like Helmholtz. Otherwise,
I must humbly suggest that my consistency stuff is key. This has been
covered on the list before, contact me and I'll try to dig up the
relevant TDs. The syntonic comma is traditionally defined both as the
difference between a Pythagorean minor third and a just minor third, and
as the difference between a Pythagorean major third and a just major
third. When using the closest approximations to the consonant intervals
(one constructs the Pythagorean intervals from chains of perfect fifths)
in an ET not consistent within the 5-limit, these definitions may not
agree, so the comma itself, and the question of its disappearance, may
not be well-defined. Similarly, commas constructed from n-limit
intervals are uniquely defined in all tunings consistent within the
n-limit. Given consistency, the determination of the size of a comma in
an ET is fairly trivial.

In meantone tuning, the syntonic comma disappears by definition. That
is, even though it may be possible to find a better perfect fifth above
C than G, by extending the meantone tuning to 100 or so notes per
octave, and using such a perfect fifth to define a non-vanishing
syntonic comma, that's not really a kosher use of meantone. Other
non-closed tuning systems may also have "canonical" representations for
the consonances, and one should use these, not some distant
improvements, to define the commas. Going back to meantone, one can make
a case for defining the canonical 7/4 as the augmented sixth, and all
septimal commas can be determined from there. But that is already in
conflict with the interpretation of the dominant seventh chord as some
form of 4:5:6:7. Going to higher primes (odd composites don't lead to
any new commas) in meantone tuning or other non-closed systems is really
questionable unless those systems were designed with those primes in
mind.

On Tue, 9 Jun 1998, Paul H. Erlich wrote:
> I have only been able to find two books of Fokker's writings -- _New
> Music with 31 Notes_ and the big one with all his compositions in it,
> edited by Rudolf Rasch, I think. The matrix/determinant stuff sounds
> like something I need to seriously study. Can someone tell me more about
> what Fokker wrote, or perhaps send me some photocopies, for educational
> purposes only, and all other disclaimers relevant to copyright
> violation?

Here are the relevant publications. If you can't find them any other
way they should be Interlibrary-loan-able from us.

AUTHOR Fokker, A. D. (Adriaan Daniel), 1887-
TITLE Selections from the harmonic lattice of perfect fifths and major
thirds containing 12, 19, 22, 31, 41 or 53 notes.
PUBLISHED Amsterdam, Koninkl. Nederl. Akademie van Wetenschappen, 1968.
DESCRIPTION 251-266 p. 26 cm.
NOTE Caption title.
"Reprinted from Proceedings, Series B, 71, No. 4, 1968."
Communicated at the meeting of June 29, 1968.
SUBJECTS Musical intervals and scales.
Musical temperament.

AUTHOR Fokker, A. D. (Adriaan Daniel), 1887-
TITLE Unison vectors and periodicity blocks in the three-dimensional
(3-5-7) harmonic lattice of notes.
PUBLISHED Amsterdam, Koninkl. Nederl. Akademie van Wetenschappen, 1968.
DESCRIPTION 153-158 p. 26 cm.
NOTE Caption title.
"Reprinted from Proceedings, Series B., 72, No. 3, 1969."
Communicated at the meeting of February 22, 1969.
SUBJECTS Musical intervals and scales.

> My first question would be which ETs can and cannot be
> generated by this procedure

You can generate any damn ET you want this way, although sometimes the
unisons/commas chosen to vanish aren't very small.

> and by analagous procedures with axes 5/3
> and 3, or 5/3 and 5,

Since the space spanned by 5/3 and 3 or 5/3 and 5 is the same one
spanned by 3 and 5, any ET generable by one should be generable by the
others, with the appropriate matrix transformations.

> and analagously in higher dimensions.

Yes, as the above title implies, it works in 3 dimensions, I have used
it in 4 dimensions at least, and there's no reason why it wouldn't work
in arbitrarily high dimensions.

--pH http://library.wustl.edu/~manynote
O
/\ "Churchill? Can he run a hundred balls?"
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Paul Hahn wrote:

>You can generate any damn ET you want this way, although sometimes the
>unisons/commas chosen to vanish aren't very small.

Correct me if I'm wrong, but Fokker used octave invariant matrices.
That means you can't use them to define 88CET, for example.

I've just noticed there's nothing about this on my website. I'll add a
page sometime, including a derivation of 88CET which is particularly
interesting.

>I've just noticed there's nothing about this on my website. I'll add a
>page sometime, including a derivation of 88CET which is particularly
>interesting.

Yes, that does sound interesting, if not surprising.

What's your website's URL?

Gary Morrison wrote:

> What's your website's URL?

Aha! I've updated the website this evening so, among other
things, there is a quick explanation of matrices for equal
temperaments. It's at the bottom of the "Linear
Temperaments" page that has always been there. URL for this
is http://www.cix.co.uk/~gbreed/lintemp.htm .

I'll add an Equal Temperament page, including a discussion
of consistency and that matrix description of 88CET, maybe
at the weekend.

Graham Breed
gbreed@cix.co.uk www.cix.co.uk/~gbreed

>> What if you add the restriction that the resulting ET can have no
better
>> approximations of the just intervals than the ones represented by one
>> step along the axes? I guess I had that in the back of my mind when I
>> asked the above questions.

>I'm having difficulty imagining an ET like this--can you give me an
>example of such an ET, and then I'll try Fokker's method on it?

In all the cases you've mentioned so far, it is the case that the ETs
have no better approximations to the just intervals than the ones
represented by one step along the axes.

On Thu, 11 Jun 1998, Paul H. Erlich wrote:
> >I'm having difficulty imagining an ET like this--can you give me an
> >example of such an ET, and then I'll try Fokker's method on it?
>
> In all the cases you've mentioned so far, it is the case that the ETs
> have no better approximations to the just intervals than the ones
> represented by one step along the axes.

No, I meant can you give me an example of an ET which _does_ have a
better approximation.

--pH http://library.wustl.edu/~manynote
O
/\ "Churchill? Can he run a hundred balls?"
-\-\-- o
NOTE: dehyphenate node to remove spamblock. <*>

On Wed, 10 Jun 1998, Graham Breed wrote:
> >You can generate any damn ET you want this way, although sometimes the
> >unisons/commas chosen to vanish aren't very small.
>
> Correct me if I'm wrong, but Fokker used octave invariant matrices.
> That means you can't use them to define 88CET, for example.

Sorry. I should have said you can derive any _integral_ ET this
way--but note that the interval divided by the integral doesn't have to
be an octave. So you can derive the Bohlen-Pierce scale, for example,
by working in the 5-7 plane and assuming tritave equivalence when you
generate your intervals and choose your unison vectors.

On Wed, 10 Jun 1998, Paul H. Erlich wrote:
> What if you add the restriction that the resulting ET can have no better
> approximations of the just intervals than the ones represented by one
> step along the axes? I guess I had that in the back of my mind when I
> asked the above questions.

I'm having difficulty imagining an ET like this--can you give me an
example of such an ET, and then I'll try Fokker's method on it?

--pH http://library.wustl.edu/~manynote
O
/\ "Churchill? Can he run a hundred balls?"
-\-\-- o
NOTE: dehyphenate node to remove spamblock. <*>

On Thu, 11 Jun 1998, it was written:
> > In all the cases you've mentioned so far, it is the case that the ETs
> > have no better approximations to the just intervals than the ones
> > represented by one step along the axes.
>
> No, I meant can you give me an example of an ET which _does_ have a
> better approximation.

Never mind, here's one: in the 3-5 lattice, let the 9/8 (2 0) and the
128/125 (0 3) vanish. The determinant of the matrix is 6; in 6TET
the closest interval (just barely) to the 3/2 is actually 4 steps, but
it's the 3-step interval which must be used as the 3/2 if the 9/8 is to
vanish.

Um--so now that we have the example, what was the question again?

--pH http://library.wustl.edu/~manynote
O
/\ "Churchill? Can he run a hundred balls?"
-\-\-- o
NOTE: dehyphenate node to remove spamblock. <*>