Reply to John deLaubenfels

>> I too believe slight glissandos are necessary for JI transcription of
>> ordinary music, particularly to avoid drifting or shifting by a
>> syntonic comma in the course of a I-IV-ii-V-I or I-vi-ii-V-I
>> progression. How does your program handle these?
>In fact, though I mentioned only two, there are three things my program
>considers:
> . The pain of changing the tuning of a note previously (or
> continuously) sounding.
> . The pain of any chord being out of tune (as measured by
considering
> all pairs of notes sounding).
> . The pain of "drift": when the average tuning shift of all notes
> sounding moves either above or below zero.
>This last counteracts the tendency of the progressions you mention to
>pump the tuning downward (or upward, when reversed).

Excellent!

>I need to review the math of 22TET, but, assuming that each interval
>there has a 1:1 mapping with a corresponding JI interval, such a
>translation would be very easy, and would not involve actually casting
>about for different JI interpretations.

Well, it's about as easy for the 7-limit as 12-tET is for the 5-limit,
but there are always tricky cases. Two stacked fourths in 22-tET give
you a near-7:4; three give you a near-7:6; and four give you a near-9:7.
Two major triads a half-octave apart (Petroushka chord) have good
approximations to one another's harmonic 7ths. Etc.

>Though, the "comma pump" chord sequence you mention
>can't be easily resolved in any ET that "supports a comma" (which, it
>appears, 22TET DOES, and 31TET, oddly, does NOT).

Not odd at all -- that is why 31-tET is so popular for diatonic music.
22-tET is HORRIBLE for diatonic music. I've found 10-tone scales that
function "diatonically' in 22-tET and give you rich 7-limit harmonic
resources (tetrads instead of triads):
http://www-math.cudenver.edu/~jstarret/22ALL.pdf. There are errors of 17
cents involved in the 22-tET approximation of 7:5 (the maximum error in
the 7-limit), so a dynamic tuning approach would help about as much as
it would help 5-limit music written in 12-tET. There's also the 26-tET
thing I mentioned earlier tonight.

>> I need to review the math of 22TET, but, assuming that each interval
>> there has a 1:1 mapping with a corresponding JI interval, such a
>> translation would be very easy, and would not involve actually casting
>> about for different JI interpretations.

While all the 5-limit consonances have a 1:1 mapping in 12TET, there are
a couple of exceptions when using 22TET for 7-limit:

- 7:5 is represented by the same interval (11/22TET) as its
inversion, 10:7.
- 8:7, a 7-limit consonance, is represented by the same interval
(4/22TET) as 9:8, which is _not_ a 7-limit consonance.

Of course, these should be resolvable through harmonic context; however,
I think it will make the translation a little less straightforward,
since the mapping is not quite 1:1.

--pH <manynote@lib-rary.wustl.edu> http://library.wustl.edu/~manynote
O
/\ "Well, so far, every time I break he runs out.
-\-\-- o But he's gotta slip up sometime . . . "

NOTE: dehyphenate node to remove spamblock. <*>

John deLaubenfels wrote,

>>> I need to review the math of 22TET, but, assuming that each interval
>>> there has a 1:1 mapping with a corresponding JI interval, such a
>>> translation would be very easy, and would not involve actually
casting
>>> about for different JI interpretations.

>Paul Hahn wrote,

>While all the 5-limit consonances have a 1:1 mapping in 12TET, there
are
>a couple of exceptions when using 22TET for 7-limit:

> - 7:5 is represented by the same interval (11/22TET) as its
> inversion, 10:7.
> - 8:7, a 7-limit consonance, is represented by the same interval
> (4/22TET) as 9:8, which is _not_ a 7-limit consonance.

The second item is not an exception. In 12-tET, 6:5, a 5-limit
consonance, is represented by the same interval as 7:6, which is _not_ a
5-limit consonance. So only the first item is an exception. However, if
you use 22-tET for 9-limit, the second item is an exception, because 9:8
is a 9-limit consonance.

On Mon, 8 Mar 1999, Paul H. Erlich wrote:
>> Paul Hahn wrote,
>> - 7:5 is represented by the same interval (11/22TET) as its
>> inversion, 10:7.
>> - 8:7, a 7-limit consonance, is represented by the same interval
>> (4/22TET) as 9:8, which is _not_ a 7-limit consonance.
>
> The second item is not an exception. In 12-tET, 6:5, a 5-limit
> consonance, is represented by the same interval as 7:6, which is _not_ a
> 5-limit consonance. So only the first item is an exception. However, if
> you use 22-tET for 9-limit, the second item is an exception, because 9:8
> is a 9-limit consonance.

Well, but if you're working in strict 5-limit, the 7:6 simply never
arises. OTOH, in 7-limit, 9:8 still comes up as a secondary interval,
so you still may have to resolve the ambiguity.

--pH <manynote@lib-rary.wustl.edu> http://library.wustl.edu/~manynote
O
/\ "Well, so far, every time I break he runs out.
-\-\-- o But he's gotta slip up sometime . . . "

NOTE: dehyphenate node to remove spamblock. <*>

>On Mon, 8 Mar 1999, Paul H. Erlich wrote:
>>> Paul Hahn wrote,
>>> - 7:5 is represented by the same interval (11/22TET) as
its
>>> inversion, 10:7.
>>> - 8:7, a 7-limit consonance, is represented by the same
interval
>>> (4/22TET) as 9:8, which is _not_ a 7-limit consonance.
>>
>> The second item is not an exception. In 12-tET, 6:5, a 5-limit
>> consonance, is represented by the same interval as 7:6, which is
_not_ a
>> 5-limit consonance. So only the first item is an exception. However,
if
>> you use 22-tET for 9-limit, the second item is an exception, because
9:8
>> is a 9-limit consonance.

>Well, but if you're working in strict 5-limit, the 7:6 simply never
>arises. OTOH, in 7-limit, 9:8 still comes up as a secondary interval,
>so you still may have to resolve the ambiguity.

Well, in 12-tET, 25:16 comes up as a secondary interval, and is
represented by the same interval as 8:5, so by your reasoning you can
have ambiguity in the 5-limit. So the half-octave is really the only
"new" problem for 7-limit 22-tET. As I mentioned before, this interval
actually sounds more consonant when used as a 7:5 then as a 10:7, so a
good "extra" compositional guideline for 22-tET would be to voice all
chords so that the half-octave always represents 7:5.

John A. deLaubenfels wrote,

>I've been frustrated trying to realize retuned .midi files with a really
>nice piano voice

Hi John, good to hear from you again! Anyway, my new computer has a Yamaha
DS-XG built in, and the piano sounds are AMAZING. All your retuned MIDI
files sound infinitely better to me than they did before due to this change.
Don't know how much it would cost you to get one.

-Paul

>I am now thinking of trying to map any piece of music onto 31-tET,

That's what Dave Keenan's adaptive tuning program does!

>with
>the goal of deciding whether or not it makes sense; if it does, then
>the world of more than 12 notes is open; if it does not, then a 12
>note
>universe (howbeit with the possibility of adaptive tuning) seems to
>be
>more fitting.

Well, 31 certainly won't make sense for Beethoven, Schubert, and
Romantic composers, who frequently make use of the diesis (or
equivalently, the diaschisma or Pythagorean comma) in their work.

[I wrote:]
>>I am now thinking of trying to map any piece of music onto 31-tET,

[Paul Erlich:]
>That's what Dave Keenan's adaptive tuning program does!

Kyool. Of course, the particular choice of 31-tET is simply a closed-
system convenience; the analysis would work equally well if mapped onto
an open-ended system. What is important is to keep track of fifths,
whether they're moving up (toward sharps) or down (toward flats).

[JdL:]
>>with the goal of deciding whether or not it makes sense; if it does,
>>then the world of more than 12 notes is open; if it does not, then a
>>12 note universe (howbeit with the possibility of adaptive tuning)
>>seems to be more fitting.

[Paul:]
>Well, 31 certainly won't make sense for Beethoven, Schubert, and
>Romantic composers, who frequently make use of the diesis (or
>equivalently, the diaschisma or Pythagorean comma) in their work.

Even Mozart was known to go 'round the circle of fifths every now and
then just for fun, wasn't he?

JdL

[I wrote:]
>>>Even Mozart was known to go 'round the circle of fifths every now and
>>>then just for fun, wasn't he?

[Paul Erlich:]
>>Someone correct me if I'm wrong, but I don't think he went all the way
>>around. What you may be thinking of is his frequent use of a chord
>>progression that moves clockwise along the circle, instead of the
>>usual counterclockwise.

[Daniel Wolf:]
>Paul Erlich is right. Mozart is very conservative in terms of the
>variety of tonalities through which he passes (it's his treatment of
>vertical dissonances and asymmetrical phrasing that are radical). In
>the Viennese classical repertoire, it is only with late Haydn and
>Beethoven that a circular temperament becomes advantageous.

Thanks for the correction, guys. Paul, by "clockwise", you mean, for
example, C->G->D, etc., as opposed to C->F->Bb ("counterclockwise"),
yes? And Haydn! He lived long enough to experience early Beethoven
works; was he, I wonder, influenced by them? (I'd always heard that he
didn't care much for Beethoven...).

JdL

>Thanks for the correction, guys. Paul, by "clockwise", you mean, for
>example, C->G->D, etc., as opposed to C->F->Bb ("counterclockwise"),
>yes? And Haydn! He lived long enough to experience early Beethoven
>works; was he, I wonder, influenced by them? (I'd always heard that he
>didn't care much for Beethoven...).

I believe Haydn was Beethoven's teacher, and it was Beethovan who
didn't care for Haydn!

-Carl