Reply from JdL (John deLaubenfels)

To: internet:tuning@onelist.com
Date: 03-03-99

Thanks to everyone who responded to my introduction of 03-01-99!

In Digest 72, Paul Hahn writes:
> I agree about the augmented triads, but (being a 31-toner) I kind of
> like the way full diminished 7ths come out in 31TET, with the
> "augmented second" going to 7:6 and the tritones going to 7:5s or
> 10:7s.
I need to review the math of 31TET. Could you give me the exact 31TET
intervals of the chord you're describing?
> I think most people would prefer those in something meantonish, in the
> 1/4 or 1/5 comma range, though IIRC (and I'm not positive that I do)
> Paul Erlich has advocated something closer to 2/7 comma. I lean
> towards 1/4 myself--again, me being a 31-toner, that's probably not
> surprising.
Please forgive my ignorance of terms in common use: what does IIRC stand
for?
Paul, thanks for introducing me to the term "Adaptive JI" - that
describes well what my program does.

In Digest 73, Paul H. Erlich writes:
> 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).
> When you're done, maybe we can make a program that renders 22-tone
> compositions in 7-limit JI!
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. Alternatively, the program
could be made to take 12TET into JI and from there into some other ET,
including 22TET. 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).

In Digest 73, Dave Hill writes:
> I found that in cases of four stacked fourths, making the fourths just
> and letting the minor third between the ends of the stack be 32/27 -
> a Pythagorean minor third sounded fine, even when there was some
> "bite" of beating between the notes of the 32/27.
I actually haven't tried that, but will!
> Fifths or fourths off by much more than 5 cents begin to get
> conspicuous quickly in the kind of music I've worked with - at least
> they sound conspicuous to me and to a lot of people.
I agree. The error of 2 cents in 12TET isn't bad, but I'm reluctant to
shrink fifths by much more.
Thanks for your thoughts on full diminished seventh and augmented
triads.

In Digest 74, Carl Lumma writes:
>> Chords such as full diminished 7ths and augmented triads, creatures
>> of equal temperament, my software simply leaves tuned in equal
>> temperament.
> Have you considered 10-12-14-17 (or, in some cases, 9-11-13-15) for
> the diminished 7th?
No I haven't; thanks for the suggestion!
> The augmented triad can be more tricky. Here's a list of
> possibilities, all of them smoother than 3tET to my ear...
> 1)The "literal" translation, 1/1-5/4-25/16, pretty tense
> 2)"Classical" version, 1/1-5/4-8/5, very similar to #1
> 3)1/1-5/4-19/12, still spacy enough for most applications
> 4)8-10-13, too smooth for many applications
Thanks for the suggestions; I'll try them!
>> Other problem chords, such as those with four or more notes forming
>> consecutive fifths, may also be tuned in 12-tone, in which fifths
>> aren't too bad, and thirds are at least preferable to those generated
>> in 3-limit tuning!
> True, but do the 3rds matter as much as the 5ths?
I agree that 3rds don't matter as much as 5ths, but the error in 12TET
fifths is not particularly objectionable to my ear.

In the next few months, I hope to have a web site up and running, from
which the Adaptive JI program I'm porting from NeXT to PC/Windows95 will
be downloadable. As time permits, I will try to include options for
tuning the output to xxTET, as chosen by the listener.

Again, thanks for the feedback!

JdL

On Wed, 3 Mar 1999, John deLaubenfels wrote:
> In Digest 72, Paul Hahn writes:
>> I agree about the augmented triads, but (being a 31-toner) I kind of
>> like the way full diminished 7ths come out in 31TET, with the
>> "augmented second" going to 7:6 and the tritones going to 7:5s or
>> 10:7s.
>
> I need to review the math of 31TET. Could you give me the exact 31TET
> intervals of the chord you're describing?

In 31TET, a full diminished 7th contains three 8-step intervals, which
approximate 6:5s within 6 cents, and one 7-step interval, which
approximates a 7:6 within about 4 cents. The tritones, 15 or 16 steps,
approximate either 7:5 or 10:7 within 2 cents. Thus, all intervals
approximate 7-limit consonances much more closely than in 12TET.

(Could I use the word "approximate" any more? D'oh, I just did. 8-)> )

I was just thinking about distributing the comma or "vanishing interval"
(126/125 in this case) evenly to minimize the worst-case error, but it
only gains you about a cent.

> Please forgive my ignorance of terms in common use: what does IIRC stand
> for?

If I Recall Correctly.

> Paul, thanks for introducing me to the term "Adaptive JI" - that
> describes well what my program does.

It's a fairly standard term in this forum, I think.

--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 . . . "

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