24-eq and notational consistency

[Paul Erlich, TD 163.4]
> Max Meyer (and more recently, Joe Monzo) have suggested 24-tone
> equal temperament (quartertones) as a way of achieving better
> 7-limit harmony.

I suggested 24-eq *NOTATION* as a way for musicians with
variable-pitch instruments (or voice) 'as a way of achieving
better 7-limit harmony'. NOT 24-eq *tuning*.

There is a difference.

(And in any case, I still agree with you that even for notation
only, 72-eq is far better than 24-eq or any other smaller system.)

I understand perfectly well the consistency issue you pointed
out, but as I was trying to emphasize a few weeks ago in my
discussion of 144-eq notation, using an ET's notation in a
situation where the performers are expected to approximate
JI (i.e., NON-keyboard, NON-fretted), the consistency issue
is not as much of a problem here as it would be in considering
that ET as a tuning system _per se_ (as it must be on keyboard
or fretted instruments).

24-eq notation will still lead to a better approximation
of 7-limit JI than 12-eq notation.

I agree completely with the points you make regarding consistency
in 24-eq *tuning*. The debate here, for me, is the importance
of the consistency issue in the situation of acheiving quasi-JI
(or even actual JI, if the instruments are so tuned) thru the
use of an ET notation.

As an example:

There is a serious consistency issue in regard to 12-eq's
representation of 11-limit ratios.[*] And yet, if I were to
compose an 11-limit piece for unaccompanied voices and notate
it in 12-eq, my bet is that I'd get a result that's pretty
close to what I seek.

Good musicians (again, on variable-pitch instruments) will use
their ears and their understanding of the composer's harmonic
language to adjust what they see in the notation. If I explain
to the choir that I intend for them to produce 11-limit harmonies,
(assuming they know what 11-limit ratios sound like) they will
adjust the inconsistent notation to fit what I ask of them.

Barbershop Quartets routinely sing consonant 7-limit ratios
for notes that are notated in 12-eq with 30 to 35 cents error.

[Erlich]
> The 7:4 is only 19 cents sharp of 4 3/4 tones, 7:5 is only
> 17 cents flat of 3 tones, and 7:6 is only 17 cents sharp of
> 1 1/2 tones. But this will not lead to better 7-limit tetrads.

I think it would be a little more clear if we used 24-eq
nomenclature, rather than that for 6-eq:

4 3/4 tones = 2^(19/24) = 950 cents
3 tones = 2^(12/24) = 600 cents
1 1/2 tones = 2^( 6/24) = 300 cents

[Erlich]
> For example, adding the 4 3/4-tone interval above the root of
> a 4:5:6 (major) triad (tuned as in 12-equal) leads to a
> representation of 7:5 which is an unacceptable 33 cents flat.

OK, since we're discussing problems of notational consistency,
how about if we notate the chord you're suggesting *in* 24-eq
instead of the confusing mixture of JI, 6-eq, and 12-eq you used:

4 3/4 tones = 2^(19/24) = 950 cents
'6:4' = 2^(7/12) = 2^(14/24) = 700 cents
'5:4' = 2^(4/12) = 2^( 8/24) = 400 cents
'4:4' = 2^(0/12) = 2^( 0/24) = 0 cents

Paul, I've *listened* to this and it sounds like a 7-limit tetrad
to me! This is a good approximation of 4:5:6:7. Any musician
with a good ear, confronted with this notated chord, would probably
sharpen the highest note a bit to acheive a good 7:4.

In fact, using 2^(7/24) = 350 cents for the 'major 3rd'
also gives what I think is a decent approximation of 4:5:6:7
- if the musical context allows it to be heard that way.

(Actually, this chord is a much better approximation of an
11-limit tetrad than it is any 7-limit chord, and since I'm
familiar with the sound of 11-limit chords, that's what it
really sounds like to me.)

But the point is, if a group of musicians read this and
played/sang it, chances are that it would come out sounding
an awful lot like 4:5:6:7.

[Erlich]
> Using 12-equal, the 7:5 is only 17 cents flat, but the 7:6 is 33,
> and the 7:4 is 31, cents sharp.

I understand the point of what you're saying here to be to
emphasize that 24-eq offers no improvement over 12-eq.
Mathematically, true enough.

This is an important consideration when constructing a
tuning for a fixed-pitch instrument.

But for performers on variable-pitch instruments, there is a
*suggestive* quality to reading notation that I think you're overlooking.

I think this may be the way Ezra Sims looks at the situation:

[Erlich]
> Wendy Carlos, Yunik & Swift, and Ezra Sims are among those who
> don't appear to have realized the implications of inconsistency.
> When I met Sims and we discussed the issue, he admitted that I
> raised a good point but told me that triads went out a long time
> ago. I don't know what he meant, since much of his music uses
> more than two voices.

Sims, in a letter to me, wrote:

[Ezra Sims, personal communication]
> I'm a thorough traditionalist and deeply disagree with -- and
> indeed, I think, disapprove of -- notations that try to coerce
> the *intonation* (which is phenomenal and varies with the era
> and hour), instead of transparently describing the *notes*
> (which are ideal and do not change, however they're tuned.)

In this case it's also unclear to me exactly what he meant,
but I'm pretty sure it's along the lines of my argument above:
that notation and tuning really are apples and oranges, at least
to some extent.

-monz

[*] It is interesting in this regard to consider Schoenberg's
explanation of the chromatic scale presented in his _Harmonielehre_
[1911, p 24 and 25 in the English translation].

He is describing the notes as combinations of overtones of
the three 'principal' tones: the I IV V degrees of the 'major
scale' based on 'C'.

He uses the following (I've added the ratios and cents values)
to describe the 11th partials:

11th of G = C 33:32 = 0.53 cents
11th of C = F 11:8 = 5.51 cents
11th of F = B 11:6 = 10.49 cents

The inconsistency is clear: to be notationally consistent,
the notes should have been called either B, F#, and C#,
or Bb, F, and C.

But what's really interesting to me is that he (consistently)
used the letter-name that falls on the 'wrong' side of the
(intonational) inconsistency!

The inconsistency that Erlich would point out would have called
these notes Bb, F#, and C#.

Joseph L. Monzo monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
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>[Erlich]
>> For example, adding the 4 3/4-tone interval above the root of
>> a 4:5:6 (major) triad (tuned as in 12-equal) leads to a
>> representation of 7:5 which is an unacceptable 33 cents flat.

>OK, since we're discussing problems of notational >consistency,
>how about if we notate the chord you're >suggesting *in* 24-eq
>instead of the confusing mixture of JI, 6-eq, and >12-eq you used:

>4 3/4 tones����� = 2^(19/24) = 950 cents
>'6:4' = 2^(7/12) = 2^(14/24) = 700 cents
>'5:4' = 2^(4/12) = 2^( 8/24) = 400 cents
>'4:4' = 2^(0/12) = 2^( 0/24) =�� 0 cents

>Paul, I've *listened* to this and it sounds like >a 7-limit tetrad
>to me!� This is a good approximation of 4:5:6:7.

Not significantly better than 12-eq's approximation (considering all 6 intervals).

>Any musician
>with a good ear, confronted with this notated >chord, would probably
>sharpen the highest note a bit to acheive a good >7:4.

Let's say an ensemble trained in quartertone notation is playing a very difficult score and two of the members find time to rehearse outside the group. The chord above is buried somewhere in the score. One is assigned to the "third" of the chord above and the other is assigned to the "seventh". If they are truly trained in quartertones, then when they encounter the 550-cent interval between them in the score, they aim not for a 7:5, which is twice as close to 600 cents and thus categorized as the latter, but something in the same category as an 11:8.

Think of a world where musicians either have fixed pitches on their instruments or are trained to produce exact pitches in the new system in order than they may more closely be able to observe a composer's wishes, even if their own creative taste would lead them to alter some pitches. A composer who wants to change the way people hear music would clearly prefer a world like this.

Even outside that world, there will typically be more than one musician playing the chord, and each will simultaneously attempt to adjust their tuning to improve the chord. Such efforts are not likely to be psychically coordinated, and especially in the case of inconsistency may lead to an unstable tuning situation.

>In fact, using 2^(7/24) = 350 cents for the >'major 3rd'
>also gives what I think is a decent approximation >of 4:5:6:7
>- if the musical context allows it to be heard >that way.

Again, not significantly better than 12-eq's approximation.

Joe Monzo wrote,

>There is a serious consistency issue in regard to >12-eq's
>representation of 11-limit ratios.

I wouldn't call it serious, as a slight stretching of the "gestalt" harmonic series, as documented by Terhardt and others, would imply that 12-eq consistently represents all ratios through the 11th partial, and the 12th partial would involve a mild inconsistency of less than 2�. The inconsistencies of 24-eq in the 7-limit are on the order of 16� and do not disappear with any stretching of the harmonic series.

>Barbershop Quartets routinely sing consonant >7-limit ratios
>for notes that are notated in 12-eq with 30 to 35 >cents error.

Yes, and there's no inconsistency here.

>11th of G = C 33:32 =� 0.53 cents
>11th of C = F 11:8� =� 5.51 cents
>11th of F = B 11:6� = 10.49 cents

>The inconsistency is clear: to be notationally >consistent,
>the notes should have been called either B, F#, >and C#,
>or Bb, F, and C.

>But what's really interesting to me is that he >(consistently)
>used the letter-name that falls on the 'wrong' >side of the
>(intonational) inconsistency!

>The inconsistency that Erlich would point out >would have called
>these notes Bb, F#, and C#.

Never. That's yet another kind of consistency concept that couldn't be further divorced from my own. Using the Erlich consistency that one gets with stretched partials, I would call these notes B, F#, and C#.