minor chord

Paul H. Erlich wrote:

> Helmholtz preferred 16:19:24 for its combination tones. What is the other
> element? (To be, having the root of the chord octave-equivalent to the
> fundamental of the harmonic series seems important).
I did'nt know that Helmholtz had written 16-19-24 for minor chord. But I
read in his "physiological theory..." something which involve that he
prefered unconsciously 19/16 in place of 32/27. He was listening the
third D-F to tune it on 32/27 by the means of combination tones.
"The Pythagorean minor Third d''' / f''' has a1 for its combinational
tone, which completes it into the chord d / f + a1, and this is not a
perfectly correct minor chord. But as the incorrect Fifth a1 lies among
the deep combinational tones and is very weak, the difference is
scarcely perceptible. Moreover, practically almost impossible to tune
the interval so precisely as to insure the combinational tone a1 in
place of a."
(in Dover edition, translated by Ellis, it's p. 335)
For Helmholtz, a1 is one comma (81/80) below a. If he hear an a, it
involve that the Fourth a - d is 4/3. Mathematically, the only solution
is with 19/16 for the Third d - f. But, he didn't written this fraction.

Paul H. Erlich wrote:
>However, in Bach's time
> 10:12:15 (nearly) would still have appeared on many instruments,
I'm not an expert in temperaments, but when I learned some of them, I
was a little surprise that author always spoke about major chord and
nearly never of minor chord. And then, I looked the minor chord of usual
tonality (c, d, f, g, a): I saw that minor chord were rarely 10/12/15.
It was around 19/16 for half of them (exactly between 32/27 and the
Third of the equal temperament), and one of ten was exactly 19/16.

I try to play 19/16 on my violin. It's difficult but not impossible,
with 2 high tones it's easier, then I prefere 19/16 than 6/5 for minor
chord.

One day, I realized that even if major chord is hard on a piano, minor
chord is'nt so agressive. Then, mathematically, the Third of the piano
is not far from 19/16.

An other "proof" is bound to an hypothesis I developped in my "memoire
de DEA" for explain the difference between tonality and modality. With a
big simplification, it's because 16=2x2x2x2, therefore 16 is a tonic,
that is impossible with the 5 of 6/5, or with the 27 of 32/27.

Paul H. Erlich wrote:
> Besides Helmholtz, Van Eck makes a point of the 16:19:24 chord in his book,
> _J. S. Bach's Critique of Pure Music_.
Thank you for this reference, honestly, I don't know anything of Van
Eck. May you give me the year and the edition please?

After this I must say that, even I admit 16-19-24 for minor chord do not
involve that all chords named minor by traditionnal analysis are really
minor chord. For example, the chord "d-f-a" in the tonality of C, may be
a part of dominant chord, then a 6-7-9 chord. It really depend of the
context, moreover, this problem is a big part of my futur thesis
(1998-2003...).

Xavier CHARLES

I wrote,

>>However, in Bach's time
>> 10:12:15 (nearly) would still have appeared on many instruments,

Xavier wrote,

>I'm not an expert in temperaments, but when I learned some of them, I
>was a little surprise that author always spoke about major chord and
>nearly never of minor chord. And then, I looked the minor chord of usual
>tonality (c, d, f, g, a): I saw that minor chord were rarely 10/12/15.
>It was around 19/16 for half of them (exactly between 32/27 and the
>Third of the equal temperament), and one of ten was exactly 19/16.

I don't think this is a fair analysis. Typically, temperaments from around
Bach's time were symmetrical around the d-a fifth. So if you looked at (c,
d, f, g, a) major, you should look at (b, d, e, g, a) minor. You should find
that those minor chords are as close to 10:12:15 as the former major chords
are to 4:5:6. Let me know if you don't find this to be true.

>I did'nt know that Helmholtz had written 16-19-24 for minor chord. But I
>read in his "physiological theory..." something which involve that he
>prefered unconsciously 19/16 in place of 32/27. He was listening the
>third D-F to tune it on 32/27 by the means of combination tones.

That's what I meant. So your two elements are combination tones and . . .
combination tones?

>An other "proof" is bound to an hypothesis I developped in my "memoire
>de DEA" for explain the difference between tonality and modality. With a
>big simplification, it's because 16=2x2x2x2, therefore 16 is a tonic,
>that is impossible with the 5 of 6/5, or with the 27 of 32/27.

That's the same point I already made, that the root of the minor chord being
octave-equivalent to the fundamental makes it more stable as a tonic.

>Thank you for this reference, honestly, I don't know anything of Van
>Eck. May you give me the year and the edition please?

van Eck, C. L. Van Panthaleon. 1981. _J. S. Bach's Critique of Pure Music_.
Princo, Culemborg, the Netherlands.

Paul wrote:

>
> Xavier wrote,
>
> >I'm not an expert in temperaments, but when I learned some of them, I
> >was a little surprise that author always spoke about major chord and
> >nearly never of minor chord. And then, I looked the minor chord of usual
> >tonality (c, d, f, g, a): I saw that minor chord were rarely 10/12/15.
> >It was around 19/16 for half of them (exactly between 32/27 and the
> >Third of the equal temperament), and one of ten was exactly 19/16.
>
> I don't think this is a fair analysis. Typically, temperaments from around
> Bach's time were symmetrical around the d-a fifth. So if you looked at (c,
> d, f, g, a) major, you should look at (b, d, e, g, a) minor. You should find
> that those minor chords are as close to 10:12:15 as the former major chords
> are to 4:5:6. Let me know if you don't find this to be true.

I think you are right, I'm really not an expert in temperaments...,
minor chords that I learned were essentially in a french book (LATTARD,
Jean, Gammes et temperaments musicaux, ed. Masson, 1988). He had write
(p. 54) : "9 minor thirds are cut by a syntonic comma (like fifths),
but they seemed in tune."
Did you read somewhere the same establishment for major chord?

Xavier wrote
> >I did'nt know that Helmholtz had written 16-19-24 for minor chord. But I
> >read in his "physiological theory..." something which involve that he
> >prefered unconsciously 19/16 in place of 32/27. He was listening the
> >third D-F to tune it on 32/27 by the means of combination tones.
Paul wrote
> That's what I meant. So your two elements are combination tones and . . .
> combination tones?
Yes... but I did calculus of combination tones before reading the
Helmholtz's theory. Then, for me, it's two elements.

Xavier CHARLES

>I think you are right, I'm really not an expert in temperaments...,
>minor chords that I learned were essentially in a french book (LATTARD,
>Jean, Gammes et temperaments musicaux, ed. Masson, 1988). He had write
>(p. 54) : "9 minor thirds are cut by a syntonic comma (like fifths),
>but they seemed in tune."

In quarter-comma meantone temperament, the fifths and minor thirds are
narrowed by a _quarter_ of a syntonic comma. You must have left out that
"quarter".

>Did you read somewhere the same establishment for major chord?

Rememer, a major chord contains a minor third as well as a major third, and
a minor chord contains a minor third as well as a major third. So whatever
applies to minor thirds affects minor chords and major chords equally.

Paul H. Erlich wrote:

> [Xavier wrote] : >I think you are right, I'm really not an expert in temperaments...,
> >minor chords that I learned were essentially in a french book (LATTARD,
> >Jean, Gammes et temperaments musicaux, ed. Masson, 1988). He had write
> >(p. 54) : "9 minor thirds are cut by a syntonic comma (like fifths),
> >but they seemed in tune."
>
> In quarter-comma meantone temperament, the fifths and minor thirds are
> narrowed by a _quarter_ of a syntonic comma. You must have left out that
> "quarter".

You are right, I have lost this "quarter"!

> Remember, a major chord contains a minor third as well as a major third, and
> a minor chord contains a minor third as well as a major third. So whatever
> applies to minor thirds affects minor chords and major chords equally.

Yes, of course, but, since a long time, I'm persuade that E-G in a major
chord is different from E-G in a minor chord. Beginning, it was just
with my hears, know it is also with my mind. Even these two minor
thirds are the sames, on a piano for example, they seem differents to my
hears when the context is different (C Major and E Minor for E-G). It
is'nt directly a proof for 16-19-24 as minor chord, but, excuse me for
repeating my question : do you think that a major chord can seem in tune
even if the major third is to big?
And are you agree with Jean Lattard for his judgment about minor chord?

>do you think that a major chord can seem in tune
>even if the major third is to big?

Well, 4:5:6 is clearly the "magnet" for major chords, while there certainly
might be several for the minor chords, including 16:19:24.

>And are you agree with Jean Lattard for his judgment about minor chord?

Again, Lattard doesn't say anything (in your quote) about minor chords, only
about minor thirds, which occur in both major and minor chords. Lattard is
saying that in 1/4-comma meantone, the minor thirds are about 5 cents flat
of 6:5; this is close enough to 6:5 to sound pretty good, as a 6:5, and not
as a 19:16, which is 19 cents flat of 6:5.

>"9 minor thirds are cut by a syntonic comma (like fifths),
>but they seemed in tune."
>Did you read somewhere the same establishment for major chord?

Xavier: As far as the thirds go, note that in 1/3-comma meantone (advocated
by Salinas), the minor third is a pure 6:5 and the major third is 1/3-comma
_flat_. In 2/7-comma meantone (advocated by Zarlino), the minor third and
major third are both 1/7-comma flat. By far the main reason 1/4-comma was
more popular than these in the Renaissance and early Baroque was the fact
that the fifth is tempered less in 1/4-comma (in fact, the name of the
temperament refers to how much the fifth is tempered). I would definitely
favor 10:12:15 minor chords for music from this time period, since at points
of rest the major chord was substituted anyway (via the Picardy third).