What I think about 12-et

I think that the normal 12et tuning, standard in the West is a 7-
limit system, which hints at 11 and 13, but only remotely, and also
brings in 17 and 19 (but not in a multiplicative way). I think the
ear interprets the 12et system as basically a Pythagorean system,
with fifths pure (they're so close!)

I think major chords are 4:5:6 and minor chords are (both) 6:7:8 and
also 10:12:15. Commas are all over the place, and include 64:63,
36:35, 81:80 and the like. Comparing the two versions of the minor
chords gives 72:70=36:35, which is about 50 cents, which is the
largest possible error you can have, (but I still think the ear
reduces to the most simple fractions, up to the 7-limit). Depending
on the musical context, you can produce strands like 14-15-16-17-18-
19-20-21, (with middle C=1, this is Bb,B,C,C#,D,Eb,E,F why not 22 for
F#? Well maybe you could say it hints at it. 23 is very problematic!
Don't know what to do about that one. You can continue at G=24 with
24-25-26or27-28...

Everything depends on musical context: Counterpoint, Voiceleading,
and the like. Major chords have a bright third (14c sharp) and minor
chords a dark third (16c flat or 32c sharp if you use 6:7:8...)

Musical cadences often involve working up to a 6/4 chord, like
C-F-A-C'. This usually leads to a V7, C-E-G-Bb-C. (and then of
course, to F-A-C-F') It's interesting that the ratios are 3:4:5:6 and
4:5:6:7:8 respectively. I'm still working out what augmented 6th
chords are all about, for example the German Sixth. It depends how
you treat 6#...

Diminished Sevenths: These get real messy. Tritones are 7:5 or 10:7
One possibility is 10/12/14/17/20. It's like an Escher print:
ambiguous. Another possibility it 5/6/7 for half of it, and 5/6/7
again for the top half. This leads to a comma of 50:49...

Well don't want to get too mathematical, but since this is all basic
arithmetic I thought tuning would be the best group to post this
under. I'll finish with Half-Diminished Sevenths. One interpretation
is 5:6:7:9, another is 1/7 to 1/6 to 1/5 to 1/4...can the ear hear
reciprocals? I have a number of books on Psychoacoustics, one on
Schema Theory, I'll consult that and continue this topic later

Ciao

--- In tuning@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> I think that the normal 12et tuning, standard in the West is a 7-
> limit system, which hints at 11 and 13, but only remotely, and also
> brings in 17 and 19 (but not in a multiplicative way). I think the
> ear interprets the 12et system as basically a Pythagorean system,
> with fifths pure (they're so close!)
>
> I think major chords are 4:5:6 and minor chords are (both) 6:7:8 and
> (Oops! I meant 6:7:9!)
>also 10:12:15. Commas are all over the place, and include 64:63,
> 36:35, 81:80 and the like. Comparing the two versions of the minor
> chords gives 72:70=36:35, which is about 50 cents, which is the
> largest possible error you can have, (but I still think the ear
> reduces to the most simple fractions, up to the 7-limit). Depending
> on the musical context, you can produce strands like 14-15-16-17-18-
> 19-20-21, (with middle C=1, this is Bb,B,C,C#,D,Eb,E,F why not 22
for
> F#? Well maybe you could say it hints at it. 23 is very problematic!
> Don't know what to do about that one. You can continue at G=24 with
> 24-25-26or27-28...
>
> Everything depends on musical context: Counterpoint, Voiceleading,
> and the like. Major chords have a bright third (14c sharp) and minor
> chords a dark third (16c flat or 32c sharp if you use 6:7:8...)
>
> Musical cadences often involve working up to a 6/4 chord, like
> C-F-A-C'. This usually leads to a V7, C-E-G-Bb-C. (and then of
> course, to F-A-C-F') It's interesting that the ratios are 3:4:5:6
and
> 4:5:6:7:8 respectively. I'm still working out what augmented 6th
> chords are all about, for example the German Sixth. It depends how
> you treat 6#...
>
> Diminished Sevenths: These get real messy. Tritones are 7:5 or 10:7
> One possibility is 10/12/14/17/20. It's like an Escher print:
> ambiguous. Another possibility it 5/6/7 for half of it, and 5/6/7
> again for the top half. This leads to a comma of 50:49...
>
> Well don't want to get too mathematical, but since this is all basic
> arithmetic I thought tuning would be the best group to post this
> under. I'll finish with Half-Diminished Sevenths. One interpretation
> is 5:6:7:9, another is 1/7 to 1/6 to 1/5 to 1/4...can the ear hear
> reciprocals? I have a number of books on Psychoacoustics, one on
> Schema Theory, I'll consult that and continue this topic later
>
> Ciao

Hi Paul.
Quite an interesting thought.
However, I dissaprove of your thinking 12eq as a 7-limit system.
Consider the 4:5:6:7 tetrad: it has no equivalence in 12eq.
If you think of a 12eq Dominant-seventh chord as a reference, (e.g.
GBDF), note that it's quite instable, tending to the tonic's triad,
while 4:5:6:7 is very stable.

> --- In tuning@yahoogroups.com, "Paul G Hjelmstad"
> <paul.hjelmstad@u...> wrote:
> > I think that the normal 12et tuning, standard in the West is a 7-
> > limit system, which hints at 11 and 13, but only remotely, and
also
> > brings in 17 and 19 (but not in a multiplicative way).
> > (Message follows)

I agree with Bruno. 12-et is a 5-limit just intonation temperament. Arnold
Schoenberg had suggested that the 16/9 minor seventh was really a 7/4 minor
seventh (rather than the sub-minor seventh which is what it is functionally.
The seventh limit gives an arrangement of sounds that is still apart from what
12-et offers.

best, Johnny