Re: magic chord

Paul Erlich wrote...

>In 72-tone equal temperament, take the chord C E- F#- Bb-- (where - means
>flattening by 1/72 octave relative to 12-tone equal temperament). The six
>intervals in this chord are:
>
<->snip<->
>
>One cannot tune the chord in JI and have all the intervals listed in the
>rightmost column. The best one can do is to use 28:25 instead of 9:8; then
>the other five intervals can be just as shown. 28:25 is a 25-limit interval
>but is clearly a case where the limit is of far less importance than the
>proximity to simpler ratios. It is 196 cents, and has a fairly clear
>interpretation as an 8-cent flat 9:8, but it might also be heard as a
>14-cent sharp 10:9. Tuning it closer to 9:8 would increase its consonance
>and the 72-tET version shown here does so with minimal damage to the other
>intervals.
>
>So the "magic chord" is, allowing for intervals to be tempered by up to 4
>cents, a saturated 9-limit chord. Similarly, the augmented triad in 12-tET
>is, allowing intervals to be tempered by up to 14 cents, a saturated 5-limit
>chord. Neither of these chords can be expressed adequately in just
>intonation.

I agree that there are consonant chords which do not have a natural expression in JI. But have you actually compared the two versions of this chord? I did, in root position, and they sounded almost exactly alike. Certainly they shared the same VF. In no way did the 72tET version sound more consonant, in fact the just version sounded a little smoother. I doubt anyone would resent the substitution of one of these chords for the other in a reasonably-paced piece of music.

-C.

>>So the "magic chord" is, allowing for intervals to be tempered by up to 4
>>cents, a saturated 9-limit chord. Similarly, the augmented triad in 12-tET
>>is, allowing intervals to be tempered by up to 14 cents, a saturated
5-limit
>>chord. Neither of these chords can be expressed adequately in just
>>intonation.

>I agree that there are consonant chords which do not have a natural
expression in JI. But have you actually >compared the two versions of this
chord? I did, in root position, and they sounded almost exactly alike.
>Certainly they shared the same VF. In no way did the 72tET version sound
more consonant, in fact the just >version sounded a little smoother. I
doubt anyone would resent the substitution of one of these chords for >the
other in a reasonably-paced piece of music.

What's the VF here? Anyway, I think that 9:8 is too weak a "consonance" to
have an audible effect here, but with the augmented triad, I definitely
prefer the tempered over the just version.

Paul Erlich wrote...

>What's the VF here? Anyway, I think that 9:8 is too weak a "consonance" to
>have an audible effect here, but with the augmented triad, I definitely
>prefer the tempered over the just version.

Of course the VF is given as a distribution, and a fairly even one in this case, which is why the chord sounds as it does. What is it? I listened to all inversions of the just version, rather in haste...

1/1 5/4 7/5 7/4 (1) 1/1 8/5 ...
c e f# a# 70 20 10

1/1 28/25 7/5 8/5 (2) 8/5 7/5 ...
c c# f# g# 70 20 10

1/1 5/4 10/7 25/14 (3) 10/7 1/1 8/7
c e g b 50 40 10

1/1 8/7 10/7 8/5 (4) 8/7 10/7 ...
c d g g# 70 20 10

...of course the 1/1 frequency must be fixed, or you'll get tricked. Here's the scala file to do it (1/1 is mapped to "c" above)...

! magic.scl
!
magic chord test
12
!
28/25
8/7
6/5
5/4
4/3
7/5
10/7
8/5
5/3
7/4
25/14
2/1

I compared only root-position just and tempered versions, but I doubt the VF for the tempered version differs in any significant way from the above.

As for the augmented triad, to which just version do you "prefer" 3tET...

7:9:11
8:10:13
12:15:19
16:20:25

...?

-C.

>As for the augmented triad, to which just version do you "prefer" 3tET...

>7:9:11
>8:10:13
>12:15:19
>16:20:25

All -- each one has at least one interval more dissonant than the tempered
major third.

>>As for the augmented triad, to which just version do you "prefer" 3tET...
>
>>7:9:11
>>8:10:13
>>12:15:19
>>16:20:25
>
>All -- each one has at least one interval more dissonant than the tempered
>major third.

Don't you mean at least one interval more dissonant than the tempered 8/5?

Chords are more than the sums of their intervals. I hear the first three of the above as clearly more consonant than the 12tET augmented triad.

Even so, I find it strange to prefer any one chord to any three.

-C.

Carl Lumma wrote,

>>>As for the augmented triad, to which just version do you "prefer" 3tET...
>
>>>7:9:11
>>>8:10:13
>>>12:15:19
>>>16:20:25

I wrote,

>>All -- each one has at least one interval more dissonant than the tempered
>>major third.

>Don't you mean at least one interval more dissonant than the tempered 8/5?

I was thinking octave equivalence, which seemed appropriate since you did
not list any inversions.

>Chords are more than the sums of their intervals.

Agreed!

>I hear the first three of the above as clearly more consonant >than the
12tET augmented triad.

Well, you may be listening in particular for the effects of otonal-in-JI
chords, which one might call consonance. there are other ways of perceiving
consonance, and these ways are important in music.