Topic: Re: magic chord
1 scales
| File | Description | Notes | Period (ยข) | Limit |
|---|---|---|---|---|
| magic | magic chord test | 12 | 1200.0 | 7 |
Thread (6 messages)
From: Carl Lumma (1999-06-22) Subject: 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.
From: Paul H. Erlich (1999-06-23) Subject: Re: magic chord >>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.
From: Carl Lumma (1999-06-24) Subject: Re: magic chord 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.
From: Paul H. Erlich (1999-06-25) Subject: Re: magic chord >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.
From: Carl Lumma (1999-06-26) Subject: Re: magic chord >>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.
From: Paul H. Erlich (1999-06-28) Subject: Re: magic chord 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.