source file: m1444.txt Date: Thu, 11 Jun 1998 16:33:51 -0400 Subject: RE: magic chord From: "Paul H. Erlich" >>Think about triads in 12-equal. Even though all the intervals in the >>augmented triad can function as consonances in major and minor triads, >>the augmented triad is dissonant. And the thirds aren't even close to >>any simple ratios other than the 5-limit ones. Can you say that all >>three intervals in an augmented triad are functioning as 5-limit >>consonances? >Thirds in 12-equal are mistuned 5-limit intervals. So, the chord as a >whole will be a mistuned 5-limit chord. And what would be the correctly tuned version? >The mistuning is as important >as the 5-limit bit. Augmented triads sound more dissonant in 31-equal, >so this must be relevant. I strongly agree that augmented triads sound more dissonant in 31-equal, or with just major thirds as in JI or meantone, than in 12-equal. Does that argue against anything I said? >The 9/8 in 31-eq is tuned much better than the 5/4 in 12-eq. So, it >should be more recognisable as a consonance. Unless you think higher >limit intervals generally require better tuning, In this context, yes. The 5/4 is by far the strongest interpretation of the 4deg12 interval, while the 9/8 and 10/9 have nearly equal claims to the 5deg31 interval. >but three out of tune >intervals will still be worse than one. I was only making an analogy, not a direct comparison. >I'll write the chord under discussion as Eb-G-A-C#. The G-A interval >_is_ ambiguous as to 9/8 or 10/9 -- I made a mistake in my working >before. The chord does sound worse in 31-eq than 1/5 comma meantone or >schismic temperament, probably because of the poor 9/8. However, that >interval isn't so bad as to be unimportant. I don't see that its >proximity to 10/9 makes it _more_ dissonant, but maybe nobody else does >either. The chord sounds worse in golden meantone, implying that G-A >really should be 9/8 and not 10/9. Can you flesh out exactly how you see that one implies the other? Don't forget that the 28/25 is naturally closer to a 9/8 than a 10/9, so tuning it to a 10/9 means you've had to distort the other intervals more. >I generally find 9/8 to be much the >more consonant interval, although they are almost equally complex. Well, 9/8 has that "rooted" stability since it has a power of two in the denominator. >I find that G-Eb-A-C# and Eb-G-C#-A sound better than A-Eb-G-C# and >Eb-A-C#-G in 31-eq. In golden meantone, they all have roughly the same >consonance. I suggest this is because of the 9/4 vs 10/5. Come again? >The 31-eq 9/8 and 28/25 both being sharp is the most relevant thing >here. Ideally, the tempered interval should be between the two just >ones for them to both be well tuned. Er, do you really think 28/25 is any kind of "just" interval in the sense of being a "target" you could really aim for without tuning the other intervals in the chord? >I need to do more listening to be sure of these things. Does anyone >have a good chord progression that exploits this comma? Anything that goes from the dominant of the dominant to the augmented sixth chord is exploiting the vanishing of the 225/224.