source file: m1443.txt Date: Wed, 10 Jun 1998 20:08 +0100 (BST) Subject: Re: magic chord From: gbreed@cix.compulink.co.uk (Graham Breed) >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. The mistuning is as important as the 5-limit bit. Augmented triads sound more dissonant in 31-equal, so this must be relevant. 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, but three out of tune intervals will still be worse than one. 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. I generally find 9/8 to be much the more consonant interval, although they are almost equally complex. 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. 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. I need to do more listening to be sure of these things. Does anyone have a good chord progression that exploits this comma?