RE: magic chord

>>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.

>You can't change patches in real-time on the samplers.

If by that you mean that there will be a delay when you select a patch -
a delay to load it from disk - then yes, that's certainly true. But the
ASR's designers didn't say something like "well, this is a sampler, so we
can't switch patches, therefore we won't even respond to patch-change
messages".