Re: 13-limit triad

>>1. 13/12 - 4/3 - 5/3
>
>Can't you reduce this to 13/4 - 4 - 5?

I think so.

>>I think in reality it's 4/5 - 1 - 5/4

I don't think so.

>>I will say that my conclusions supported my long-standing belief that
>>octave-invariance is out the window when you go above the 7-limit.
>
>Although I basically agree with you, I wouldn't say it goes out the
>window at a particular limit. Octave invariance is never complete no
>matter how low the limit...

I agree. I do think there may be some uneveness in the curve when you
cross over to 9, since for the first time the basic chords insist on spanning
more than an octave in every voicing...

>At the 7-limit, it is certainly true that 7:5 is more consonant than
>10:7 and chords that include the former are perceived as more stable
>than those that include the latter. Even in 22-tET where these intervals
>are both represented by a half-octave, chords where the half-octave
>represents a 7:5 tend to sound a good deal more stable than those where the
half-octave represents a 10:7.

Could you give an example or two?

>In most contexts, 7:6 is quite a bit more consonant than 12:7.

I think I can agree with that.

>Even in 12-tET, a minor sixth alone can be heard as a dissonance,
>in sharp contrast to a major third. Triadic harmony will tend to
>clarify the meaning of the minor sixth to the point where it can
>no longer be considered dissonant.

Gee, I don't know about this one... Harmonizing a melody with sixths, for
example?

Carl