Re: [tuning] Digest Number 929

I had written:
>>
>> octave divided harmonically
>> ______________________________________
>> | | |
>>
>> 5th divided harmonically
>> ______________________
>> | | |
>> C E F G A C
>> |____________|________|
>> 5th divided harmonically
>>
>> |________________|_____________________|
>> octave divided arithmetically
>>
>>
>> You can't get this starting on any other note.

and Paul wrote:

> I beg to differ: you could do this starting on G (the notes would be G B
> C D E G).

correct - I'll look again at the notes I made from Zarlino, but I think
what he liked about C, beyond the stuff he shows in a diagram like the one
above, was that you also got a perfect harmonic triad on G, the harmonic
mean of the octave. *This* you don't get in a mixolydian mode. (Also about
this triad on G: requiring the 5th to be pure is what differentiates
Zarlino's tuning from Fogliano's - doesn't Fogliano tune C-D 10:9, so that
D-A is pure and the wolf is between G and D?)

Speaking of major/minor and triads, it might be important to remember that
Zarlino considered E-G-C, say, a *minor* sonority because it has a minor
3rd and 6th above the bass. There is still no recognition in Zarlino of
inversional equivalence of triads. Lippius has been mentioned by Monz and
Margo, and he's the guy to thank for that.

It's also interesting that Zarlino can't escape the conclusion that the
6/4 chord 3:4:5 "should" be more consonant, according to his theory, than
the 5/3 4:5:6 - just like Helmholtz, 300 years later. I can't quite
remember how he squirms out of it - does anyone have it handy to check?

Jon

Jon Wild wrote,

>It's also interesting that Zarlino can't escape the conclusion that the
>6/4 chord 3:4:5 "should" be more consonant, according to his theory, than
>the 5/3 4:5:6 - just like Helmholtz, 300 years later.

Clearly, triadic harmonic entropy can't escape that either. Hence I invoke
an additional concept, "rootedness", which is justified partially by the
masking phenomena in the ear which make the lowest note of each of these
triads sound the loudest.