RE: [tuning] Re: Zarlino's harmonic theory (Jon Wild, Paul Erlich )

Margo wrote,

>First let us take a look at Zarlino's scheme of six basic varieties of
>three-voice sonorities, with the numbers next to each note showing
>string lengths:

>------------------------------------------------------------------
>Interval set: (5,M3,m3) (M6,M3,4) (m6,m3,4)
>------------------------------------------------------------------
>
> | G3 10 | E4 12 | C4 15
>Natural | (m3) | (M3) | (4)
>arrangement 5 | E3 12 M6 | C4 15 m6 | G3 20
> | (M3) | (4) | (m3)
> | C3 15 | G3 20 | E3 24
>
>------------------------------------------------------------------
>
> | A3 4 | D4 3 | C4 5
>Artificial | (M3) | (4) | (m3)
>arrangement 5 | F3 5 M6 | A3 4 m6 | G3 6
> | (m3) | (M3) | (4)
> | D3 6 | F3 5 | E3 8
>
>------------------------------------------------------------------

The last "artifical arrangement" seems especially artificial, if you know
what I mean. Shouldn't that be A3-D4-F4?

Margo wrote,

> C5 15
> 8:5
> G4 20

that should be 8:6 . . .

So in other words, Zarlino had to unknowingly reconstruct the harmonic
series in order to provide a basis for harmony as he knew it. Fascinating!

>Lippius (1610, 1612) interestingly agrees with Zarlino about the
>fourth, classifying it as a perfect concord, and saying that musicians
>who regard it as a dissonance suffer from "delirium."

Many of my disgruntled fellow students in music class would have agreed too.

>[Zarlino] does conclude in
>Chapter 60 of Book III that the "natural" C4-F4-A4, for example, with
>the fourth below and the major third above, is more harmonious than
>the converse arrangement of these intervals, e.g. C3-E3-A4.

>> 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.

>Might this tie in with Zarlino's opinion that C4-F4-A4 is "good," but
>C3-E3-A4 is "better"?

This contradicts what you wrote above. Which was did Zarlino see it?