Response to Erlich's reply to me in TD 1420

[Erlich:]
>>> I think the fact that the tritone in major nearly forms a 4:5:6:7
>>> with the dominant, and that the tritones in minor nearly form
>>> an 8:10:12:14:17 with the dominant, were not inconsequential
>>> for the development of tonal harmony.

[Monzo:]
>> I don't think the fact that the tritones are close to septimal
>> consonances has anything to do with the development of
>> tonal harmony...
>>
>> The 64/45 would normally be considered the tritone which appears
>> in the 5-limit Dominant 7th chord of 36:45:54:64. 4:5:6:7 is the
>> same as 36:45:54:63.

[Erlich:]
> This whole argument smacks of what I object to in the prime-limit
> theoreticians' approach. Harmonically speaking, an interval is never
> more likely to be interpreted as a higher-odd-limit, lower-prime-limit
> ratio than a higher-prime-limit, lower odd-limit ratio, holding the
> approximation error constant.

[Monzo:]
I'd be interested in seeing some quantifiable info about this.
Have there been experiments which prove this statement?
I think this is an important and overlooked aspect of the prime/ odd
debate in this forum. Lemme see some numbers.

[Erlich:]
> I think the four 5-prime-limit ratios you
> listed have very little to do with the way a tritone is heard
> harmonically, even if tuned to exactly those ratios. Although I agree
> that the Pythagorean tritones, despite their low prime limit, are very
> dissonant, I don't think the JI tritones are "much" more consonant. In
> fact, they may even be more dissonant, since the Pythagoean tritones
> approximate simple 7-limit ratios better.

[Monzo:]
Well, Paul, the reason you say these things is because you favor
ETs, where you must be concerned with consistency, approximations,
etc. In real just tuning, the differences are quite audible. The
5-limit
tritones provide a biting dissonance in the "dominant 7th" chord
that *demand* resolution onto the major or minor triad on the "tonic".

A perfectly tuned 4:5:6:7 chord is only a tiny bit less consonant
than a plain old 4:5:6 triad. There's a big difference in the sound
and feel from the 3- and 5-limit "dominant 7th"s, provided they are
in perfect tune also. And we're not talking about the tritone dyad
by itself, we're talking about a 4-part chord.

[Erlich:]
> Anyway, I maintain that the fact that the tritones reinforced the
> root of the dominant chord when combined with it harmonically
> helped to define major and minor as the "tonal modes", while
> other modes without such harmonic-melodic focus fell out of use.

[Monzo:]
It's important to remember that the 36:45:54:64 dominant 7th chord
arose as a result of using a circumscribed set of pitches. In a
5-limit system which is using only the 7 diatonic scale pitch-classes,
this particular set of proportions could only occur over the dominant.

I really think that in perfect just tuning, the 3- or 5-limit chords as a
whole beat too much to be nailed down as a 4:5:6:7

However, I'm am willing to grant the possibility that Erlich's idea
may have some validity. I certainly agree that in hearing the
music the 3- or 5-limit tritones would normally be interpreted as
the much simpler 7:5 _if we are isolating the tritone_.

------------------------------------------------------------------------
[Me:]
>> Chords can be tuned in JI with a variety of ratios
>> that are close enough to a "target" that many of
>> these ambiguities can be explored rationally.

[Erlich:]
> Yes, Joe, you are correct about that, but it seems
> that you may have been implying otherwise in your
> post about tritones. That is, I don't think 64:45 can
> itself be a "target" but is instead close to several
> possible targets. Other notes in the chord can clarify
> which target that is, and in the case of dominant
> seventh chords, only 7:5 can fit with all the other
> ratios to define a simple "target chord".

[Monzo:]
I'm glad you didn't misinterpret me here: I should have
specified that the "target" would be a lower-prime/
lower-odd ratio. You're exactly correct that any of
these more complicated ratios would imply a 7:5
(or 10:7, depending on context).

Joseph L. Monzo
monz@juno.com

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