RE: Re: Numerical accuracy conceptions of past music the orists

Carl Lumma wrote,

>The intervals between melodic
>notes tend towards ratios of small whole numbers for the same reason that
>harmonic intervals do -- the hearing system likes to fit things into a
>harmonic series. The accuracy is far less than with harmonic intervals, at
>least because the roughness mechanism is not involved (so melodies produced
>on a tempered instrument will sound okay).

And in fact, for ratios of 5 or higher, the accuracy with which ratios of
_melodic_ intervals are reckoned is so low that factors of "custom" or
"familiarity" are far more important than the harmonic series. Furthermore,
if your first sentence above is true, why are minor arpeggios at least as
common as major arpeggios in non-harmonized music from around the world (at
least what I've heard)?

>I reject the idea that
>only ratios of 3 are important in melodic tuning. Often, the desire to
>simplify the DM makes the global scale definable in terms of a single just
>interval, and this interval has historically been the 3/2, because 3/2 is
>such a strong consonance. But chains of any strong consonance could work,

Speaking again strictly of melodic tuning, I fail to see any evidence of
that.

>Let's not forget Wyschnegradsky, Carillo, Haba, Mandelbaum, Blackwood, or
>Haverstick!

Surprisingly similar to what I just posted:

>In this century alone, we
>have counterexamples in Wychnegradsky, Haba, Carillo, Ives, Stockhausen,
>Xenakis, Badings, Darreg, Blackwood, Maneri, Penderecki, Carlos, and
>Reinhard.

Kraig Grady wrote,

>Despite the worth of [those composers], they are not as well known!

Kraig, that may be true on the West Coast, but here on the East Coast,
things are quite different. For example, when I submitted my proposal for a
microtonal independent study course to the chair of the music department at
Yale, the only microtonal composer she had ever heard of was Wyschnegradsky!

>That's hard to determine. And what are the driving forces behind custom,
>then?

I don't know; join the sociology list.

>I'd call +/-10 cents a fair estimate of the accuracy of singing the
>"base-two under" 7-limit melodic intervals, which is closer than 19-tone
>equal temperament.

Come again?

>Far greater accuracy is possible when a drone is present.

Absolutely.

>Granted, these
>would be "harmonic" intervals by our current convention,

Yup.

>but... clearly a
>melody over a drone is still a melody in some important ways (otherwise we
>would not find approximations to the linear series in Indian and Persian
>melodies).

Right . . . except I'd replace "the linear series" with "diatonic
structures" -- usually altered ones.

>[Reference Bobby McFerrin's "The Voice"]

That's a great album -- which track are you thinking of?

>You're assuming they approximate a utonal structure? Stuff I've heard is
>usually closer to 7/6 or 19/16. That may be because of the linear series,
>or the harmonic series, but in neither case are subharmonics implicated.
>There is the exception of ethnic flutes, which often play subharmonic
>series melodies, and after hearing such, it isn't too hard to sing them...

Are you buying into the Schlesinger equally-spaced holes argument? No,
flutes aren't quite like strings. But ethnic frettings often incorporate
equal spacings, producing subharmonic scales. Maybe that's one source of it.
I think they could arise on other instruments and even vocally, though; if
the common overtone happens to coincide with a resonance, a subharmonic
arpeggio would really stand out.

By the way, Carl, you've referred to "linear series" twice now. Are you
being Wilsonified?

>Historical evidence, maybe not (although Wilson may disagree).

Any examples?

>But
>certainly the melodic validity of Keenan's chain-of-minor-thirds scales --
>or the 22-out-of-41 scale you proposed -- is not in dispute,

I recall much dispute over the latter from yourself, and please give us an
mp3 of a nice melody with the former.

>to say nothing
>of the myriad of proper MOS's of the 5/4, 7/6, 7/5, and 7/4.

Ditto.

>>>I'd call +/-10 cents a fair estimate of the accuracy of singing the
>>>"base-two under" 7-limit melodic intervals, which is closer than 19-tone
>>>equal temperament.
>
>>Come again?

>7/4, 5/4, 3/2

7:4 is 21 cents off in 19-tone equal temperament. And the point is?

>What do equal-spaced holes on flutes produce, then?

If the top hole is placed in just the right spot (tricky!), the notes with
the bottom hole open will produce a decent approximation of a subharmonic
series, but you still have to worry about edge effects, etc.

>But maybe it's true
>not because of the virtual pitch mechanism, but because of the spectral
>mechanism -- maybe the ear just hears just melodic intervals better because
>it is aware of a common harmonic.

Maybe!

>>By the way, Carl, you've referred to "linear series" twice now. Are you
>>being Wilsonified?

>I was Wilsonified long ago.

So tell me, why does he see modulus-n as having two primary representations,
one as a linear series, and one as a high-limit diamond or CPS with added
notes? What about representations with intermediate numbers of dimensions?

>I disputed the scale only because it's so big -- most melodies would be
>subsets of it. And due to perceptual limitations, even a melody that did
>use the entire scale would probably fail to convey a 22-member melodic
>pitch set. But it surely contains a wealth of great subsets.

That was the point of it. Under those qualifications, it would be hard to
dispute the melodic validity of any large scale with regularities.

>At last, you may try the following, in the proper 8-tone subset of
>Keenan's scale...

Cool! On first hearing, my culturally-conditioned ear wants to hear them as
the usual octatonic scale in 12-equal (C Db Eb E F# G A Bb C) despite the
fact that your scale is a linear chain. If I listen more, I may learn to
hear them differently. You certainly get more harmonic contrast in
your/Keenan's scale.