RE: Open letter to Ken Wauchope and Dave Hill

Ken wrote,

>Another example is the pair of neighbors 24/13 - 13/7, which I heard
>as respectively "medium" - "easy" despite 24 being just an even
>multiple of 3. So I would go a step beyond odd/prime and allow as how
>evenness can't be ignored in this either -- it all just seems to boil
>down to how many partials are involved and how faint and far out in the
>spectrum they are.

I have to agree. The "integer-limit" seems an even better
characterization of harmonic fusion. However, given the pervasiveness of
octave equivalence on the level of musical composition, the
considerations that go into designing a tuning system should typically
consider all inversions and extensions of an interval to be an
equivalence class. Thus the odd-limit concept gains its practical
relevance.

But see the "BUT" below.

Gary wrote,

>I personally am pretty much a fence-sitter on the prime vs. odd
>question, but perhaps it's worth asking: Do you perceive that there's any
>mechanism in our auditory system for detecting powers of two (i.e.,
>octaves)? If so, then why not powers of three or five?

There does seem to be a brain-based mechanism for detecting powers of
two, even in some animals exposed to pure sine waves. It may be that as
a way of efficiently processing auditory information, the factor of two
was chosen by evolution as a period of repetition; that way, (a) the
redundant information contained in the 2nd, 4th, 8th, etc. partials need
not confuse the system; and (b) the pitch space is reduced from some 10
octaves down to one octave, feasible since any particular stimulus
(especially a given human voice) tends to stay within a one-octave range
anyway.

BUT, I read that the cochlea actually winds around once per octave. If
there is any cross-stimulation between one turn of the cochlea and
adjacent ones, then there is a mechanism for at least some degree of
octave equivalence even on the level of the auditory stimulus, which
would of course affect harmonic fusion as well as pitch recognition. If
such a phenomenon exists, then there is no reason to postulate a
brain-based mechanism for recognizing powers of two.

Ken wrote,

>However this was a very narrowly defined exercise concentrating only
>on beating, roughness and harmonic fusion, without addressing any
>other aspects of interval recognition, such as whether it's easier to
>tune a 15/8 than a 13/7 based on affect, and if so, why. On the prime/odd
>controversy, I'm still an agnostic. I've certainly noticed how 7, 11 and
>13 sound exotic to me and 9 and 15 sound familiar, but just why that is,
>I haven't decided.

Fair enough. I attribute this familiarity to the diatonic grammar with
which we were brought up. This grammar is very powerful and has some
amazing properties, as I point out in my paper. The psychological effect
of these properties is profound. 7 and 11 are usually not understandable
in terms of this grammar and so sound exotic, regardless of the degree
of harmonic fusion of these intervals.