Pitch Relations and Frequency Ratios

Paul Rapoport wrote:

> [The] Syntonic comma may be defined algebraically and
> then applied to any tuning system in which that formula makes sense.
> To say, for example, that 53-tET has no syntonic comma isn't helpful.
> Sure it's tempered, but so are all ET intervals, including the perfect
> fifth, a term which is not restricted to 3/2 in common usage. What's
> so special about 81/80?

When this discussion cropped up earlier (in May, I think), I was unable
to send out mail due to certain technical problems, so I'll take the
opportunity now to mention what I had in mind then.

Paul, I think this problem admits of a resolution if we agree to treat
pitch relations as a phenomenal, secondary-quality discourse, operating
within the conceptual system of a given musical tradition; on the
physical, primary quality level, we can speak of frequency ratios. I
would suggest that the problem you have encountered arises from the
confusion of these distinct levels of discourse: the perfect fifth
belongs to the phenomenal level, the syntonic comma to the physical. If
I were to say "pure fifth", however, it would be natural to assume that
a 3/2 was intended; "pure fifth" is opposed to "impure fifth", i.e.
tempered, whereas "perfect fifth" is opposed to "imperfect fifth", i.e.
augmented or diminished. To specify an interval on the phenomenal level
leaves the physical realisation of the interval underdetermined: a
perfect fifth could be a 3/2, or any meantone or ET analogue; what
matters is that the perfect fifth, regardless of the tuning system in
which it may be realised, should _function_ as such.

The syntonic comma is already well-defined as 81/80, and its etymology
specifies the context in which it arises: the syntonic diatonic of
Ptolemy, and implicitly the difference between this and the ditonic
diatonic. Its use, established since the 16th century, for categorizing
the various shades of meantone entrenches this sense further.
Nevertheless, there is probably no harm in saying, at the beginning of
an article on ETs, that you will be discussing analogues of the syntonic
comma, and that for convenience you will thereafter refer to these as
"syntonic commas" or simply "commas", on the understanding that the
meaning of the terms is derived, not literal. But to treat "syntonic
comma" as if it were a phenomenal term such as "perfect fifth" is to
merge the phenomenal and physical levels, and open the way to all manner
of confusion.

Consider how this applies to Greek theory: Ptolemy was, in part, arguing
at cross purposes with Aristoxenus. Ptolemy, giving an empiricist slant
to Pythagorean theorising, was discussing intervals as frequency ratios,
while Aristoxenus, trying to supplant the Pythagorean tradition, was
trying to argue that intervals should be defined purely within the
concepts native to music, as pitch relations on the phenomenal level.
(Aristoxenus then brought trouble upon himself by constructing a
numerical model which was too heavy for such a foundation, but we can
leave this aside for present purposes.) In spite of the exclusive claims
of each theorist, there is no logical incompatiblity between the two
approaches, if they are understood to operate on different levels:
intervals qua pitch relations, and intervals qua frequency ratios.

> Any JI term that has an ET equivalent is probably OK to use in the
> tempered sense, as long as that use is understood. This is where
> "quartertone" is different: it's already a tempered term.

Whether quarter-tones are defined functionally as pitch relations, or
physically as frequency ratios is a moot point. I would judge that
"quarter-tone" is context-dependent in this respect: if I were reading
through a Vyshnegradsky score, I would understand them as 2^(1/24)/1
frequency ratios because the tuning system in which the score is to be
realised is specified by the composer. But if I were reading, say, a
Ferneyhough score, I would not make any such assumption: the flautist,
violinist, etc. is required to produce a sound which is sufficiently
distinct from its neighbours to have its own audible function. In such
music, tones are not specified to be 2^(1/6)/1, and semitones are not
specified to be 2^(1/12)/1, so we can hardly suggest that quartertones
are somehow to be assigned a precise frequency ratio. Audible function
is all that counts in such contexts, placing quarter-tone notation on
the phenomenal level, and not the physical.

> I know we've been over this before, but I think I'm being consistent.
> As far as I'm concerned, anyone is welcome to talk about quartertones
> in 31-tET; you just won't find them in the tuning.

Lowinsky spoke of quartertones when discussing Vicentino's enharmonic
writing; since Vicentino's assignment of ratios to intervals was highly
confused, this usage might not be quite so reprehensible as it would
seem. If Vicentino had unequivocally specified, say, 31TET, then
"quartertone" would be out of place; but considering Vicentino's
vagueness, I think we can charitably allow Lowinsky the functional,
phenomenal sense of quartertone.

Interestingly enough, 14th and 15th century music requires two levels
for specifying pitch relations (one for notation, one for singing), as
well as another for specifying frequency ratios. But that's another
story -- if anyone's interested, I'll explain what I mean.



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