Reply to Dave Hill

Dave Hill wrote,

>I just wanted to respond briefly to Paul Erlich's
>suggestion that mean tone temperament rather than
>just intonation was most probably regarded as the
>"real" basis for musical tuning up through the
>19th century and early 20th century.

> It's my impression based on my reading of earlier
>writings that most musical theorists before the 20th
>century actually believed that there was a correct way for
>musical tuning based on nature. In ancient times it
>had been discovered that there was a certain
>psychological "resonance" aroused in listeners when
>sounds produced on otherwise identical strings having
>length ratios 2:1 or 3:2 and compound ratios derived
>from these ratios were heard. Then by the 16th
>century ratios of 5 in string lengths were recognized
>as being significant in musical harmony (Zarlino). Until the
>17th century, when the overtones were discovered by
>Galileo and Mersenne, it was believed that somehow
>the numbers themselves had been assigned a mystical
>importance by the Creator. After the discovery of
>the overtones, it was realized that there were physically
>unique properties - not always completely understood -
>possessed by pairs of sounds having small integer frequency
>ratios one to another.

So far, we are in complete agreement.

>In a "Dictionary of the Arts..."
>quoted from in Jorgenson which had been published in
>1764, the author states that in the quarter comma
>mean tone temperament, which he speaks of as the common
>or vulgar temperament, there are small departures from
>the true frequency ratios for the fifth, the minor
>third, and the diatonic 16/15 and chromatic 25/24
>semitones of 1/4 comma or in modern terms, about 5.4
>cents. He states that although these departures from
>the just ratios can be clearly heard, they do not
>impair the quality of the music performed in this tuning
>system.

Here it must be noted that any theorist who actually bothered to _listen_ to
these ratios (as Partch is fond of pointing out) would notice that the fifth
and minor third would sound more "locked" in just intonation than in
1/4-comma meantone, while there would be no qualitative difference whatsover
between the diatonic and chromatic semitones in meantone on the one hand,
and in just intonation on the other -- only a quantitative difference in the
rate of beating caused by the deviation from 1:1 itself.

>The UCSD music library has quite a few 19th
>century musical tomes and they usually have rather
>elaborate schemes of integer frequency ratios depicted
>in articles on musical harmony and intonation - some
>are in languages which I don't understand.

Many of the more elaborate schemes of frequency ratios can be considered
worthless for the reasons above and below.

>I may be being misunderstood as to what I mean by music
>in just intonation. I mean music in which the notes are
>produced at pitches called for by the harmonies being
>sounded. A 12 per octave keyboard tuned in a "just intonation"
>scheme can only produce a limited number of harmonies where
>the frequencies of all the notes are in just intonation.
>If one tries to play many other chords as if the keyboard
>were in equal temperament, one will sound many dissonant
>chords so that the result would not be very harmonious and
>it certainly wouldn't be in just intonation, although there
>have been people who have done just that in order to
>demonstrate how terrible just intonation sounds! Such a
>"demonstration" could be presented very convincingly, but
>it would be deceptive.

No, that is not the type of just intonation I was talking about. I was
talking about the type of just intonation for which 53-tone keyboards were
devised in the 19th century. The problem with this type of just intonation
is that the structural syntonic comma, an interval clearly assumed
melodically inaudible by Western composers since 1500 (see Mathieu, _The
Harmonic Experience_, and Blackwood, _The Structure of Recognizable Diatonic
Tunings_), becomes audibly large -- and virtually any substantial piece of
Western music since 1500 would require arbitrary judgments on the part of
the "arranger" as to what to do with the comma -- either shifts or drift
would have to result. Such renditions ultimately offend the musical ear; the
term "slimy" is one I recall from the New Grove Dictionary. By the time of
Beethoven even extended meantone is insufficient to capture all the
equivalencies assumed in the music -- other commas such as the diesis and
diaschisma make a closed system of 12 (within melodic discrimination)
pitches indispensable for the music.

Now, that is not to say that meantone temperament isn't ultimately derived
from an ideal of just intonation in vertical sonorities. It is, with the
additional constraint of syntonic comma equivalency. But meantone
temperament is not the only way to reconcile these desiderata. Vicentino's
second tuning of 1555 does so as well, by using pure JI sonorities for
vertical triads but forming these from _two_ 1/4-comma meantone chains tuned
1/4 comma apart. Thus any comma shifts are restricted to 1/4 comma in size,
melodically unobjectionable as a unison. John deLaubenfels' current adaptive
tuning work takes this idea a step further, ideally allowing any piece
written in any tuning system (for now only 12-tone ones) to approach JI in
its verticalities while eliminating any comma-like melodic shifts or pitch
drifts. Both approaches, however, would have been impracticable on an
actual, human-operated keyboard without computerized gadgetry, so would have
been of relatively little interest, as compared with meantone, in writings
of past theorists. Theorists, that is, who went beyond elaborate schemes of
frequency ratios and saw the full musical implications of tuning systems --
theorists like Woolhouse, for example.

Getting back to the original point, any composer from Monteverdi to Mahler
might think of G# and Ab as two different notes, but would think of D as D,
a single pitch, not a comma higher here, a comma lower there. Western
musical reality until the early 20th century had, at its center, the
diatonic scale, the scale of all known music in that world and the basis of
its notation. Elaborate schemes of frequency ratios may have profligated at
the periphery, but for any Western musician actually making music, there was
only one size of whole tone; modulating to the dominant meant only one note
changed, not two; and progressions like I-vi-ii-V-I and I-IV-ii-V-I had no
small shifts of sustained pitches, nor overall drifts of pitch level. Though
such curiosities may have been ackowledged in theory by otherwise
knowledgable theorists like Rameau and Zarlino, they played no part in the
historically important music of their era or of any other in our history.