Just Intervals?

[Robin Hufford]

Bradley,
     The M2, I believe, should be 9/8 which, expressed decimally is
1.125.  The equal tempered value is 1.12462048, indicating that 9/8 is a
better simple ratio than 7/6 which is 1.11666667.  Similarly the m2
should be 16/15, again decimally expressed as  1.066666667, the tempered
value, that is,  2^1/12 is 1.059463094.  The 8/7 and 7/6 intervals,
although musical ones nevertheless, are not used on the piano, and, of
course, there are numerous other small  intervals similarly situated.
     The ratio of an interval, if smaller than an octave, taken
ascending,  when mutiplied by the ratio of the interval that is the
musical  inversion of the first, also taken ascending,  must equal two,
that is an octave.   For example, the ratio of an ascending fifth
multiplied by the ratio of an ascending fourth equals two (3/2 * 4/3) =
12/6=2.  The M3 and m6 (5/4 * 8/5) = 40/20 = 2; the m3 and M6 (6/5 *
5/3) = 30/15 = 2,  etc.  This is the same as taking the multiplicative
inverse of the ratio of an interval,
multiplying it by two and then multiplying the two ratios together.
Carrying this further,  the m7 is (x * 9/8  = 2, that is 16/9).
Similarly, the M7 is (x *16/15 = 2, that is 30/16).  The tritone is (7/5
* x  =2, that is 10/7) and could, rightly, be considered to be either an
interval of 7/5 simple ratio or 10/7, as one wishes, although for the
sake of simplicity, it is perhaps better to label it a 7/5 ratio.
     In general, I agree with your opinion, if I understand it
correctly, as to the fundamental nature of just intervals; they are the
paradigm of the tempered intervals we are accustomed to.  Whether these
intervals are just or not, or tempered equally or unequally to some
degree or the other, whether through habituation or inadvertency,  the
just values are, speaking both acoustically and psychologically,  the
fundamental, "special relationship", to use Benande's expression, of
frequencies which have a psychoacoustic value to the human mind.
       There is a more complex aspect of this question, however, than
the simple numerical relationships of the of two frequencies comprising
an interval, or whether and to what degree these are approximated in
various tunings,  and that flows from auditory adaption, habituation,
and that great imponderable, emotionality.  I myself, am aware on
occasion, perhaps due to emotional or psychological reasons that
tempered intervals on a piano when played may one day sound beautiful
and then, somewhat later, unpleasant without any significant change
occuring in tuning. I attribute this to on the one hand hearing the
tempering of the intervals, that is the degree of dissonance induced in
them by tempering, particularly thirds and sixths, and experiencing this
dissonance, and, on the other, hearing the same intervals sound
beautifully, at which time, apparently, I am able, as it were, to sense
past the dissonance and perceive the residual beauty of the
approximated, just,  interval.  This may something of a glass either
half full or half empty thing.   Listening to choral music, great
orchestral performances and string music and, in particular, string
quartets, I don't sense the unpleasant aspect of the tempered values; it
is obvious to me that their tuning of harmonic values, if not completely
just,  is substantially closer than tempered values.
     Another aspect to this, is that of learned frequencies, which is
far more common than frequently thought.  I believe that people in
general are conditioned on an unconscious level by the vast amount of
music they hear, the great bulk of which is in equal temperament, so
that, when asked to make sounds or sing a pitch level, the sounds so
generated will be centered around the  frequencies generated by Equal
Temperament at A-440 to a significant degree such that this cannot
possibly be random. Musicians are even better at this but their
competence is simply a matter of degree and differs from the skills of
untrained individuals only in quantity and not in substance.
     As people have an unconscious, conditioned,  memory of pitch, they
similarly have a conditioned sense of tempered intervals, particularly
thirds and sixths, while at the same time another impetus exists, and
that is toward the sense of just values for these intervals.  I believe,
that, in general, the preponderance of trend is toward the just side of
the equation, especially in ensemble work.
      More variable tuning than can be expressed on a twelve note
keyboard  can be encountered in performance of instruments tuned as they
are played, though, as I am sure you know,   this is both understandable
and desirable. So, even though I personally believe in the fundamental
nature of just intervals, I can see how habituation and adaptation can
lead to differing opinions which typify the very contradictory
perceptions I, at times, experience myself.
Regards, Robin Hufford
"Bradley M. Snook" wrote:

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