More on Just Intonation 1

I apologize for the extemely primitive character of this post, but I
desire my meaning to be crystal-clear to all.

When a singer sings a note, and wishes to proceed upward or downward by
a disjunct interval, he will find the following notes (C used as tonic
here) most convenient, because they are consonant (those who choose to
wage the war to make 7:4 consonant I advise to do some reading,
thinking, and listening; even if they manage the feat, they will find
themselves involved in the thankless task of attempting to erect a
system of tonal relations outside the senario, and based on very weak
consonance at best):

Eb E F G Ab A _C_ Eb E F
G Ab A
5 6
4 5
3 4
2 3
5 8

3 5
5 6
4 5
3 4
2 3
5 8
3 5

The numbers give the ratios of the intervals corresponding to the ratio
of the vibrations.

Once a singer has proceeded upward to Eb, say, he can again proceed
surely by consonant intervals from Eb to any one of the six intervals
above or the six intervals below. Of course, singers do not sing a long
succession of disjunct intervals; they outline them with intercalated
conjunct dissonances. Occasionally too, they sing disjunct dissonances,
but with great difficulty, and less accuracy. Once a singer gets to Eb,
here are his new consonant options, as it were:

Gb G Ab Bb Cb C _Eb_ Gb G Ab Bb Cb C

We can say that a tone such as Gb is related to C, the original tonic
here, _through_ Eb. To find the ratio between C & Gb, we multiply 6/5 x
6/5 = 36/25.

If we perform a like calculation for every one of the notes at a
consonant distance from the initial tonic (here C) we get 12 'tonal'
dissonances that are related via one consonant interval to the original
tonic. Such dissonances can without exception be reached by at least six
(usually eight) different consonant progressions. They, together with
the consonant intervals, form a closed, highly coherent tonal system.
This 'tonal net' is woven exceedingly tight - though it has a hole,
which it is one of the prime objectives of temperament to patch. There
can be _no other_ basis of tonal or musical coherence. Let those who
question this find another such mathematical basis of music; there is
none.

>From this system different selections of tones and sub-systems of tones
can be made (the modes, and the modal genera consisting of modes related
to each other in the same way that the seven diatonic modes are
related). As is well-known, each mode possesses a unique ethos, in the
sense that the ionian differs in effect from the phrygian, and a melody
transposed from ionian to phrygian sounds very different in its general
spirit, but in a predictable way. But all these sub-systems are part of
the original matrix of just intonation, or rather of that temperament,
the 19-tone equal , which most closely corresponds to just intonation.

The question arises: could one take Gb in the above example as a new
jumping-off point for the voice and understanding? Indeed one can - but
the memory of the original tonic fades, and the old tonal relationships
give way to new ones ? on a different tonic, but still using the same
consonances and tonal dissonances. Furthermore, because the voice and
mind can reliably perceive only about 19 melodic pitch classes in the
octave, we here involve ourselves in intervals (the 'atonal' disonances
such as 6/5 x 6/5 x 6/5 = 216/125 or 3/2 x 3/2 x 3/2 = 27/16) which have
no real melodic independence. 27/16 for example is confused with 5/3
melodically ? it sounds like a mistuned 5/3, in harmony a horribly
mistuned 5/3.

There are many just intonations, but only this version describes with
fair accuracy what a singer actually confronts when he tries to sing.

Arthur Benade has some interesting passages on the acoustics of why
singers can sing consonances with such surprising accuracy, but find
disjunct dissonances harder to sing, and this not merely in harmony, but
in melody as well.


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