Worthy & Worthless

My qualification of some temperaments as 'worthless' seems to have
caused much offense to a few. By 'worthless' I do not mean 'morally
reprehensible' or 'indicating criminal insanity in those who use them'.
I mean that certain temperaments grossly reduce the number of
expressible, distinct melodies as compared with other temperaments _or_
deprive us of even so much consonant harmony as is found in the senario.
(I find septimal intervals quite valuable as dissonances, but happen to
believe the senario includes all consonance... that is another subject
though.) Actually, every one of the worthless temperaments that have
some following (17- 22- 29- 34- 41- & 53-tone equal) impoverish _both_
melody and harmony. This is too complex to even begin to prove here.
This will have to wait for its own post.

Some assume that every system has at least a few interesting features,
and therefore, I presume, that music should use an infinity of different
temperaments, with perhaps the best (if they can bring themselves to
make any judgements at all) enjoying slightly greater use. This is the
opposite error to that made by previous generations of theorists, who
with the enormous resources of the 19-tone equal staring them in the
face, looked at its flat fifth for a second or two, dismissed this
temperament without taking five minutes to inquire if this could be
remedied, and went on to lose themselves in the serpentine coils of 53
tones in the octave. I am thinking of Ellis in particular, but thousands
more made the same mistake. Every generation seems to delight in making
the opposite mistakes from those made by its predecessors.

This universal, unconditional tolerance betrays an astounding naivet?
about human nature, and also about musical reality. For those who
haven't noticed, 12-tone equal rules our musical life perhaps more than
any musical system has ever tyrannized over a culture. One is reminded
of the Chinese with their sacred bamboo pipes and later, their edicts
against wicked temperers of the pythagorean scale. Those who expect
instrument-makers to provide us with thirty different fundamentally
different kinds of guitar or keyboard or oboe, and who imagine music
teachers will be found to instruct pupils on them, inhabit the happy
realm of Faerie, where all things are possible.

Happily however, anyone who dispassionately investigates the subject of
temperament with the view to expanding our musical resources will at
once (I mean, after 10 years or so) find one temperament - and one only
- that _dramatically_ expands the melodic and harmonic resources of the
12-tone equal... and of just intonation as well! This is 19-tone equal.

I would like to now begin to explore why this is so, at least in
outline. Here we have a temperament that opens up to us an immense world
of enharmonic melody, to begin with. This, as the great theorist
Francisco Salinas long ago noted, is the _only_ system that does so, for
it alone provides us with an interval, the 1/3 tone, that is wide enough
to always effect a change in melody when a given melodic member is
altered thereby, but narrow enough never to be confused with the
diatonic semitone.

To give some idea of just how immense this universe is - and remember
these are _real_, usable resources, not purely theoretical, aurally
imperceptible variations - compare the number of modes (I here use the
term loosely merely to indicate a collection of seven notes used in a
melody) consisting of 12 notes taken 7 at a time:

nCr where n = 12 & r = 7
= 792
with the number of modes consisting of 19 notes taken 7 at a time:
nCr where n = 19 & r = 7
= 50,388
Most of the difference consists of 19-tone enharmonic modes. Later, if
there is an interest, I will provide some summaries of some of my
studies of chromatic modes (those modes that are neither enharmonic nor
diatonic) which should make it clear that in that respect as well, the
19-tone equal is far, far richer melodically than the 12-tone equal.

The harmonic contrast between 12- & 19-tone equal is no less dramatic.
It is not merely that 19-tone gives much more consonant chords. It also
gives much more powerfully, wrenchingly
_dissonant_ chords. These are not whining, commatically mistuned
consonances such as one finds in 53-tone equal, but outright dissonances
that possess full _melodic_ independence from any consonance. This power
of the 19-tone dissonances is obvious when one recalls that the
augmented primes, augmented & diminished fifths, and augmented &
diminished octaves of the 19-tone equal are roughly 1/3 tone from the
just consonances, instead of about 1/2 tone as in the case of the
12-tone dissonances, and consequently much more dissonant. These 19-tone
dissonances happen to fall in precisely the most dissonant regions that
surround these most powerful consonances. One proof of this is
Helmholtz' famous graph of the smoothness of violin tone; when the
19-tone dissonances are superimposed thereon it will be found as I have
observed above.

12-tone equal has weakly consonant, rather simpering chords and weakly
dissonant chords; the contrast between the two classes is feeble, and
the overall effect leaves the heart stone-cold. It is scarcely
surprising that under the leaden weight of such a temperament, harmony
has fallen into disrepute.

19-tone equal has, especially if its octave is stretched by two or three
cents to equalize the roughness of its fifths and fourths, extremely
smooth consonances. The fourth is scarcely more sensitive to mistuning
than the third, for reasons having to do with beating partials present
in the fourth. Consequently one can mistune the fourth much more than
the fifth without noticeably impairing its consonance. The consonances
of the 19-tone equal are just mistuned enough to be brilliant, but of
course not quite so smooth or quiet as those of just intonation.

Hence, 19-tone equal gives rise to a powerful contrast between
consonance and dissonance, both melodically and harmonically speaking.
The effect is to give music a kind of elastic, forward impetus quite
unknown in the 12-tone equal, which has a more static, uncertain
esthetic.

This contrast between consonance and dissonance affects the emotions
very poignantly. Composers with a reputation for formalism or mannerism
such as Gesualdo, Haydn and Mozart seem bursting with suppressed
vitality in the 19-tone equal. Gesualdo in particular is virtually a
different - and a far better - composer in the 19-tone equal, lending
credence to the story that he owned a 19-tone clavier (not necessarily,
but quite possibly more or less equally tempered.)

No other temperament, and certainly not the 31-tone equal with its
continual problems of interval confusion in melody, can remotely compare
to the 19-tone equal for sheer tension and force.

In a famous passage, Fokker has spoken of 31-tone equal having two
faces, one turned to the past, one to the future. This is far, far more
profoundly true of the 19-tone equal. For the 19-tone equal, at the same
time that it renders the ancestral music of our own culture more present
to the imagination and the emotions, also opens up to us the unsuspected
melodic riches both of rock music and of non-Western cultures, of which
the former however is of more vital concern to our own people than the
latter. For any song that can be sung and reliably reproduced by
singers, can be notated and played in 19-tone equal. This is a
consequence of the fact that, providentially, the human melodic limen
(55-60 cents) is just narrower than the tuning degree of 19-tone equal
(~63 cents).
This may not, by the way, be _entirely_ providential. Our species may
have evolved as a 19-tone-equal and 5-limit interpreter of musical
experience. I merely suggest this by the way... obviously this is very
speculative.

All this means that, for the first time, this temperament permits us
to notate all musical traditions - especially our own living, popular,
rock tradition - with as much precision as singers (and listeners) can
reliably reproduce (and esthetically understand) their own native
melodies.

All this is very exciting, and at least merits determined
popularization. I make no doubt that popular, Renaissance, and classical
musicians alike have very much to gain from the commercial development
of 19-tone instruments. The academic 12-tone equal abortions of our
century are of no concern. Undoubtedly the museums could preserve a few
12-tone instruments so that the odd musicologue could study these
compositions, if he has nothing better to do.

Instead of embracing the 19-tone equal temperament however, we seem,
mostly through ignorance (the subject of temperament is at once very
difficult, and poorly remunerated, and so progress has been slow) but
partly from less excusable motives, more concerned to endlessly prove
how better we are at hearing than the poor, stupid masses.

Thank God we are all so wise.

But among those masses are popular musicians, some of whom have more
creativity in their little fingers than many a whole conservatory.
Certain of the most creatively sterile academic composers are the most
eager, either to perpetuate the 12-tone equal as some kind of apostolic
chrism, or else to have us adopt some utterly lifeless, dissonant,
emotionally neutral temperament, whose only claim to recognition is the
certainty that no paying, demanding audience would ever sit still to
hear such aimless, random impudence for five minutes. In other words,
they seem to adopt temperaments other than 12-tone equal only as a means
to out-Schoenberging Schoenberg.

I would like to emphasize however that our 12-tone equal mandarins are
far more malevolent (as will as far more powerful) than our most
inconsequent microtonal devotees. Absurdities that everyone politely
ignores are harmless, and beneath the civilized to condemn; it is
absurdities enshrined in dogma that stunt and maim.


SMTPOriginator: tuning@eartha.mills.edu
From: "Bob Lee"
Subject: Singing small intervals
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