Just Intonation

The great principle of music theory in its pitch aspect is to define
only those theoretical systems which we are profoundly convinced that
singers and practical musicians can actually use, or which are necessary
for the derivation of such systems.

Disjunct intervals which are highly dissonant with the tonic, and which
cannot be reached by consonant intervals, or by conjunct intervals
themselves defined by consonant intervals, have _no_ musical existence.
They are as irrelevant to music as the third planet of alpha Centauri.

Consonances are not playthings or inventions of man, which we can
hopefully extend to the 9th or 11th harmonic; they have an objective,
physical existence, and music without them is inconceivable and
impossible. They are as necessary to music, either melodically or
harmonically conceived, as the numerals are to mathematics, and their
existence is just as certain.

I apologize if all this seems too elementary to be worth stating. But
these principles have been repeatedly denied by a certain school of
modern theory, at least by implication.

Having now set the stage, I should like to inquire what just intonation
is, and what is the form of just intonation that a singer (any singer,
not just a Western singer of some particular tradition) can use, and
which it is the business of temperament to imitate, both for instruments
and for the training of singers.

Within the octave the consonances are six in number, and _only_ six: 3:2
4:3 5:3 5:4 6:5 & 8:5. The septimal intervals 7:4 7:5 & 7:6 are
dissonances, both because their partials beat with each much more
severely than is the case for the consonances, and because they are
melodically confused with closely adjacent dissonant intervals directly
derived from the consonances (e.g. 16:9 with 7:4.) The septimals have no
harmonic connection with the six consonances, and have undoubtedly
dissonant inversions 8:7 10:7 & 12:7. (Septimal 'harmony' therefore
gives rise to dissonances.) But the six consonances do have a
mathematical interrelationship with each other, as will be seen.

All music begins with a tonic, the initial note. The singer in creating
for himself a body of tones that are interrelated acoustically, and
which he can remember and internalize, can proceed by the narrower,
conjunct intervals (closer than 6:5, say,) and here he is limited only
by the skill of his vocal chords and mind, which make it very difficult
to sing or remember pitches closer than about 55-60 cents.

As soon as music becomes disjunct, however (a compass of more than about
6:5) the singer (or listener or instrumentalist) runs the very real risk
of becoming totally confused; pitches cease to have anything in common
that one can remember, and music readily disintegrates into a welter of
random noise, that all sounds much alike, and so can have little power
to illustrate language or affect our emotions.

There is a great deal of evidence that non-Western cultures are at least
as intolerant of what they conceive to be aimless or random as the most
hidebound Western conservative, although certainly they define this
somewhat differently.

None of these ideas are new. I seek only to make myself the purveyor of
what the common sense of mankind knows almost instinctively. There is no
objection if some wish to challenge these principles. If they can find
an enthusiastic audience for music that violates these principles, I
certainly do not object. But they should be intimately aware of these
principles. So far composers who have sought to enshrine randomness or
pure dissonance have only discredited the musical art.

Fortunately, disjunct music (or what ultimately amounts to the same
thing, music of compass wider than about 6:5) is possible. It is the
phenomenon of consonance that makes it so.

In our century consonance has acquired a reputation for vaguely
indicating hypocrisy or insincerity or immaturiy in those who love it;
one thinks of sugary-sweet childrens' choirs singing one of the more
uninspired polyphonic masters - or something much worse. This is a very
dangerous mistake. Consonances are not the enemy of dissonance. They
only are what make dissonance itself emotionally meaningful. Not only
that, but consonances alone make disjunct melody itself possible. And
this is as true for the Indian or Arab or Turk as for the cultural heirs
of the Greeks and Romans.

Providentially - or perhaps this is the result of human evolution itself
- just as melody becomes too disjunct to keep in order by the
'dead-reckoning' of conjunct intervals alone, we find an undoubted
consonance, 6:5. We then find a series of consonant, beat-free intervals
that provide us with sure resting points for the voice (and for the
understanding) and which melody can invest with the most profound
emotional significance. Again, this is as true for the rock singer as
for the opera singer or an Indian singer.

Once we reach any one of these consonances, we can take it for a new
tonic, though in subordination to the original tonic. (I have no space
here to discuss how the tonic can be changed, but I will consider that
elsewhere in a more appropriate forum.) By this means we can, for
example, if we wish, sound a minor third above the original minor third,
so that we reach 6/5 x 6/5 = 36/25 which is dissonant with the original
tonic, and disjunct, but - and this is a kind of miracle - nevertheless
singable with fair accuracy and comprehensible to us.

Conjunct intervals intercalated between these disjunct consonances
further assist the voice and the understanding.

Such dissonances as 36:25 (the diminished fifth) number twelve within
the octave. They hold a quite unique place among the dissonances,
because they are, in a sense, 'consonant with the tonic at one remove',
as it were, because they can be sung entirely by singing consonances,
though since a subtonic (6:5 in this example), that is to say, an
intermediate tonic, separates them from the original tonic, they are not
so closely related to the tonic as are the six consonances.

I call these intervals 'tonal dissonances', because they more than
anything else bind disjunct music together and make it tonal, that is to
say coherent and consistent with itself. I am not of course referring to
the so-called 'tonal' era of Western music, which might be better called
the period of its decadence.

The human voice does not sound these intervals with the same confidence
and accuracy that it sounds the consonances, or even those conjunct
dissonances made familiar to it by cultural training. Nevertheless it
can readily distinguish the tonal dissonance 25:18 (the augmented
fourth) from the tonal disonance 36:25 (the diminished fifth) because
these are 63 cents apart, because they indicate quite different
relations to the tonic, and also because there is enough 'tonal space'
between 4:3 and 3:2 so that two (but not three) pitches can be kept
melodically distinct, both from each other and from the adjacent
consonances:

4:3 25:18 36:25 3:2

The 25:18 occurs in the lydian mode, and has an entirely different
psychological effect eery and weird, but relaxed) from 36:25, which has
a menacing, tense effect, and occurs in the locrian mode. The septimal
dissonance 7:5 (the septimal tritone) also lies between 4:3 and 3:2, but
becasue it lacks any relation with the consonances, and has a dissonant
inversion 10:7, it is melodically and harmonically confounded with 25:18
(only 14 cents narrower than 7:5.) The tonal dissonance 25:18 is in fact
much more easy for a singer to sing than 7:5, because the former is part
of the fabric woven from the undoubted consonances, whereas 7:5 occupies
the shadowy, weird realm between consonance and dissonance.

Here are the six consonances and twelve tonal dissonances, together with
the prime and octave:

1:1 25:24 16:15 10:9 9:8 6:5 5:4 32:25 4:3 25:18 36:25 3:2 25:16 8:5 5:3
16:9 9:5 15:8 48:25 2:1

Or in cents:

0 71 112 182 204 316 386 427 498 569 631 702 773 814 884 996 1018 1088
1129 1200

or by name:

augmented prime, diminished second, minor tone, major tone, minor third,
major third, diminished fourth, perfect fourth, augmented fourth or
tritone, diminished fifth, perfect fifth, augmented fifth, minor sixth,
major sixth, subminor seventh, superminor seventh, major seventh,
diminished octave, octave

Every music theorist has these values by heart, as well he should have -
they are part of the basic vocabulary of music.

Many of these intervals are inconveniently close for the voice. For
example, 25:24 and 16:15 are separated by only the doubly diminished
second, alias the diesis, 128:125, 41 cents. The melodic limen of 55-60
cents suggests that the voice (and mind) cannot reliably use more than
about 20 pitches in the octave. Perhaps our species will one day come
into contact with creatures who have a different limen.

Now it happens that the consonances and tonal dissonances number 19
pitches within the octave.

The obvious (and only) solution is to divide the octave into about 20
equal degrees. It happens that only one temperament within 12 and 31
equal tones in the octave gives reasonably accurate values for the
consonances, and also conciliates the three consonant cycles in such a
manner that all consonances are available not merely on one tonic, but
on all possible tonics.

This is the 19-tone equal temperamnt. The 12-tone equal temperament
gives reasonable (or halfway reasonable) values only for the
consonances. The 19-tone equal gives much better values for the
consonances, although its fifths are rougher, and also gives good values
for the tonal disonances, which is no less important, for music is not
only consonance and conjunct dissonance, but disjunct dissonance also.

It may be instructive to pursue the method further, and ask ourselves,
could singers reliably relate dissonances to the tonic through a
sub-subtonic? For example, could one proceed from 6:5 to 36:25 and again
by a minor third to 216:125, the greater diminished seventh? Probably
not. But even if one could, the resulting 18 additional dissonances -
I call them 'atonal dissonances' - are so close that no singer could
ever keep them melodically distinct. The 31-tone equal is the
temperament that corresponds to the introduction of the 'atonal
dissonances'. But the 19-tone equal accepts four of these into its
fabric, namely the diminished third 144:125 and the augmented minor tone
125:108, which it merges into a single tone that is also confounded with
7:6; and the greater diminished seventh 216:125 and augmented sixth
125:72 which it also merges, and likewise confounds with the septimal
interval 7:4.

Here are the 18 atonal dissonances:

128:125 27:25 144:125 125:108 75:64 32:27 125:96 27:20 45:32 64:45 40:27
192:125 27:16 128:75 216:125 125:72 50:27 125:64

and in cents:

41 133 245 253 275 294 457 520 590 610 680 743 906 925 947 955 1067 1159

or by name:

doubly diminished minor tone, diminished major tone, diminished third,
augmented minor tone, augmented major tone, subminor third, augmented
third, superperfect fourth, superaugmented fourth, subdiminished fifth,
subperfect fifth, diminished sixth, supermajor sixth, lesser diminished
seventh, greater diminished seventh, augmented sixth, submajor seventh,
augmented seventh

Note that four different, incompatible terminologies are used: the
10:9/9:8 and 16:9/9:5,
as the commatically separated intervals, by tradition have each their
own terminology, as do the thirds and sixths on the one hand (major and
minor) and the fourths, fifths, primes and octaves on the other
(perfect). No authorities use quite the same terminology, and none that
I have found include quite all the intervals.

There is a direct and exact relationship between just intonation and the
three best temperaments of all, namely 12- 19- & 31-tone equal. The
12-tone equal loosely approximates the consonances, but radically
distorts the tonal dissonances, so much so indeed that certain of these
(e.g. the diminished fourth 32:25) are confounded with consonances. This
particularly disturbed Helmholtz, and he was right to be disturbed.
Certain nineteenth century composers used this in their modulations,
with results that quickly became a clich?. The 19-tone equal gives close
approximations to the consonances and the tonal dissonances, while
confounding the atonal dissonances with the tonals. Finally, the 31-tone
equal gives close approximations to the consonances, tonal dissonances
and atonal dissonances, the last to no good or useful effect.

The method can of course be pursued yet further, to 24 'ultra-atonal
dissonances', and leads to 50-tone equal temperament (virtually
Zarlino's 2/7 comma mesotonic.) The 43- & 55-tone equal are also fully
cyclic, and constitute alternate, though less logical methods of
conciliating the divers categories of intervals.

Our notation is closely in accord with 19- 31- & 50-tone equal. The
first of these may be called the 'system of the flat and sharp', the
second the 'system of the double flat and double sharp', and the last
the 'system of the triple flat and triple sharp'. The 12-tone of course
confounds both flat and sharp, and also, at two points, accidental with
natural. The 43- & 55-tone equal are also mixed systems.

There are many systems of just intonation. But I daresay this system
corresponds most nearly to what is actually possible for the voice, and
present to the mind.


SMTPOriginator: tuning@eartha.mills.edu
From: Gregg Gibson
Subject: Reply to Graham Breed
PostedDate: 14-12-97 02:27:27
SendTo: CN=coul1358/OU=AT/O=EZH
ReplyTo: tuning@eartha.mills.edu
$MessageStorage: 0
$UpdatedBy: CN=notesrv2/OU=Server/O=EZH,CN=coul1358/OU=AT/O=EZH,CN=Manuel op de Coul/OU=AT/O=EZH
RouteServers: CN=notesrv2/OU=Server/O=EZH,CN=notesrv1/OU=Server/O=EZH
RouteTimes: 14-12-97 02:25:34-14-12-97 02:25:35,14-12-97 02:25:16-14-12-97 02:25:17
DeliveredDate: 14-12-97 02:25:17
Categories:
$Revisions:

Received: from ns.ezh.nl ([137.174.112.59]) by notesrv2.ezh.nl (Lotus SMTP MTA SMTP v4.6 (462.2
9-3-1997)) with SMTP id C125656D.0007D103; Sun, 14 Dec 1997 02:27:18 +0100
Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA14772; Sun, 14 Dec 1997 02:27:27 +0100
Date: Sun, 14 Dec 1997 02:27:27 +0100
Received: from ella.mills.edu by ns (smtpxd); id XA14769
Received: (qmail 29027 invoked from network); 13 Dec 1997 17:27:24 -0800
Received: from localhost (HELO ella.mills.edu) (127.0.0.1)
by localhost with SMTP; 13 Dec 1997 17:27:24 -0800
Message-Id: <3493987D.855@ww-interlink.net>
Errors-To: madole@mills.edu
Reply-To: tuning@eartha.mills.edu
Originator: tuning@eartha.mills.edu
Sender: tuning@eartha.mills.edu

Gregg Gibson
> Alas, the incommensurability of the thirds and fifths in 22-tone
> equal does not arise only when modulation is employed, which I have
> always been very careful to note. If, taking C as tonic, we tune by
> fifths of 709 cents, we have a G of 709 cents and a B of 1145
> cents.=20

What is `C'? What is 'G'? What is `B'? What is the sound of
forcing a seven-sided peg into a ten-sided [for sake of argument]
hole?

When I first got into tuning, I was locked into the 7 tone diatonic
thing, too -- problem was I could never reconsile that with what
I wanted to hear. Once I let go of my security blanket and opened
up my ears, a whole world opened up around me.

I had absolutely no trouble finding a 10 tone scale in 22 by ear,=20
with no theory at all to guide me (until after the fact, that it :).
Maybe that makes me special (of full of it), since it's more than
the magic 7, but what of it? Lots of people listen to 12 serial
rows and to them it is comprehensible [I'm not there yet, myself].

At any rate I make music for myself, and whomever is open enough
to listen to what I might have to say. I've no interest whatever
in forming a band to please the masses [and I'm not that good=20
yet, anyhow ;-)].

> The idea that a static scale =E0 la just intonation can be employed
> melodically so long as one does not modulate, fails to understand
> that it is not only in modulation that commensurable consonance is
> necessary; the very tissue of melody itself falls apart unless the
> consonances are conciliated by temperament. 22-tone equal does not
> do this, which is one reason why it is such a poor system.

I found this yesterday, while noodling around on my worthless^H^H^H^H
22TET guitar:

To make sense of it, you should read Pauls paper on 22. I will say
now that I found this aurally, and I have yet to sit down and grind
the numbers out. Frankly, numbers don't mean squat if it sounds
good, IMHO as a musician.

I've been working with Pauls Decatonic Minor modes, notably the=20
Standard Pentachordal form. [I haven't seen the edition of the
paper with the graphics, so I'll use the following notation:
=09
=09I(10) - Maj decatonic tetrad
=09i(10) - min " "
=09I-(10) - Maj-Min tetrad
=09i+(10) - min-Maj tetrad
=09I*(10) - Augmented tetrad.
]

What I've found is that the three _consonant_ chords that completly
define the mode as suggested by Paul are i, II- and III. =20

I was playing with various patterns of chords like this:

i(10) -> v(10) -> i(10) -> IV(10) -> III(10) -> II-(10) -> i

when I came up with the following modulation to the subdominant (v):
[assume the `(10)' subscript here]:

First, firmly establish the sense of tonality:

i -> v -> i -> IV -> III -> II- -> i

Then, go to the subdom:

i -> v -> i -> IV -> III+ -> IV+ -> v (of i)

[IX+] -> [X+] -> i (of v)

This was every bit as powerful, if not more so, to my ears, as the
usual diatonic modulation to the dominant. =20

Now, as soon as I can find an equally cool way to get back again,
I'm going to plop a reel on the 8 track and start working on a=20
bass line. ;-)

Steve





=20


SMTPOriginator: tuning@eartha.mills.edu
From: Steven Rezsutek
Subject: Correction Re: Reply 2 to Graham Breed
PostedDate: 15-12-97 18:57:18
SendTo: CN=coul1358/OU=AT/O=EZH
ReplyTo: tuning@eartha.mills.edu
$MessageStorage: 0
$UpdatedBy: CN=notesrv2/OU=Server/O=EZH,CN=coul1358/OU=AT/O=EZH,CN=Manuel op de Coul/OU=AT/O=EZH
RouteServers: CN=notesrv2/OU=Server/O=EZH,CN=notesrv1/OU=Server/O=EZH
RouteTimes: 15-12-97 18:55:20-15-12-97 18:55:21,15-12-97 18:55:00-15-12-97 18:55:01
DeliveredDate: 15-12-97 18:55:01
Categories:
$Revisions:

Received: from ns.ezh.nl ([137.174.112.59]) by notesrv2.ezh.nl (Lotus SMTP MTA SMTP v4.6 (462.2
9-3-1997)) with SMTP id C125656E.00626F95; Mon, 15 Dec 1997 18:57:07 +0100
Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA16139; Mon, 15 Dec 1997 18:57:18 +0100
Date: Mon, 15 Dec 1997 18:57:18 +0100
Received: from ella.mills.edu by ns (smtpxd); id XA16137
Received: (qmail 17163 invoked from network); 15 Dec 1997 09:57:14 -0800
Received: from localhost (HELO ella.mills.edu) (127.0.0.1)
by localhost with SMTP; 15 Dec 1997 09:57:14 -0800
Message-Id: <199712151753.MAA07535@doghouse.hq.nasa.gov>
Errors-To: madole@mills.edu
Reply-To: tuning@eartha.mills.edu
Originator: tuning@eartha.mills.edu
Sender: tuning@eartha.mills.edu