Practical Bases of Just Intonation

One traditional means of deriving just intervals is of course to simply
embark upon a series of perfect fifths:

3/2 x 3/2 = 9/4 x 1/2 = 9/8
9/8 x 3/2 = 27/16
27/16 x 3/2 = 81/32 x 1/2 = 81/64
etc

Bringing the ratios back within the octave as one proceeds. This is
Pythagorean intonation.

But does a singer ever sing this many successive fifths (or fourths)?
Evidently not. In highly modulated music such a chain might occur, but
would be only an almost undiscernable trickle of tonality in the great
flood of tones more directly related to the tonic (new or old.)

The system is therefore without practical relation to music, and is one
of those theoretical delusions which so plague and confuse musicians.
But this cycle of fifths does nevertheless have an important r?le to
play in the derivation of systems which are fit for practical music.

If the cycle of fifths/fourths is used merely to ensure that each one of
the tones of a temperament has a consonant fifth above and below it, and
that the other consonances are likewise related to the fifths such that
consonant chords are present, and the maximum number of consonant
melodic intervals likewise present (this is the point I have earlier
made in connection with the harmonic and melodic deficiencies of the
22-tone equal) then the cycle of fifths/fourths is very useful - in the
context of temperament, not just intonation - to ensure that the cycle
of the fifth/fourth shall be harmonically and melodically congruent with
the cycles of the major third/minor sixth and minor third/major sixth.
That is to say, by congruent, I mean that every pitch of the temperament
has both consonant fifths and thirds above and below it. If even one
tone lacks this, then we have at a stroke severely restricted both
modulation, and also gravely weakened the melodic coherence of music,
which depends on a seamless fabric of consonances and dissonances
related by consonance. Within limits, music can survive a poor
temperament, as the 12-tone equal daily attests, but its variety and
expressivity will be found to be greatly constrained.

But if the cycle of fifths is used as a basis for just intonation, it
excludes the thirds and sixths from consonance, which is absurd. There
are two other consonant cycles besides the cycle of the fifths/fourths,
to which cycles I referred in the previous paragraph. The thirds and
sixths are more common in song than either fifth or fourth, so although
they are less consonant, they are but little less likely to be sung
accurately, the more so as the thirds are much narrower than the fifth
and fourth.

It follows that any just intonation which hopes to be of practical use
in describing what singers can sing, or what they find easiest to sing,
and in describing how closely a given dissonance is related to the
tonic, and hence how "tonal" that dissonance is, must include all three
consonant cycles in the calculation.

These principles follow from acoustics, and not primarily from art or
culture. Renaissance polyphony, Rock music, Western 'Classical' music,
Arab music, Indian music, etc all share, at least for singers, in these
principles. No singer accurately sings the 20th just fifth from the
tonic (unless by chance this corresponds to a consonant interval.)

As it turns out, even if one does give greater weight to the cycle of
the fifth/fourth, calling dissonances "tonal" that are related to the
tonic through _two_ successive fifths or fourths (and not one only) we
have only two additional tonal dissonances, both of them a mere comma
removed from consonances, and so perfectly irrelevant melodically, and
worthless for harmony. These are 32:27 (4/3 x 4/3 x 4/3) and 27:16 (3/2
x 3/2 x 3/2), and occur of course among the regular atonal dissonances
that I have enumerated.

These principles have nothing to do with any one temperament; they hold
true for all temperaments and systems; I certainly did not dream them up
to promote the 19-tone equal temperament, which however closely conforms
to the resulting intervals.

No matter how one may try to avoid it, no matter what argument one may
devise, it cannot be denied that the 19 degrees of the 19-tone equal
correspond remarkably well (though not of course perfectly) to the 7
consonances and 12 tonal dissonances. Of course, as I have just noted,
it is possible to discuss _at the margins_ in a manner of speaking. One
may discuss whether the septimals should be included as consonances, but
even were we so unwise as to take these intervals for consonant, we
would do nothing except set up a very limited exception to the general
tonality of music. The manner of deriving the tonal dissonances is also
debatable, though again, within the realms of reasonable debate, the
number of 12 tonal dissonances can be altered only slightly if at all.

This is not an arbitrary system, any more than the multiplication table
is arbitrary. The tonal dissonances are associated to the consonances
through 94 different consonant progressions, which form the fabric of
tonality which it is the business of temperament to knit yet more
strongly together. By this means dissonance acquires an inevitable,
highly memorable character that marvelously contributes to the
expressive power of music.

I believe I have posted the tonal progressions proper to one of the 19
intervals. I don't think this is the place to post them all, but here is
one more:

25/24 , 70.7 cents, the chromatic semitone:

5/4 x 5/3 , 5/4 x 5/6, 5/3 x 5/4, 5/3 x 5/8, 25/18 x 3/2, 25/18 x 3/4,
25/16 x 4/3, 25/16 x 2/3


SMTPOriginator: tuning@eartha.mills.edu
From: Aline Surman
Subject: 19 tone/20 tone
PostedDate: 18-12-97 03:48:00
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: 18-12-97 03:45:54-18-12-97 03:45:54,18-12-97 03:45:31-18-12-97 03:45:32
DeliveredDate: 18-12-97 03:45:32
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 C1256571.000F2EC3; Thu, 18 Dec 1997 03:47:46 +0100
Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA18430; Thu, 18 Dec 1997 03:48:00 +0100
Date: Thu, 18 Dec 1997 03:48:00 +0100
Received: from ella.mills.edu by ns (smtpxd); id XA18621
Received: (qmail 9585 invoked from network); 17 Dec 1997 18:46:50 -0800
Received: from localhost (HELO ella.mills.edu) (127.0.0.1)
by localhost with SMTP; 17 Dec 1997 18:46:50 -0800
Message-Id: <34989783.57EC@dnvr.uswest.net>
Errors-To: madole@mills.edu
Reply-To: tuning@eartha.mills.edu
Originator: tuning@eartha.mills.edu
Sender: tuning@eartha.mills.edu

>I want to say that guitarist
>Dan Stearns uses 20 eq, and creates music of great depth and power with
>it. In my opinion, he is possibly the most innovative rock guitarist at
>this time (that I have personally heard).

Dan is indeed impressive alright. And almost completely self-taught I
think, right?




Thanks for the thanks Neil, but I'll definitely say that helping to get
"Acoustic Stick" out was well worth it! I'm giving several copies for
Christmas.

And when it comes to microtonal Christmas CDs, another good possibility
would be Rod Poole's CD. And of course I wouldn't forget Easley
Blackwood's classic.


SMTPOriginator: tuning@eartha.mills.edu
From: mr88cet@texas.net (Gary Morrison)
Subject: Re: Call me a "Zartist"
PostedDate: 20-12-97 06:29:02
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: 20-12-97 06:26:54-20-12-97 06:26:55,20-12-97 06:26:30-20-12-97 06:26:30
DeliveredDate: 20-12-97 06:26:30
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 C1256573.001DEB95; Sat, 20 Dec 1997 06:28:44 +0100
Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA22499; Sat, 20 Dec 1997 06:29:02 +0100
Date: Sat, 20 Dec 1997 06:29:02 +0100
Received: from ella.mills.edu by ns (smtpxd); id XA22519
Received: (qmail 13614 invoked from network); 19 Dec 1997 21:28:05 -0800
Received: from localhost (HELO ella.mills.edu) (127.0.0.1)
by localhost with SMTP; 19 Dec 1997 21:28:05 -0800
Message-Id:
Errors-To: madole@mills.edu
Reply-To: tuning@eartha.mills.edu
Originator: tuning@eartha.mills.edu
Sender: tuning@eartha.mills.edu