MetaMeantone, etc.

Wilson's MetaMeantone: While it is true that the major third and
the fifth beat at the same rate in this tuning, the actual genesis
is quite different (for more tunings with equally-beating consonances
see Rasch's article and posts by Manuel and me). Erv invented this
tuning by solving for tempered values for the fifth and major third
in which the first order difference tones would be would be equal.
In this case, the triad is 1.0, 1.247265 and 1.4945302. By multiplying
these by 400 hz (for convenience), one gets the frequencies 400,
498.906 and 597.81207 hz (0 382.52175 and 695.63044 cents). The
first order difference tones between the major third and the tonic
and the fifth and the third are both 98.906 hz. Erv believes this
relationship gives this triad and its near approximations (such as
the Lucy-Harrison tuning) a degree of unity and consonance not shared
by other meantone-like tunings.

These difference tones are not submultiples of the tonic (400 hz),
as they would be in just intonation (400 500 600 hz, 0 386.3137
and 701.955 cents). Nevertheless, Erv claims that when accurately
tuned the effect is perceptible, perhaps because the synchonous
beat rates lead to a greater sense of fusion.

While Erv used an iterative algorithm of his and Larry Hanson's own
devising, the MetaMeantone fifth may be computed in a more standard
manner. Let the tonic, mediant and dominant of the triad be 1.0 X and Y.
Therefore by the major third X(Y^4)/4 according to the usual relation
between "negative" fifths and major thirds (4 fifths up minus 2 octaves).
The difference town relation is X-1 Y-X (absolute pitch of the triad
is not important). Therefore X(Y+1)/2 and by setting this equal
to the first relation, one gets (Y+1)/2(Y^4)/4. This reduces to
Y^4-2*Y-2. At this point, I applied the Newton-Raphson method to
get 1.494530181 for the fifth. The major third may be obtained by
substituting back into the first equation.

Other "MetaTunings" may be found the same way by defining a triad in
terms of one cyclic interval and a difference tone relation.

There actually are two tunings in which the beat rates of the Major
Third and Perfect Fifth are equal. These are this tuning and 697.2784
cents as negative and positive beats sound the same (set the beats
equal or equal and opposite). The second tuning is also one in which
the 3 4 5 triad has equal difference tones.

To sume up, MetaMeantone sets the DT's of the P5-Tmaj and Tmaj-Tonic
equal and also the beat rate of the P5 and Tmaj. 697.2784 sets the DTs
of the 6Maj and P4-tonic (3 4 5 triad,1st inv. etc.) equal, and sets
the beats of the P5 equal to the Major Third (Tmaj). A third tuning,
696.2958 sets the beats of the P4(fourth) equal to the 6maj, but
the DT's are not equal for eithe the P4 and 6maj or P5 and Tmaj.
There is another tuning in which the P4 has the same beat rate as
the 6maj, but the fifth is a rather flat 692.1612 cents.The DT's
are unequal in this case. The conclusion is that setting the beat
rates of triadic intervals equal does not necessarily set the first
order difference tones equal as well.



--John


Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl
with SMTP-OpenVMS via TCP/IP; Tue, 4 Feb 1997 03:57 +0100
Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA23893; Tue, 4 Feb 1997 03:57:53 +0100
Received: from ella.mills.edu by ns (smtpxd); id XA23912
Received: from by ella.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI)
id SAA22812; Mon, 3 Feb 1997 18:56:18 -0800
Date: Mon, 3 Feb 1997 18:56:18 -0800
Message-Id:
Errors-To: madole@mills.edu
Reply-To: tuning@ella.mills.edu
Originator: tuning@eartha.mills.edu
Sender: tuning@ella.mills.edu