source file: mills2.txt Date: Mon, 3 Feb 1997 17:04:42 -0800 Subject: MetaMeantone, etc. From: John Chalmers 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