source file: mills3.txt Date: Sat, 3 Jan 1998 17:37:40 +0100 Subject: Positive and negative temperaments From: gbreed@cix.compulink.co.uk (Graham Breed) The formula given relates the Pythagorean comma to the step size. If 50 equal is a doubly negative temperament, this isn't such a useful method of classification. Better is to state the interval that's defined out of existence. For generalised keyboard mappings, you need to define two small intervals for two directions. The way these add up to give 5-limit harmony is determined by which temperament class you choose. The nthly positive definition is useless here. It is also undefined for unequal temperaments. 79, 91 and 103 equal are singly negative, but not meantone. There are two different kinds of triply positive temperaments. With 15, 27 and 39 equal, the enharmonic diesis (7 0 -3)H is zeroed. With 51, 63, 75 and 87 equal, zero 3*(13 -4 -1)H, or set 2 Pythagorean equivalent to 3 syntonic commas. Now, look at this: Triply negative 9, 21, 33, 45, 57, 69, 81, 93, 105, 117 Doubly negative 2, 14, 26, 38, 50, 62, 74, 86, 98, 110 Singly negative 7, 19, 31, 43, 55, 67, 79, 91, 103, 115 Zeroly positive 12, 24, 36, 48, 60, 72, 84, 96, 108, 120 Singly positive 5, 17, 29, 41, 53, 65, 77, 89, 101, 113 Doubly positive 10, 22, 34, 46, 58, 70, 82, 94, 106, 118 Triply positive 3, 15, 27, 39, 51, 63, 75, 87, 99, 111 The difference between each scale size in a class is 12. This follows from the definition of the Pythagorean comma. A better nomenclature for the same classes would simply be to take the remainder of the scale size when divided by 12. Defining the sizes of the octave(T) and Pythagorean comma(X) uniquely defines the size of the fifth(P5) 12 * P5 - 7 T = X P5 = (X + 7T)/12 T 2 3 5 7 9 10 12 X -2 3 1 -1 -3 2 0 P5 1 2 3 4 5 6 7 1*ln(2)/ln(1.5)=1.7 2*ln(2)/ln(1.5)=3.4 3*ln(2)/ln(1.5)=5.1 4*ln(2)/ln(1.5)=6.8 5*ln(2)/ln(1.5)=8.5 6*ln(2)/ln(1.5)=10.3 7*ln(2)/ln(1.5)=12.0 When you add 12 to the octave size, you add 7 to the fifth size. So, this is a complete set of consistent scale classes. If you add the octave sizes of any two scales in this superset, you also add the fifth sizes. That means: T_3 = T_1 + T_2 P5_3 = P5_1 + P5_2 For perfect octaves, the fifth is P5/T-ln(3/2)/ln(2) octaves sharp. The mistuning in terms of scale steps is then: d = P5 - T*ln(3/2)/ln(2) When adding scales, d_3 = P5_3 - T_3*ln(3/2)/ln(2) d_3 = (P5_1+P5_2) - (T_1+T_2)*ln(3/2)/ln(2) d_3 = P5_1 - T_1*ln(3/2)/ln(2) + P5_2 - T_2*ln(3/2)/ln(2) d_3 = d_1 + d_2 So, adding scale sizes means adding the amount of mistuning of fifths in terms of scale steps. Paul Erlich stated this on the tuning list last month, but didn't prove it. SMTPOriginator: tuning@eartha.mills.edu From: gbreed@cix.compulink.co.uk (Graham Breed) Subject: Horizontal and vertical PostedDate: 03-01-98 17:38:20 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: 03-01-98 17:35:49-03-01-98 17:35:50,03-01-98 17:35:09-03-01-98 17:35:09 DeliveredDate: 03-01-98 17:35:09 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 C1256581.005B2AE7; Sat, 3 Jan 1998 17:37:44 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA07297; Sat, 3 Jan 1998 17:38:20 +0100 Date: Sat, 3 Jan 1998 17:38:20 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA07281 Received: (qmail 3215 invoked from network); 3 Jan 1998 08:37:55 -0800 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 3 Jan 1998 08:37:55 -0800 Message-Id: Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu