source file: mills3.txt Date: Wed, 30 Jul 1997 01:01:02 +0200 Subject: RE: theories From: "Paul H. Erlich" Neil, Like you, I am a guitarist who plays best when feeling, not thinking. But due to my physics background I tend to often think in mathematical terms. The math involved in tuning theory is quite simple, really. Some of us like to make it kind of complicated just to stretch our intellectual muscles. But you don't need anything beyond a basic understanding of logarithms to grasp any of the really good ideas in tuning. Many of us have read Harry Partch's "Genesis of a Music" and throw around terms like "tonality diamond" and assume that others will understand us. This book does not assume any mathematical ability and is written by a musician for musicians. A tonality diamond is just a scale formed by picking a central pitch and including all notes that have simple frequency ratios with that pitch. Partch dealt mainly with odd numbers because dividing by 2 is just transposing down an octave, so if you're talking about non-octave-specific pitch classes, you can dispence with all factors of 2. So by "simple frequency ratios" we mean that both the numerator and denominator of the ratios are odd numbers not exceeding a pre-specified limit. It is easy to find all possible such ratios by allowing the numerator to increase along one axis and allowing the denominator to increase along the other. So the tonality diamond is just a two-dimensional grid, which Partch liked to rotate by 45 degrees to make it look like a diamond. Let's work out the first few tonality diamonds. I will call the central pitch 1/1 and G and will give the scale tones best approximating the tones of the diamond in 72-tone equal temperament (all errors are 4 cents or less) notated using 1/6ths of the conventional 12-tone sharps and flats (e.g., 1/2#, 5/6b). Tonality Diamonds ----------------- 1-limit 1 - 1 or G 3-limit 1 3 - - 1 1 1 3 - - 3 3 or G D C G 5-limit 1 3 5 - - - 1 1 1 1 3 5 - - - 3 3 3 1 3 5 - - - 5 5 5 or G D B1/6b C G E1/6b E5/6b B5/6b G 7-limit 1 3 5 7 - - - - 1 1 1 1 1 3 5 7 - - - - 3 3 3 3 1 3 5 7 - - - - 5 5 5 5 1 3 5 7 - - - - 7 7 7 7 or G D B1/6b F1/3b C G E1/6b A2/3# E5/6b B5/6b G C5/6# A1/3# E1/3# D5/6b G 9-limit 1 3 5 7 9 - - - - - 1 1 1 1 1 1 3 5 7 9 - - - - - 3 3 3 3 3 1 3 5 7 9 - - - - - 5 5 5 5 5 1 3 5 7 9 - - - - - 7 7 7 7 7 1 3 5 7 9 - - - - - 9 9 9 9 9 or G D B1/6b F1/3b A C G E1/6b A2/3# D E5/6b B5/6b G C5/6# F1/6# A1/3# E1/3# D5/6b G B1/3# F C A1/6b D2/3# G 11-limit 1 3 5 7 9 11 - - - - - -- 1 1 1 1 1 1 1 3 5 7 9 11 - - - - - -- 3 3 3 3 3 3 1 3 5 7 9 11 - - - - - -- 5 5 5 5 5 5 1 3 5 7 9 11 - - - - - -- 7 7 7 7 7 7 1 3 5 7 9 11 - - - - - -- 9 9 9 9 9 9 1 3 5 7 9 11 -- -- -- -- -- -- 11 11 11 11 11 11 or G D B1/6b F1/3b A C1/2# C G E1/6b A2/3# D F1/2# E5/6b B5/6b G C5/6# F1/6# A1/3b A1/3# E1/3# D5/6b G B1/3# D5/6# F C A1/6b D2/3# G B1/2b D1/2b A1/2b F1/3# B1/6# E1/2b G Look at the 5-limit diamond for a familiar case. Each row is a major chord, and each column is a minor chord. So tradiationally speaking, the diamond is a compact way of seeing all the consonant chords that contain the note G. (Of ourse, the 1/1 could mean anything, not just G.) So if you were writing a song and you knew you wanted G in the melody, you could just try out the different chords in the tonality diamond of G and see which one you liked the best. For higher limits, we are talking about more complicated chords, but the concept is exactly the same. How's this so far for basic concepts? SMTPOriginator: tuning@eartha.mills.edu From: petalsinpetaluma@hotmail.com Subject: Re: TUNING digest 1139 (?'s for Gary Morrison) PostedDate: 31-07-97 05:50:05 SendTo: CN=coul1358/OU=AT/O=EZH ReplyTo: tuning@eartha.mills.edu $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: 31-07-97 16:42:05-31-07-97 16:42:06,31-07-97 16:40:50-31-07-97 16:40:50 DeliveredDate: 31-07-97 16:40:50 Categories: $Revisions: Received: from ns.ezh.nl by notesrv2.ezh.nl (Lotus SMTP MTA v1.1 (385.6 5-6-1997)) with SMTP id C12564E5.00280236; Thu, 31 Jul 1997 09:17:01 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA25866; Thu, 31 Jul 1997 05:50:05 +0200 Date: Thu, 31 Jul 1997 05:50:05 +0200 Received: from ella.mills.edu by ns (smtpxd); id XA25864 Received: (qmail 26356 invoked from network); 31 Jul 1997 03:49:58 -0000 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 31 Jul 1997 03:49:58 -0000 Message-Id: <199707310346.UAA23121@f40.hotmail.com> Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu