source file: m1554.txt Date: Fri, 16 Oct 1998 15:17:09 +1000 Subject: A meantone tuning with six 7-limit tetrads From: Dave Keenan Hi, I'm new to this list. Greetings from Brisbane, Australia. Has anyone seen the following tuning before? It's a choice of 12 from 31-TET which has many 7-limit harmonies. It has the advantage of mapping to an ordinary keyboard, approximating equal tempered, embedding some ordinary diatonic scales within it (C, G, Em-harm) and having its note names conform to the familiar western system, since it is syntonic (4 fifths = 1 third). C Db D D# E F F# G Ab A A# B C 3 2 2 3 3 2 3 3 2 2 3 3 (number of 31-TET steps between notes) As a sequence of 5ths it is: Db Ab - - F C G D A E B F# - - D# A# It gives the following 7-limit tetrads Augmented sixths (7:6:5:4) F A C D# C E G A# Db F Ab B Minor diminished sevenths (1/4:1/5:1/6:1/7) B D F# Ab E G B Db D# F# A# C All but one of its 12 notes (either Db or A#) can be covered by three of these tetrads. All can be covered by four. Major sevenths are also available on F and C while dominant sevenths are available on G and D and of course minors on some other degrees. There are 3 wolf fifths (A#Db, AbD#, F#Db) however they may be considered as 11-limit intervals with an accuracy of 10 cents, if you believe in such things. I think most of (if not all of) what would otherwise have been wolf thirds have become 7-limit intervals with an error of at most 4.1 cents. In case you want to dial it up on a synthesizer and give it a try, here are the optimum offsets from 12-TET given one cent resolution. C Db D D# E F F# G Ab A A# B 10 26 4 -19 -3 13 -9 7 23 0 -22 -6 This gives errors relative to the just intervals of only: fifth 3:2 -5.0 cents (-6.0 for D A) third 5:4 0.7 cents minor third 6:5 -5.6 cents (-6.6 for D F and F# A) augmented sixth 7:4 -0.8 cents augmented fourth 7:5 -1.5 cents augmented second 7:6 4.1 cents I developed it after reading Paul Erlich's criteria for 7-limit generalised-diatonic scales (in which he finds that only a 10 of 22-TET is suitable) and seeing a 7-limit just tuning given in the manual for Bill Cooper's excellent RealTime Tuner as: "1, 15/4, 9/8, 7/6, 5/4, 4/3, 7/5, 3/2, 14/9, 5/3, 7/4, 15/8" where 15/4 was no doubt intended to be 15/14. Of course the above 12 of 31-TET (or meantone-type) scale has more 7-limit tetrads and fewer wolves than this just scale. Regards, -- Dave Keenan http://uq.net.au/~zzdkeena