________________________________________________________________________ To: tuning@yahoogroups.com From: "Paul Erlich" Date: Tue, 23 Sep 2003 19:47:19 Subject: [tuning] reply to rick mcgowan warning -- this post contains lots of numbers! --- In MakeMicroMusic@yahoogroups.com, Rick McGowan wrote: > >> I've heard Easley Blackwood's music in 13 through 24 note >> equal temperament. > > Just FYI, we generally distinguish temperament from tuning. So these > are 13-24 equal tunings. They're not properly temperaments of > anything; they just consist of a number of equal sized intervals per > octave. why would you say that? certainly the way blackwood used them, many of these tunings *are* temperaments. in the abstract, you can look at the chart and table on this page http://www.sonic-arts.org/dict/eqtemp.htm and determine how to derive *most* of these equal tunings as 5-limit temperaments. presumably it's familiar to many readers that 12-equal can be derived by tempering out the usual "comma" that's tempered out in all meantones -- the syntonic comma or 81:80 -- and then tempering out an additional small interval representing the ratio between "enharmonic equivalents" like G#:Ab -- such as 648:625 (the "major diesis"), 128:125 (the usual "diesis"), 2048:2025 (the "diaschisma"), 32805:32768 (the "schisma"), or 531441:524288 (the "pythagorean comma"). on the chart, you'll see that the green lines corresponding to these intervals, labeled 'meantone', 'diminished', 'augmented', 'schismic', and 'aristoxenean' respectively (in accord with the table below the chart), all intersect at the big number *12*, making it easy to read this information off the chart. one instance of *13* is at the intersection of the 'magic' and 'beep' lines, so 13-equal can be derived from 5-limit JI by tempering out 3125:3072 and 27:25. another instance of *13* is at the intersection of the 'father' and 'tetracot' lines, so 13-equal can be derived from 5-limit JI by tempering out 16:15 (missing from monz's table) and 20000:19683. 27:25 and 16:15 are unlikely intervals to temper out when using harmonic timbres, but the next two are solidly derived as temperaments and reflect the patterns of blackwood's highly triadic usage: *15* is at the intersection of the 'blackwood', 'augmented', 'kleismic', and 'porcupine' lines, so 15-equal can be derived from 5-limit JI by tempering out any two of 256:243, 128:125, 15625:15552, and 250:243. *16* is at the intersection of the 'diminished', 'pelogic', and 'magic' lines, so 16-equal can be derived from 5-limit JI by tempering out any two of 648:625, 135:128, and 3125:3072. one instance of *17* is at the intersection of the 'schismic' and 'dicot' lines, so 17-equal can be derived from 5-limit JI by tempering out 32805:32768 and 25:24. blackwood's usage actually corresponds to the *another* instance of 17, on the extreme left on the 'meantone' line. *18* is at the intersection of the 'augmented' and 'semisuper' lines, so 18-equal can be derived from 5-limit JI by tempering out 128:125 and 6115295232:6103515625. admittedly, situations where the latter interval needs to be tempered out will probably be quite rare, and when they occur, will usually be handled with more accurate temperaments. *19* is at the intersection of the 'meantone', 'negri', 'magic', 'kleismic', and 'semisixths' lines, so 19-equal can be derived from 5-limit JI by tempering out any two of 81:80, 16875:16384, 3125:3072, 15625:15552, and 78732:78125. one instance of *20* is at the intersection of the 'blackwood', 'diminished', and 'tetracot' lines, so 20-equal can be derived from 5-limit JI by tempering out any two of 256:243, 648:625, and 20000:19683. one instance of *21* is at the intersection of the 'augmented' and 'escapade' lines, so 21-equal can be derived from 5-limit JI by tempering out 128:125 and 4294967296:4271484375. admittedly, situations where the latter interval needs to be tempered out will probably be quite rare, and when they occur, will usually be handled with more accurate temperaments. *22* is at the intersection of the 'diaschismic', 'magic', and 'porcupine' lines, so 22-equal can be derived from 5-limit JI by tempering out any two of 2048:2025, 3125:3072, and 250:243. *23* is at the intersection of the 'pelogic' and 'kleismic' lines, so 23-equal can be derived from 5-limit JI by tempering out 135:128 and 15625:15552. *14* and *24* appear on the chart too, superimposed upon 7 and 12 respectively, since they temper out the same commas as their halfling kin. but only half the notes of these larger tunings can be reasonably said to derive from a single instance of 5-limit JI -- the other half are "incompatible" and truly result, as you imply, from constructing equal-sized intervals bisecting those of 7- and 12-equal respectively. i'll be more than happy to answer questions if anyone's confused (i assume someone must be!), and also to describe how the *next* ("small 5-limit intervals") chart on that webpage helps one to understand the information above when used in conjuction with the "honeycomb" lattices of the ETs: http://groups.yahoo.com/group/tuning/files/perlich/15.gif http://groups.yahoo.com/group/tuning/files/perlich/22.gif (there are more like these on the tuning-math list)