source file: mills3.txt Date: Sun, 14 Dec 1997 17:31:22 +0100 Subject: 61 Intervals of Just Intonation From: Gregg Gibson Here is a table of the 61 intervals of Just Intonation, according to my posting of yesterday. Class A comprises the six consonances plus the octave, Class B comprises the twelve tonal dissonances, Class C the eighteen atonal dissonances, and Class D the twenty-four ultra-atonal dissonances. Cents Ratio Class 21.5 81:80 D 41.1 128:125 C 62.6 648:625 D 70.7 25:24 B 92.2 135:128 D 111.7 16:15 B 133.2 27:25 C 141.3 625:576 D 182.4 10:9 B 203.9 9:8 B 223.5 256:225 D 245.0 144:125 C 253.1 125:108 C 274.6 75:64 C 294.1 32:27 C 315.6 6:5 A 345.3 625:512 D 356.7 768:625 D 364.8 100:81 D 386.3 5:4 A 407.8 81:64 D 427.4 32:25 B 448.9 162:125 D 457.0 125:96 C 498.0 4:3 A 519.6 27:20 C 539.1 512:375 D 560.6 864:625 D 568.7 25:18 B 590.2 45:32 C 609.8 64:45 C 631.3 36:25 B 639.4 625:432 D 660.9 375:256 D 680.7 40:27 C 702.0 3:2 A 743.0 192:125 C 751.1 125:81 D 772.6 25:16 B 792.2 128:81 D 813.7 8:5 A 835.2 81:50 D 843.3 625:384 D 854.7 1024:625 D 884.4 5:3 A 905.9 27:16 C 925.4 128:75 C 946.9 216:125 C 955.0 125:72 C 976.5 225:128 D 996.1 16:9 B 1017.6 9:5 B 1058.7 1152:625 D 1066.8 50:27 C 1088.3 15:8 B 1107.8 256:135 D 1129.3 48:25 B 1137.4 625:324 D 1158.9 125:64 C 1178.5 160:81 D 1200.0 2:1 A Whew! I do hope I made clear the quite deliberate and _non-arbitrary_ manner in which these values are derived. One takes a given pitch, calls it the tonic, then fixes the six intervals that are consonant with it, 3:2 4:3 5:4 5:3 6:5 & 8:5 both ascending and descending (not really necessary, because three of these are inversions of the other three, but this makes the process clearer, and nearer to the viewpoint of a singer). To find the tonal dissonances one ascends and descends by the same six consonant intervals from the pitches consonant with the tonic. This gives 3/2 x 3/2 = 9/4 (or 9/8), 3/2 x 4/3, 3/2 x 5/4, etc which yields 12 additional pitches, called tonal dissonances. One can then use each of these 12 pitches as new points of departure, proceeding from each via the six consonances, to find 18 atonal dissonances. And so on ad infinitum. This manner of constructing just intonation is open to the important objection that it may well be possible to relate a dissonance more closely to the tonic via fifths than via minor thirds, say. So an atonal dissonance such as 27:16 or 45:32 (the latter often used in the just diatonic scale!) might be considered more closely related to the tonic than a tonal dissonance such as 36:25. On the other hand, two factors militate against such an objection. First, fifths often give rise to dissonances quite near to consonances (e.g. 27:16 or 81:64) and such dissonances are not likely to acquire any very stable melodic existence at all, for they are susceptible to being drawn into the orbit of adjacent consonances, and drained of any possible melodic individuality. Second, we are not really dealing with the fifth versus the minor sixth, say, but with the three consonant cycles: fifth/fourth, major third/minor sixth and minor third/major sixth, and each of these cycles is more or less equally important to harmony. If one accepts the three septimal intervals 7:4 7:5 & 7:6 as consonant, just intonation becomes immeasurably more complex. But the additional values are melodically identical with those defined via the six traditional consonances of the senario. Futhermore, the septimal intervals have only the most tenuous mathematical or acoustic relationship with the intervals of the senario. And inasmuch as no informed person doubts that the six consonances of the senario are , even if not the only consonances, at the least much more consonant _as a system_ than the septimal intervals _as a system_, whether one defines the septimals as weak consonances (as some do) or as weak dissonances (as I and many others do) is largely irrelevant to just intonation qua theoretical basis for temperament. It need scarcely be observed that this or any other just intonation is _not_ fit for practical use in music; it is the indispensable basis for tuning theory, but requires to be tempered before it can be used in actual music. SMTPOriginator: tuning@eartha.mills.edu From: mr88cet@texas.net (Gary Morrison) Subject: Quick Note on Tape-Swap Orders PostedDate: 14-12-97 20:39:12 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: 14-12-97 20:37:18-14-12-97 20:37:18,14-12-97 20:36:59-14-12-97 20:36:59 DeliveredDate: 14-12-97 20:36:59 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 C125656D.006BC49C; Sun, 14 Dec 1997 20:39:03 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA15056; Sun, 14 Dec 1997 20:39:12 +0100 Date: Sun, 14 Dec 1997 20:39:12 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA15055 Received: (qmail 29566 invoked from network); 14 Dec 1997 11:39:08 -0800 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 14 Dec 1997 11:39:08 -0800 Message-Id: Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu