source file: mills2.txt Date: Wed, 15 Nov 1995 05:15:22 -0800 Subject: Tritriadics, etc. From: "John H. Chalmers" Re Tritriadics :The idea for tritriadics is from A. J. Ellis's appendix to Helmholtz's On the Sensations of Tone in which he derives 56 tonal 7-tone "trichordal" modes in 5-limit JI by allowing the tonic, dominant and subdominant triads to either major or minor and then circularly permuting each scale (8 x 7 = 56). The concept goes back to Rameau. What I did was to generalize the type of triad and apply the idea to MOS and similar scales in ET as well. One is not limited to only 3 triads or to triads. Some scales are best chorded with 5 triads, 4 tetrads, etc. Max Mathews, Linda Roberts and John R. Pierce have also experimented with this kind of scale, especially those based on 3:5:7 and 5:7:9. The Pierce-Bohlen scale in based on 3:5:7:9 triads mapped into the 13th root of 3/1 (This ET has also been proposed by Enrique Moreno). Unfortunately, none of their published papers actually give the ratios or notes of the 4 scales they discovered. Does anyone have the M357a, M357b (the P-B scale in JI?), M579 and P3579a scales? JI: One of the major themes of Brian's recent posts on psychoacoustics was that subjects tested in the laboratory prefer sharp to JI intervals. While there is considerable data supporting this idea for single voice melodic intervals, it is not necessarily true for intervals in chords. Roberts and Mathews (1984) found that for the middle tone of various triads (major, minor, 5:7:9 and 3:5:7) some listeners preferred JI and some preferred intervals either sharp or flat by about 15 cents. Preference correlated with musical experience, violinists, cellists and tympanists preferred JI, untrained musicians, singers and pianists preferred the "rich" non-JI intonation. In these experiments the outer tones were tuned pure and the inner tones varied. In the case of minor triads, the "rich" intonationists preferred the flat 300 cent interval to the JI value. Similar results were obtained for chords in I IV I progressions. The roots of the triads were 262 hz. In the case of the progressions, the subdominant was taken from a major or natural minor scale on 262 hz. The 3:5:7 and 5:7:9 progressions appear to be taken from the corresponding tritriadics, but I have not been able to verify this yet. The point is that while musical experience has a strong effect on preference, though how one separates musical experience clearly from musical talent has never been clear to me, some listeners not only prefer JI for major and minor chords, they also do so for 7-limit chords as well. Secondly, while some listeners prefer sharp intervals, others prefer flat ones or have no preference for the direction of mistuning. They just like beats. These data imply that JI has a distinct "mood" which many, but not all, listeners prefer. The principal reference is Linda A. Roberts and Max V.Mathews. 1984. Intonation Sensitivity for traditional and nontraditional chords. J Acoust. Soc. Am. 75(3): 952-959. --John Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Wed, 15 Nov 1995 20:29 +0100 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id KAA16182; Wed, 15 Nov 1995 10:28:52 -0800 Date: Wed, 15 Nov 1995 10:28:52 -0800 Message-Id: <9511151028.ZM28007@frollo> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu