source file: mills2.txt Date: Fri, 21 Jun 1996 11:58:47 -0700 Subject: Re: Post from McLaren From: PAULE It is also well documented that stretched octaves are preferred to just octaves. But all these studies tend to focus of melodic rather than harmonic intervals, or else tones with weak harmonics. The fact that "stretched" intervals are preferred to just intervals in a context where beating between partials is not the most prominent determinant of consonance may indicate that Terhardt's hypothesis about a "stretched" harmonic template is correct. The ear, when presented with a multiplicity of pure tone components (in Parncutt's version, the prenatal ear hearing the mother's voice), is known to slightly stretch the intervals between them, and one hypothesis is that this stretching is "expected" even when a meager stimulus is presented. Goldstein studied the precision with which frequency information is transmitted by the brain's central pitch processor by measuring residue ("virtual") pitches produced by two pure tones, but I think his approach assumed symmetrical errors and did not pay attention to systematic biases in the allowable mistuning of the pure tones. I wonder if such a study would reveal that judgments of harmonic relationships are easier to make when the distances are stretched somewhat. In any case, it may be a case of the archers aiming for the bullsye, but a steady wind blowing the arrows off course. The aesthetic of stretched intervals must be somewhat compromised in practice: phenomena such as second-order beating and combination tones can be disturbing for significant departures from harmonicity. The "small integers" concept of consonance that McLaren "disproves" is a rather extreme, Partchian version. Which is more consonant, 300001/200000, or 11/9? While the former clearly involves larger integers, it is also very close to 3/2, and is likely to be heard as such. He also extends the small number theory to chords of more than two notes, where the use of frequencies vs. periods can determine which chord is calculated to be more consonant. Observe: Frequencies: Major - 4:5:6 Minor - 10:12:15 Suspended - 6:8:9 Periods: Major - 10:12:15 Minor - 4:5:6 Suspended - 6:8:9 Using either frequencies or periods, the suspended triad comes out in between major and minor; the fact that the suspended triad is more dissonant is related to the fact that when the intervals are looked at individually, it requires higher numbers. Moreover, McLaren seems to love neutral thirds and hate tritones; I have to strongly disagree with his consonance judgments here. But consonance is a very subjective judgment, and the art of composition can be just as effective (or more so) when based on flawed or subjective premises. . . . Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Fri, 21 Jun 1996 22:18 +0100 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id NAA16477; Fri, 21 Jun 1996 13:18:33 -0700 Date: Fri, 21 Jun 1996 13:18:33 -0700 Message-Id: Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu