source file: mills2.txt Date: Mon, 4 Nov 1996 12:17:22 -0800 Subject: Re: Intervals list From: PAULE Incidentally, Plomp and Levelt did find that even with sine waves, in-tune octaves were more consonant than their out-of-tune neighbors. This has been attributed to second-order beating, which can be thought of as beating between the lower tone's summation tone with itself and the upper tone. However, at a low enough volume, second-order beating should disappear. I vaguely remember reading that it doesn't disappear, which was taken as evidence for periodicity pitch detection. Plomp and Levelt's main results need to be taken seriously in any theory of consonance. K&K (two Japanese workers -- I can't remember their names) calculated consonance curves based on Plomp and Levelt's results. K&K assumed some typical harmonic series, and assumed (I think) that total roughness is the sum of the individual sine-wave roughnesses calculated according to P&L's results. They derived the usual consonance/dissonance relations by defining consonant intervals as local minima on the graph. Clearly, with inharmonic timbres, none of the usual consonant intervals would appear (except the unison), and a whole new set would arise. This is the basis of Sethares' work, I believe. However, there is an additional component to consonance that is a somewhat separable issue. It goes under many names, such as "spectral fusion," "rootedness," etc., and is the degree to which the entire set of sine waves, belonging to more than one complex tone, approximates a single harmonic series. The harmonic series is privileged in our auditory system as the information is "compressed" into a single pitch (the fundamental, even if it is physically absent) and timbre (which represents the relative amplitudes of the harmonics). For instruments with harmonic timbres, this phenomenon explains the difference in consonance between "otonal" and "utonal" chords, whose roughness (if equal to the sum of the roughnesses of the individual intervals) is identical, but whose consonance is not. For minor vs. major triads, this may be arguable, but for complete 11-limit hexads, it's pretty easy to hear that the utonal chord is less consonant. This is because the otonal chord forms a single harmonic series, while the utonal chord can only be considered a harmonic series if four- and five-digit harmonic numbers are allowed. Tones with completely inharmonic spectra may be formed into chords that have the same degree of roughness as an "otonal" or "utonal" chord of harmonic spectra, but this other component of consonance will favor both of the chords with harmonic spectra over the inharmonic one. This is because any pair of tones, even in the utonal chord, forms a harmonic relationship, and, particularly for the perfect fifth, the set of harmonics resulting from the pair of tones forms a single harmonic series (in the perfect fifth, this harmonic series is 2/3 complete). Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Mon, 4 Nov 1996 21:34 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA01146; Mon, 4 Nov 1996 21:35:32 +0100 Received: from eartha.mills.edu by ns (smtpxd); id XA01149 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id MAA00486; Mon, 4 Nov 1996 12:35:27 -0800 Date: Mon, 4 Nov 1996 12:35:27 -0800 Message-Id: <199611042038.UAA24942@gollum.globalnet.co.uk> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu