source file: mills2.txt Date: Fri, 3 Nov 1995 17:56:16 -0800 Subject: 88CET #24: 88CET-Mapped Timbres From: Gary Morrison <71670.2576@compuserve.com> At various times in these posts, I've attached a caveat to my statements about 88CET, that they apply to essentially harmonic timbres. Many of those statements relate to the "taboo", off perfect consonances, like the off octaves, and how important it is to avoid them. I also briefly aluded to Bill Sethares' work with 88CET using timbres optimized for 88CET. That's what I'm going to discuss briefly here. As most tuning-listers are well aware, almost all musical tones consist of composite of frequency components (partials). Although we often speak in terms of relationships between the fundamental pitches of the notes in a chord, all of the frequency components in all notes within a chord ultimately interact. The pitches of all of the partials ultimately determine how a chord will sound. So my repeated statements then that the "taboo" off perfect consonances sound awful, are all relative to the timbre in which you play them. If the partials are essentially harmonic, then those "taboo" intervals are realistically unusable. But Sethares took this tuning-timbre relationship in reverse. Rather than taking the overtone structure for granted, and accepting certain intervals as unfortunate casualties, Sethares instead took the 14-step stretched octave as a basic building block of his 88CET music, and remapped the partials so that sounds good. In particular, he came up with a mapping so that the 1232c off-octave creates an identity sensation much like that of an octave for the exact harmonic timbres. Not surprisingly, Sethares has this process down to a science, or at least to an art, and 88CET is just one of many tunings to which he's applied the technique. Two general comments are in order. First, this pseudo-octave aside, this mapping process produces some really fascinating-sounding timbres. They sound somewhat like hybrid timbres - hybrids of whatever they were before mapping, with a bell-like sound. The commonly-held belief that only harmonic overtone structures "fuse" to what our ears perceive as single tones, turns out to be mostly myth. Secondly, it does work; on their own terms, these timbres really do produce an octave-like identity sensation. This paper is not the correct forum for explaining the theory behind Sethares' mapping process, but this much is worth telling: The mapping is built upon empirical curves elating the dissonance of two sinewaves to frequency differences, measured by Plomp and Levelt in 1965. Their experimental data was based on statistical surveys of many people. Sethares curve-fitted this experimental data to a mathematical relationship, and derived a procedure for applying the results to all partials of a complex tone, thereby measuring the overall dissonance of an given interval as rendered in a given tonal spectrum. He then turned the process around so that he could derive tonal spectra for minimal dissonance of a given set of intervals. In the dissonance curve for 88CET, the troughs (local minima) in the curve are places of minimum dissonance, and thus maximal consonance. The overtone retuning for making 88CET's 1232c off-octave into a pseudo-octave, starting from a hypothetical purely-harmonic tone, uses the following frequency multipliers for the first ten partials: 1, 2.03, 3.06, 3.94, 5.087, 5.92, 6.9, 8.04, 9.36, 10.36. Perhaps not surprisingly, this isn't without its trade-offs, both good and bad. To my ears, these nonharmonic overtone structures make the exact pitch of the notes a little hard to pinpoint. As such, they somewhat blur 88CET's interval approximations that are nearly just. The fifth loses some of its steely-cold simplicity, the subminor third and seventh some of their septimal "zap", and the major tenth some of its sweet richness. But at least one intervals benefits: the 9:7 supramajor third, often cited as harsh, loses some of its edge. The dissonance curve has a trough on 88CET's 9:7 approximation. Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Sat, 4 Nov 1995 07:04 +0100 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id VAA12354; Fri, 3 Nov 1995 21:04:50 -0800 Date: Fri, 3 Nov 1995 21:04:50 -0800 Message-Id: <199511040504.AA03370@world.std.com> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu