source file: mills2.txt Date: Thu, 9 Nov 1995 10:28:53 -0800 From: "John H. Chalmers" From: mclaren Subject: A new rhythmic vocabulary --- "All things come from thence, from that universal rule either directly proceeding or by way of sequence." [Marcus Aurelius, "Meditations"] The previous post explored the outer edges of rhythm by suggesting an extension of the waveform periodicity in the frequency domain of non-just non-equal-tempered scales up into the beat level of the individual measure. As mentioned, there is a complete and utter lack of generalized rhythmic vocabulary to talk about these kinds of divisions of the beat. We can talk sensibly in Western music about "half notes," "3 eighth notes in the time of 2," and so on, and we can even (with some difficulty) talk about "40% of a half note" (notated as the N-tuplet `8 in the time of 5'), but when it comes to something like 1/1.61803399 of a half note, conventional Western notation falls mute. Indeed, there is not even the ghost of a clue where to start talking about such divisions of the beat except in raw numbers--which are tremendously hard to deal with intuitively, or manipulate as ensembles without huge amounts of gratuitous calculation. What does this have to do with tuning? It seems clear that the natural way to compose in non-just non-equal-tempered scales is to extend the frequencies into divisions of the beat. But what we want is a simple way of manipulating such rhythms. Ideally, we should be able to easily and simply apply such familiar concepts as rhythmic augmentation and diminution to sets of rhythms derived from the pitches of non-just non-equal-tempered scale frequencies. If we can't do this, it cripples us at the start in composing with non-just non-equal-tempered scales. First, let me suggest a generalization of the standard Western rhythmic vocabulary. Traditionally, divisions of the beat are handled with a descriptive vocabaular that directly specifies the division of the beat: half note lasts one half a whole note, quarter note lasts one quarter of a whole, triplet quarter packs 3 quarters into the time of 2, and so on. This is fine as far as it goes. But extending to anything other than simple integer divisions of the beat is impossible: there's no such thing as a "1.618034-note." Instead, permit me to suggest what the computer programmers have christened a "call by reference," rather than the "call by value" of conventional Western rhythmic vocabulary. Instead of using words for the divisions of the beat that describe the actual values, suppose we use a rhythmic vocabulary which describes the successive position of the rhythm in a hierarchy from long to short. This kind of rhythmic vocabulary could be applied to an unlimited range of different divisions of the beat, rather than the extremely limited set of integer divisions of the beat which can be described by traditional Western usage. PRIMARY -- longest duration within the measure SECONDARY -- next longest duration TERNARY -- next longest QUATERNARY -- next longest ..and so on. With this change of vocabulary, it suddenly becomes possible to write down a set of rhythms derived from our non-just non-equal- tempered scale: P S S P T P Given the non-just non-equal-tempered scale described in the previous post, this is a set of notes of duration: 1.0 1/1.618034 1/1.618034 1.0 1/2.414213 1.0 In the context of this new rhythmic vocabulary, all of the traditional techniques of Western rhythm can be applied. Here, augmentation refers to multiplying all notes by the value of 1/SECONDARY beat duration. In traditional Western usage, the secondary beat duration is always 1/2, so augmentation is always a simple doubling of note durations. Contrariwise, diminution is a simple havling of note durations. In the context of the rhythms derived from our non-just non-equal-tempered scale, however, augmentation means multiplying all note durations by 1.618034, while diminution means multipying all note durations by 1/1.618034. Embedded tuplets can also be carried over into the new rhythmic scheme, with a concomittant increase in rhythmic complexity. A triplet in traditional Western usage is obtained by adding the primary to the secondary duration; here it's obtained by doing the same thing, but the result (instead of being a 3:2 duration) is a 2.618034:1 duration. And so on. This gives us at least some reasonable way to *talk* about rhythmic divisions derived by time-scaling our non-just non-equal-tempered micro-level of frequency up into the level of the individual measure. Because of the obvious implications for new kinds of compositional techniques, this derivation of rhythm from non-just non-equal-tempered scale frequencies clearly demands further exploration. The next post will discuss generalizations of vertical structures in non-just non-equal-tempered scales, along with an examination of consonant vertical structures in a representative non-just non-equal-tempered scale, along with several kinds of near-equivalents to conventional modulation between keys (equal temperaments) or 1/1s (JI). ---mclaren Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Fri, 10 Nov 1995 00:38 +0100 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id OAA22718; Thu, 9 Nov 1995 14:38:39 -0800 Date: Thu, 9 Nov 1995 14:38:39 -0800 Message-Id: Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu