source file: mills2.txt Date: Sun, 15 Oct 1995 21:06:48 -0700 Subject: Computer-aided composition From: MMCK@delphi.com >Perhaps Marion can expound a little further on his >techniques for exploring a wide variety of tunings (perhaps >comparing them to algorithmic composition techniques?), and why >he finds that JI systems are easier to explore in that manner. There are two dimensions (at least) to this problem. One is theoretical, and one is practical. In the theoretical dimension, we have at our disposal a musical instrument of infinite resolution. We are faced with the problem of composing in a transcendental or irrational scale. We need to decide that certain combinations of notes are acceptable, or consonant, and others are not consonant. We know that factors such as absolute pitch, volume, and harmonic content will affect consonance, but for purposes of this simplified discussion, I will ignore those. There may be many valid ways of making this judgement, but approximations of JI intervals are the usual. You will hear phrase like "this interval is a little flat from the major third". Judgements like this may be fine for humans, but are far to general for a computer to understand. It takes a much more precise definition to program a computer to make judgements about harmony. It is fairly easy to guess that 14983/10000 fairly closely approximates the 3/2, but other situations are not so clear cut-- especially if we intend to sound 3 or 4 notes at the same time. After all, the purpose of tempering is to introduce ambiguity into the tuning so that intervals may be interpreted by the ear in more than one way. For example, we can take the ET approximation for the third, and reduce it in steps to the JI third 5/4. 1259921/1000000 527253/418481 205415/163038 88992/70633 27431/21772 6699/5317 635/504 286/227 63/50 29/23 5/4 At each successive approximation, we are further from the original note of the ET scale. So we have a trade-off between accuracy to the original model, and simplicity of tuning. This is a complex decision for only two notes, and if we make different choices for different combinations of notes, which is apparently what the ear does, then it is even more complex, and what if we are to eschew this simple Euclidian series and opt to only include 5, 7, or 11 limit numbers, or use some other criteria altogether. One can come up with thousands of JI scales that approximate any irrational scale. For example, there is no single JI scale that is generally agreed upon to represent the harmonic characteristics of 12ET, surely the single most analyzed scale in human history. The purpose of tempering is to introduce ambiguity, and the degree to which it is successful in its purpose is also the degree to which it becomes difficult to analyze. OTH, JI scales contain none of that kind of ambiguity. Each combination of notes represents only one point on a gradient of consonance, and can be interpreted only one way. Some may find harmonies acceptable that others do not, but it is possible a computationally straightforward index of consonance without having to search a vast tree of possibilities. Therefore, the space that must be searched by a heuristic designed to measure consonance is almost infinitely smaller for a JI scale than an irrational scale. LCM is the heuristic of choice for me, but I think any of the others I have seen discussed here would have the same characteristics. In the practical dimension, one must also consider tuning granularity. Since all present day electronic instruments use digital oscillators, they have nearly exact accuracy, and great stability, but, because tuning has not been an important design criteria, they have poor tuning granularity compared to what could be done with very little more cost and engineering. In addition, the tuning characteristics of wavetable synthesis are a function of the sample length. Tuning granularity may be different for a piano or flute voice, or be a function of the absolute pitch. Manufacturer's of such instruments do not release the sample length data, so, when you use one of these devices, you are buying pseudo-random tuning, unless you want to reverse engineer their firmware, or measure the output with a frequency counter while changing voices etc. FM synthesis at least produces predictable tuning with a published spec. But it is certainly not an implementation of anything like infinite resolution. Tuning numbers can only range up to 1024. I think tuning numbers of 10^10 are easily possible. But the Sound Blaster can still produce an astounding number of scales. For example, it can play some 72,000, 12-tone, five- limit scales with more than 5 LCM60 chords. For those who are into non-exact JI, or tempering, the tuning characteristics of the chosen instrument introduce yet another variable to further fuzzify computations required to evaluate a particular combination of notes. This is not to say that those computations can't, or shouldn't be done, but it is a much, much harder problem than exact JI. For exact JI, tuning variations do not exist, so they do not need to be considered in the harmonic analysis. I have done experiments in pure algorithmic composition where you just program some machine, and then let it produce music. As a matter of fact, in 1983, I built such a machine out of about 75 medium scale logic chips, and I still have it in my garage. I gradually came around to the view that computer-aided composition is much better. I usually use a minimum chord set, and in many cases my compositions are made up entirely of LCM60 chords, which means the harmony is much simpler than most. If I want to use more complex chords, I usually generate a more complex scale and still use a minimum chord set. This seems to me to give better results than using a maximally consonant scale, but choosing more complex chords. If you are interested in learning more about my approach to this problem, you can ftp FasTrak from the ccm/tuning/software/pc/fastrak directory at mills.edu. Even if you don't have an IBM PC with a Sound Card, you might like to read the on-line documentation. You will need a PC to unzip the file though. Marion Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Mon, 16 Oct 1995 13:00 +0100 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id DAA04717; Mon, 16 Oct 1995 03:59:38 -0700 Date: Mon, 16 Oct 1995 03:59:38 -0700 Message-Id: <00997F4FA2788C45.52D8@ezh.nl> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu