source file: m1400.txt Date: Wed, 29 Apr 1998 19:54:20 +0000 Subject: Re: JI Tuning Resolution From: david first Paul H. Erlich says: > > For an interval in perfect just intontation, whatever phase difference > was present at the beginning of the sound would persist for the duration > of the sound. Let's say two instruments are producing tones, both of > which have an equally loud partial at 1000Hz. For someone standing in a > particular location with respect to the two instruments, there is some > delay time between the beginning times of the two instruments for which > the 1000Hz components will interefere constructively, resulting in four > times the energy at that frequency than what one instrument alone would > produce. For a delay time just 1/2000 of a second longer, you get > destructive interference, or no energy at that frequency. Obviously no > human instrumentalist can control their onset time to within 1/2000 of a > second relative to the onset time of another instrument. So you end up > with a random number between zero and four to describe the energy of the > 1000Hz component -- probably not the musical effect you were looking > for. Even an individual performer playing both instruments can't do it > -- try tuning two synth keys to the same note and notice how repeatedly > playing both keys "simultaneously" leads to random fluctuations in the > loudness. > > Now if the onset times are controlled really accurately by MIDI, and > let's say a 90-degree phase shift is chosen so that the energy at 1000Hz > is twice that from one instrument, then you're doing okay, right? But > for someone standing in a different location in the room, though, the > differece in the path lengths from the two instruments, and thus the > phase, will be different. There are nodes of destructive interference > and antinodes of constructive interference in the room no matter what > the original phase shift was. So some members of the audience may be > receiving no energy at 1000Hz, while others may be receiving four times > what they would from one instrument. Again, probably not the effect you > were looking for. > > If both instruments are electronic, and their signals are mixed into one > speaker, there will be no spatial nodes or antinodes. So finally you can > control the musical effect. But other than monophonic electronic > MIDI-sequenced music, perfect JI can lead to big problems. In the real > world, thankfully, acoustic instruments are slightly out-of-tune with > each other, so that whatever the onset delay and whatever the position > in the room, one gets a signal that frequencies common to both tones > have an average energy twice that of one instrument. There will not be > noticeable, let alone startling, differences in the musical effect > depending on minute variations in the onset delay or position in the > room. > In addition, it has been my experience that when creating JI related sample loops, that it is not enough that the frequency relationships be exact - the period of the wave cycles involved all must have a simple divisible integer relation with one common compound period/sample length. Further, I believe that it is best to have a simple divisible relationship with the sampling rate one is employing. As an example, for a recent set of 21 JI pitch relationships I developed, I used a common 1/1 period number/sample base length of 21870. I then doubled this number to achieve the sampling rate of 43740. 21870 divided by 1/1 = 21870 21870 " " " 9/8 = 19440 21870 " " " 6/5 = 18225 21870 " " " 5/4 = 17496 21870 " " " 27/20 = 16200 21870 " " " 45/32 = 15552 21870 " " " 3/2 = 14580 21870 " " " 27/16 = 12960 21870 " " " 9/5 = 12150 21870 " " " 15/8 = 11664 21870 " " " 81/40 = 10800 21870 " " " 135/64 = 10368 21870 " " " 729/320 = 9600 21870 " " " 1215/512 = 9216 21870 " " " 81/32 = 8640 21870 " " " 729/256 = 7680 21870 " " " 243/80 = 7200 21870 " " " 405/128 = 6912 21870 " " " 2187/640 = 6400 21870 " " " 3645/1024 = 6144 21870 " " " 243/64 = 5760 Obviously, this base number gets larger with each added interval and can very quickly go out of bounds even when one is using the simplest of source materials. I, myself, have found it most unwieldy, if not impossible, to create viable samples using anything besides sawtooth, square, etc. More complex waveforms, or anything remotely resembling the physical model of an acoustic instrument would take more sample RAM than I,for one, currently have! At any rate, if I read (the slightly truncated) explanation above correctly, JI is an impossible goal in the real, 3D/air based world we reside in. That makes perfect sense to me, but one can only try... David F.