source file: m1396.txt Date: Sat, 25 Apr 1998 16:45:35 -0400 Subject: RE: JI Tuning Resolution From: "Paul H. Erlich" Gary, >If you zoom it in on a single period of the tone, you see the peaks and >troughs of the vibration bobbing up and down, and sometimes moving around >too. Regardless of what you attribute that to, it's certainly not a >strictly periodic wave. But if you change the time per division so as to >zoom out to show 10 or so cycles, you can see that the bobbing up and >down >of the peaks and troughs follows a regular progression from one cycle to >the next. Sometimes you can even actually see a peak moving through the >lower frequency components. I grant you that typically the wave will not be strictly periodic, because the player's bowing, lipping, or blowing will not be entirely steady. The problem is that if one of the harmonic partials is changing in amplitude due to this unsteadiness, and you do a Fourier analysis on a segment of time in which these changes are significant, the Fourier transform will only be allowed to assign one amplitude to each frequency. Therefore, the result of the analysis will be that the frequencies are slightly altered, usually both upwards and downwards simultaneously, so that interference can simulate the changes in amplitude. (Try it!) However, as the ear can only resolve one pitch within each critical bandwidth (and there are nerve-firing periodicities which supplement the cochlear mechanisms), it will indeed be able to interpret the signal as having harmonic partials of changing amplitudes. I'm not saying that one interpretation or the other is more correct; the problem of describing a waveform as a sum of frequencies each of whose amplitude changes over time has many equally valid solutions. I'm saying that the solution relevant for how we hear the tones and how dissonance might arise with other tones is, in most cases, one where the partials are harmonic and of changing amplitude. (Looking at peaks and troughs can be misleading. The way we hear a wave has surprisingly little correlation with how the wave looks.) Add the variablity of the overall pitch level and you have another complication for a traditional Fourier analysis. If there are indeed overtones of constant amplitude that do not fall within the harmonic series of the fundamental, there must be an independent source of energy producing them. I don't think this occurs too often in any significant proportion in the types of instruments we're talking about. >>If you study the mechanics of >>a 1-component driver, such as a bow, reed, or lips, you will understand >>that regardless of the extent to which the resonant modes of vibration >>of the instrument deviate from a harmonic series, the driving mechanism >>will force the waveform to become periodic, >Would you apply that to a flute, for example? I don't think the turbulent airflow that drives a flute is well enough understood to make this characterization. However, this airflow does appear to be characterised by "periodic" and "chaotic" regimes of behavior. >It sounds like you're suggesting that lips, or a reed for example, can >only vibrate periodically. I personally am aware of no evidence to that >effect. They certainly don't have the physical characteristics of a system >that vibrates only in harmonics, notably a mechanical transmission line. Under a steady airflow, the Bernoulli effect that explains their motion predicts a periodic behavior. Even without the rest of the instrument to dictate where the fundamental will resonate with a minimum of input energy, the mouthpieces of these instruments can be used to create a musical tone with harmonic overtones (albeit highly variable overall pitch). The timbre of instruments, as I'm sure you know, is highly dependent on the ability of the instrument to resonate at harmonic overtones of a given resonance. This is becuase the lips/reed are highly non-linear, creating a host of harmonic overtones along with the sinusoudal fundamental that a simple Bernoulli model would predict. Now the extent of this non-linearity is dependent on the physical characteristics of the reed/lips, which can be constantly altered by the player, even if he/she is not meaning to do so. Hence the amplitude of the partials can vary, sometimes wildly, rendering the waveform aperiodic. However, for the time domains relevant for the ear's analysis, this does not lead to any relevant alterations in the frequencies of the partials. >>> . . . JI does what it does more in the lower harmonics >>>than the upper ones >>I think Harry Partch would disagree with you there. >That of course is why I said "to the degree that JI does what it >does...": To include the possibility that that degree might be zero. Or >very high. >In other words, I was not expressing an opinion along those lines; just >stating that if that were true, then the fact that the lower harmonics are >closer to harmonic than the would be relevant. >Boy, am I confused!