source file: mills2.txt Date: Thu, 2 Jan 1997 13:55:19 -0800 Subject: RE: To Paul Erlich From: Daniel Wolf <106232.3266@compuserve.com> You wrote: >Fourier proved that any periodic waveform can be expressed as the sum >of sine waves whose frequencies form an EXACT harmonic series. I quote from the Encyclopedic Dictionary of Mathematics (MIT 2nd Ed 1993) ''Fourier also stated (without rigorous proof) that an arbitrary function could be represented by trigonometric series, a statement that gave rise to subsequent developments in analysis. (Vol 1, p 620)'' At Santa Cruz (way back when) we did a series of sprectral analyses of real instruments and deviations from harmonic seria (mostly stretching but also some more eccentric behavior) were the norm. (This is certainly familar to anyone who has worked with real string instrument harmonics). At Darmstadt in '90 I saw some similar work from IRCAM done with extreme amplification so that the deviations through - say - the 64th partial were evident. Again: whether voices, bowed strings, brass, or woodwind, all had harmonics deviating from the expected integer ratios. Deviations and stretching in the strings and vocal chords seemed to account for the situation there, but what is with the wind instruments? A possible explanation: In a trombone, for example, you have several systems working at once whose net effect is quite complex, indeed aperiodic. The basic tube, and the bell have separate resonances (which vary with the slide and with muting), the mouth cavity of the player has its own variable resonances, and the oscillating lips are a very complicated sound with a large noise component and are variable in the extreme. The upshot of all this is a constant negotiation between the elements to get something acceptable. One short example: if I am to play d1 as the tonic of a major triad on a tenor trombone, I have to choose between buzzing d1 and trying to resonate the 5th partial of Bb in the first position or the third partial of G in the fourth position. Each of which produces a distict timbre and both of which may be problematic in the given triad, so I can change the shape of my mouth, or mute, or - yes - add either slide or lip vibrato (and I have heard players who could lip vibrato over an octave), or add or remove noise from the lip (''growl''). (I know this will damage my reputation, but I recommend listening to any recordings of Jackie Gleason's ''Velvet Brass'', of Spike Jones recordings with Frank Rehak, or of Stuart Dempster's performance of Robert Erickson's _General Speech_ to hear what trombonists can do with pitch and timbre.) About vocal ''subharmonics'', I find the easiest way to learn to produce them is to start by singing while inhaling (the ''fry''), eventually adding ex- haling, so that one can eventually do both and sing continuously (I remem- ber Robert Ashley doing something similar in the early 70s so that he could speak without pausing. The Tuvan singers here in Frankfurt on the main shopping street do the same. You wrote: >Daniel, do you agree with LaMonte and if so, how do you justify doing JI >with beatless pure tones without invoking combination tones? Yes, I agree with La Monte (while recognizing that some people have excellent pitch height memory) and thus combination tones are indispensible - if you are making reference to my spacing model, the harmonic series is the model and structures are described in terms of just intonation but only to simplify matters as I assume some degree of tolerance, likewise I assumed sine waves (need I repeat it again) to avoid masking. Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Thu, 2 Jan 1997 23:16 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA00908; Thu, 2 Jan 1997 23:19:37 +0100 Received: from eartha.mills.edu by ns (smtpxd); id XA00906 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id OAA13106; Thu, 2 Jan 1997 14:19:31 -0800 Date: Thu, 2 Jan 1997 14:19:31 -0800 Message-Id: <199701021717_MC1-DFD-DB02@compuserve.com> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu