source file: mills2.txt Date: Thu, 2 Jan 1997 09:38:56 -0800 Subject: RE: First Published Treatise in English From: PAULE >One phenomenon that I recall from trombone playing was the constant >correction of pitch (by lip and slide) so that the deficient harmonic >series of the tube would sound better. This may be taking place in vocal >performance as well. Daniel, Fourier proved that any periodic waveform can be expressed as the sum of sine waves whose frequencies form an EXACT harmonic series. A brass instrument is crafted so that its RESONANCES approximate a harmonic series, but this relationship is not exact (a lot of further misconceptions surround brass instrument physics). However, when playing a single note, the resonances are DRIVEN by the lips of the player, which normally produce a periodic waveform during the "sustained" part of the note. Therefore, an EXACT harmonic series is available to drive the resonances of the instrument. The closer these resonances are to harmonic, the more the overtones will be amplified -- this is the principle behind successful brass instrument design. Likewise, the human vocal cords, when employed in a steady, sustained manner, produce a periodic waveform (though recently I have been able to DOUBLE or occasionally TRIPLE the period of this waveform and sing an octave or twelfth lower than would be normal -- undertone singing! I can then emphasize different overtones of these really low notes for an amateur version of throat singing). Overtone singing is just the process of shifting the resonance to one or another of the harmonic overtones -- again, an exact harmonic series results (of course, no one can maintain the pitch of the fundamental perfectly, but that is a separate issue). Inharmonic series are possible in large, essentially infinite-element vibrating systems like a piano string, but brass and wind instruments and human voices are essentially elaborate amplification devices for a single-element periodic oscillator. >When ''overtone singing'' had its heydey, I recall >being disturbed by how unstable the fundamental was, and the constant >intrusion of beats, perhaps resulting from attempts to tune the harmonics >at the expense of the fundamentals. In light of the above, this doesn't really make sense. >La Monte >contends that one can sing Just intervals only through elimination of >beating Daniel, do you agree with LaMonte and if so, how do you justify doing JI with beatless pure tones without invoking combination tones? Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Thu, 2 Jan 1997 19:26 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA00810; Thu, 2 Jan 1997 19:29:23 +0100 Received: from eartha.mills.edu by ns (smtpxd); id XA00808 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id KAA09284; Thu, 2 Jan 1997 10:29:21 -0800 Date: Thu, 2 Jan 1997 10:29:21 -0800 Message-Id: <009ADC642B092160.0354@vbv40.ezh.nl> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu