From jzmckay@MIT.EDU Mon Sep 7 23:09:38 1998 Return-Path: Received: from MIT.EDU (PACIFIC-CARRIER-ANNEX.MIT.EDU [18.69.0.28]) by clinch.math.niu.edu (8.8.5/8.8.5) with SMTP id XAA06871 for ; Mon, 7 Sep 1998 23:09:37 -0500 (CDT) Received: from MIT.MIT.EDU by MIT.EDU with SMTP id AA05806; Tue, 8 Sep 98 00:09:32 EDT Received: from ELEAZAR.MIT.EDU by MIT.MIT.EDU (5.61/4.7) id AA26051; Tue, 8 Sep 98 00:09:30 EDT Message-Id: <3.0.32.19980908000914.0069d51c@po8.mit.edu> X-Sender: jzmckay@po8.mit.edu X-Mailer: Windows Eudora Pro Version 3.0 (32) Date: Tue, 08 Sep 1998 00:09:15 -0400 To: rusin@math.niu.edu From: "John Z. McKay" Subject: Mathematics and Music page--theory of chords Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Status: RO I just happened to be reading through the info on your page, and, since you say your page is used often as a resource, I thought it might be important to let you know about some inaccuracies. I have not reviewed every link, so maybe this is discussed somewhere, but the info on the tuba in the "theory of chords" link is quite incorrect (the author also presents some non-standard terminology, but that is a relatively minor point of contention). It is claimed that the three valves on the tuba raise the pitch by 1, 2, and 4 half-tones respectively. In reality, the valves *lower* the pitch--as can be guessed by simply examining the instrument and noticing that the valves lengthen the tube through which the air moves. All standard valved brass instruments--including trumpets, french horns, baritones, euphoniums, tubas, etc.--have 3 valves that function the same way (however, there are triggers on double french horns and 4th valves on some instruments to expand range). The valves do the following: 1st - lowers pitch by 2 semitones (one "whole step") 2nd - lowers pitch by 1 semitone (one "half step") 3rd - lowers pitch by 3 semitones If there is a 4th valve, I believe it lowers the pitch by 4 semitones; and the trigger on the double horn restricts the air flow pattern in the horn to shorten the length of the sound-producing tube and effectively move the sound (and all fingerings) to a different register (a perfect 4th--5 semitones--upward). Thus, there is nothing binary about a tuba's valve construction. The combinations of valves to move the sound down are as follows: 1 semitone - 2nd 2 semitones - 1st 3 semitones - 1st + 2nd (the 3rd valve is an alternative, although not quite in tune) 4 semitones - 2nd + 3rd 5 semitones - 1st + 3rd 6 semitones - 1st + 2nd + 3rd Most tubas are tuned in B-flat (rarely, in E-flat). Assuming, for sake of simplicity that they were tuned in C, the "open tones" (those played without valves) would be: C1, C2, G2, C3, E3, G3, Bb3 (way out of tune--not used), C4, D4, E4, F4 & F#4 (both out of tune), G4,... where 1,2,3,4 represent the relative octave of the pitch, and C1 represents the "fundamental" pitch of the instrument. In most brass instruments, the useful range is from a few notes below C2 up to a few notes past C4, with the majority of common notes being between C2 and C4 (4-valved instruments can go down to around C1, only missing a fingering for the half-tone above C1). As can be seen from the fingerings and the harmonic series outlined above, the only place in the useful range where the first valve can "move the pitch up" a half-step would be between E3 (open) and F3 (1st valve). The only place where the second valve can "move up" two half-steps is between E3 (open) and F#3 (2nd valve). And, the third valve can only "move up" four half-steps between C2 (open) and E2 (alternative fingering of 3rd valve), and similarly for C3 and E3. Above C4, the harmonics get quite close together, and many notes have many alternate fingerings (including those suggested by the author in some instances)... however, playing this high is rare and difficult for any novice on an instrument. French horns are somewhat different in that their fundamental is an octave further below their normal range, so horn players must deal with these complications on a daily basis. But, this certainly does not apply to tubas. I'm not quite sure where the author got his information, but if he somehow found a tuba that could produce a 1,2,4 pattern for the valves in more than the few instances I outlined above, it must be a non-standard instrument. I just thought I would offer this since persons who read the info on your site looking for musical information can be misled by the statements about tubas in that link. I would also like to say that I found your site to be a good collection of information on the links between music and mathematics. I look forward to reading anything further that you may post there. John From rusin@math.niu.edu Mon Sep 7 23:29:37 1998 Return-Path: Received: from vesuvius.math.niu.edu (vesuvius.math.niu.edu [131.156.3.93]) by clinch.math.niu.edu (8.8.5/8.8.5) with ESMTP id XAA06984; Mon, 7 Sep 1998 23:29:36 -0500 (CDT) From: Dave Rusin Received: (from rusin@localhost) by vesuvius.math.niu.edu (8.8.8/8.8.5) id XAA17669; Mon, 7 Sep 1998 23:29:36 -0500 (CDT) Date: Mon, 7 Sep 1998 23:29:36 -0500 (CDT) Message-Id: <199809080429.XAA17669@vesuvius.math.niu.edu> To: jzmckay@MIT.EDU Subject: Re: Mathematics and Music page--theory of chords Cc: rusin@math.niu.edu Status: RO >I just happened to be reading through the info on your page, and, since you >say your page is used often as a resource, I thought it might be important >to let you know about some inaccuracies. Thank you for your corrections. You laid it out quite carefully; would you mind if I simply append your letter to the one on the website? (Properly attributed, of course, unless you'd prefer to remain anonymous.) I guess I hadn't read the file carefully enough myself to catch the errors. >Most tubas are tuned in B-flat (rarely, in E-flat). Yes. Actually I was told high-school band directors liked having the Eb-tuned ones handy because trumpeters could fill in for missing tuba players. (How they would manage the lip adjustment is beyond me!) >C1, C2, G2, C3, E3, G3, Bb3 (way out of tune--not used), C4, D4, E4, F4 & ^^ Whoa! Tidbit: harp octaves are numbered from the top. But then, harpists only number four of the fingers "correctly", so I guess we can't trust them... >I look forward to reading anything further that you may post there. I'm not quite sure how the music side-line to my website got started, but I try to keep it organized since it seems to be quite popular. I don't really make any pretense that it's as comprehensive, or even as accurate, as the main math site I administer (www.math.niu.edu/~rusin/known-math/) But I'm glad you found it worth looking at. Thanks again for writing. dave From jzmckay@MIT.EDU Tue Sep 8 01:14:13 1998 Return-Path: Received: from MIT.EDU (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by clinch.math.niu.edu (8.8.5/8.8.5) with SMTP id BAA07378 for ; Tue, 8 Sep 1998 01:14:13 -0500 (CDT) Received: from MIT.MIT.EDU by MIT.EDU with SMTP id AA04040; Tue, 8 Sep 98 02:14:11 EDT Received: from ELEAZAR.MIT.EDU by MIT.MIT.EDU (5.61/4.7) id AA06440; Tue, 8 Sep 98 02:14:12 EDT Message-Id: <3.0.32.19980908021354.006915d0@po8.mit.edu> X-Sender: jzmckay@po8.mit.edu X-Mailer: Windows Eudora Pro Version 3.0 (32) Date: Tue, 08 Sep 1998 02:13:54 -0400 To: Dave Rusin From: "John Z. McKay" Subject: Re: Mathematics and Music page--theory of chords Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Status: RO You can certainly just append my e-mail, edited somewhat if you like... >>C1, C2, G2, C3, E3, G3, Bb3 (way out of tune--not used), C4, D4, E4, F4 & > ^^ Whoa! > >Tidbit: harp octaves are numbered from the top. But then, harpists only >number four of the fingers "correctly", so I guess we can't trust them... I'll admit that my makeshift notation might be somewhat non-standard. The standard musical notation I learned is "c1" (lowercase) = middle C. "C" (uppercase) is the octave below, "C1" is an octave below that, "C2", "C3", etc. The octaves above middle C are "c2", "c3", etc. Usually, the numbers are subscripted below middle C and superscripted above. Thus, the a B-flat tuba's fundamental would be Bb2, and the harmonic series would be Bb2, Bb1, F, Bb, d, f, (a-flat), b-flat, c1, d1,... I thought that writing it that way is more confusing to those unfamiliar with the notation, plus without standard flat signs you would have to write "b-flat" rather than "bb" to be clear. Just to let you know, I am a chemical engineering student at MIT who recently declared a second major in music (emphasis in composition). I've often considered majoring in math, but I never got around to pursuing it. For a long time I've been interested in the acoustical and mathematical basis of music, and at times I've questioned the legitimacy of tempering the scale into 12 equal semi-tones... especially as it is used in atonal music. Chord progressions and resolutions in tonal music are based on acoustic principals and tendencies for dissonances to resolve in certain specific ways. Those resolutions are weakened when the scale is equally-tempered. I realize that "well-tempered" instruments became popular in Bach's time due to the ease of transposition without retuning (especially on keyboard instruments). However, I question the basis for then taking these acoustically-weakened intervals and writing atonal music. The dissonance and resolution in tonal music relies heavily on the approximation to "pure" forms of chords (e.g., A-major chord of 440, 550, 660 Hz) more than it does on the fact that 12 fifths almost bring you back to the same note. Once the approximate size of a semi-tone was established and the ideas of major and minor scales and modes entered into Western music, the 12-fold subdivision seems to have taken on a secondary role until it was reintroduced in the early Baroque period for keyboard instruments. Even then, composers wrote music emphasizing acoustic resolution between intervals in chords--the 12-interval division was mainly a convenience to allow a composer to transpose and wander further away from the "home key" in a piece without it sounding way out of tune. Anyhow, when tonality is taken away, the sense of acoustic resolution to a "tonic" chord is lost. Along with it, the concept of resolution itself becomes somewhat vague--the only intrinsic acoustical property of the 12-tone system is the fact that these notes converge to an octave after going around a big circle of fifths... something that is difficult to exploit to any advantage in trying to create an acoustically-driven resolution. I've often heard my professors talk about the links between early atonal music and the music of the late Romantic period, but to me it seems a rather large departure in classical music to suddenly exploit the 12-tone system after that system had only played a secondary role to the tonalities which were built around approximations to it for at least 1000 years or so. In my opinion, Chopin and Schubert acoustically are as far away from Webern and Schoenberg as they are from 5-tone Eastern music systems. (Of course, we can talk about similarities in rhythm, form, etc.... but I am just referring to pitch.) Well, sorry to get long-winded. It's rare that I get an opportunity to share my thoughts with someone else interested in the fundamental acoustic and mathematical properties of music. Your page attracted me for the discussions of the basis for the 12-tone system, but I think a much more interesting (and much more in depth) research topic would be to discuss acoustically how tonal, modal, and finally chordal structures took over Western music after the frequency breadth of the semitone was established... and then how classical music reverted back to emphasizing the pure 12-tone system in the 20th century. I think I've talked about acoustics enough for one night... :-) John