ccastdf@prism.gatech.EDU (Dave) (12/13/90)
I am posting this for a friend who doesn't have post access yet. Please reply to him. --------cut here----- Will someone help me? I've been looking for the frequencies for standard notes for the purpose of programing sound. I realize that it's probably in EVERY reference manual that I own, but I'm not where I can get to my library. I know it's a pain to type it, but I would greatly appreciate it. Thanks SarJe gt4529b@prism.gatech.edu
jbovitz@ub.d.umn.edu (Jeffrey Bovitz) (12/13/90)
A bit ago, someone (from Georgia Tech) wanted a list of the frequencies
for musical notes. My brother has a solution. Chris...?
Thanx Jeff. The formula is based upon a Middle "A" of 440 Hz.
The formula goes:
440 * ((2)^(1/12)) ^ h
(i.e. 440 Hz * (the twelfth root of 2) raised to the h power)
where h is the number of half-steps above or below Middle A.
For example, Middle C is 8 half-steps below Middle A, so h=-8, and
the freq of Middle C is 261.6 Hz. Remember that there are 12 half-
steps in an octave, and an octave is defined as doubling of the
frequency. BTW, this formula works great on a spreasheet.
Hope this helps.
Chris Bovitz moonunit@meteor.wisc.edutpermutt@eng.umd.edu (Thomas Permutt) (12/14/90)
In article <561@ub.d.umn.edu> jbovitz@ub.d.umn.edu (Jeffrey Bovitz) writes: >A bit ago, someone (from Georgia Tech) wanted a list of the frequencies >for musical notes. My brother has a solution. Chris...? > >Thanx Jeff. The formula is based upon a Middle "A" of 440 Hz. >The formula goes: > > > 440 * ((2)^(1/12)) ^ h > > (i.e. 440 Hz * (the twelfth root of 2) raised to the h power) > where h is the number of half-steps above or below Middle A. > >For example, Middle C is 8 half-steps below Middle A, so h=-8, and >the freq of Middle C is 261.6 Hz. Remember that there are 12 half- >steps in an octave, and an octave is defined as doubling of the >frequency. BTW, this formula works great on a spreasheet. > >Hope this helps. > >Chris Bovitz moonunit@meteor.wisc.edu This is quite correct, for the "standard" equal-tempered scale. But that represents a compromise arising largely from the fact that pipe organs, for example, take a long time to tune, and need to be played in lots of different keys. If you are going to play in, say, C major, why not try C = 261.6 G = C * 3/2 F = C * 4/3 E = C * 5/4 D = C * 9/8 B = G * 5/4 I think you will hear a much nicer sound. And, since your question suggests that the frequencies are under software control, you can always reprogram for different keys.
a577@mindlink.UUCP (Curt Sampson) (12/20/90)
> mercuri@grad1.cis.upenn.edu writes: > > I'm sure that at least a half dozen people have informed you that although > your formula is "correct", it will sound like hell. Most instruments are > "stretch tuned" (higher in the high octaves, lower in the low octaves) to > accommodate for the nonlinearity in the hearing process. String players do > this "automatically" as they are playing. Actually, from what I have heard this nonlinearity is not in the hearing process but in the string instrument itself. I've been told that as the string gets shorter the harmonics get flatter, thus necessitating a "sharpening" of the string so that it sounds in tune. I've never heard this applied to anything but piano, though it must apply to all other *string* instruments as well. I don't believe that it applies to non-string instruments, especially synthesisers, which would always have their harmonics in tune with the fundemental (assuming they are programmed that way). cjs -- Curt_Sampson@mindlink.UUCP {uunet|ubc-cs}!van-bc!cynic!curt curt@cynic.wimsey.bc.ca
mercuri@grad1.cis.upenn.edu (Rebecca Mercuri) (12/21/90)
jbovitz --- I'm sure that at least a half dozen people have informed you that although your formula is "correct", it will sound like hell. Most instruments are "stretch tuned" (higher in the high octaves, lower in the low octaves) to accommodate for the nonlinearity in the hearing process. String players do this "automatically" as they are playing. If you need more info (references on this), get back to me. R. Mercuri
sandell@ils.nwu.edu (Greg Sandell) (12/22/90)
In article <35111@netnews.upenn.edu>, mercuri@grad1.cis.upenn.edu (Rebecca Mercuri) writes: > jbovitz --- > > I'm sure that at least a half dozen people have informed you that although > your formula is "correct", it will sound like hell. Most instruments are > "stretch tuned" (higher in the high octaves, lower in the low octaves) to > accommodate for the nonlinearity in the hearing process. String players do > this "automatically" as they are playing. If you need more info (references > on this), get back to me. > > R. Mercuri Rebecca, I think you are criticizing too quickly. Whether the notes will sound "like hell" or not depends on what timbre is used to instantiate them. If you did your best to create a piano-like timbre while using this formula for your fundamental frequencies, yes, your listeners will note the attempt at imitation of a piano sound but find the pitches to sound flat in the high range...but mainly for associational reasons (i.e. when you hear a piano timbre, expect it to be accompanied by stretch tuning). If the sounds involved were merely sine tones, or an impoverished harmonic sound (like a square wave), I would think that the formulaic tuning would sound appropriate. Sound right to you? - Greg **************************************************************** * Greg Sandell (sandell@ils.nwu.edu) Evanston, IL USA * * Institute for the Learning Sciences, Northwestern University * ****************************************************************
sandell@ils.nwu.edu (Greg Sandell) (12/22/90)
In article <4193@mindlink.UUCP>, a577@mindlink.UUCP (Curt Sampson) writes: > > mercuri@grad1.cis.upenn.edu writes: > > > > I'm sure that at least a half dozen people have informed you that although > > your formula is "correct", it will sound like hell. Most instruments are > > "stretch tuned" (higher in the high octaves, lower in the low octaves) to > > accommodate for the nonlinearity in the hearing process. String players do > > this "automatically" as they are playing. > > Actually, from what I have heard this nonlinearity is not in the hearing > process but in the string instrument itself. I've been told that as the string > gets shorter the harmonics get flatter, thus necessitating a "sharpening" of > the string so that it sounds in tune. I've never heard this applied to > -- > Curt_Sampson@mindlink.UUCP Yes, the phenomenon of stretch-tuning is covered by Benade in his HORNS, STRINGS & HARMONY quite nicely. Here's what I remember: not any old string stretched taught creates harmonic sounds (i.e. harmonics having frequencies in integer ratios to one another); the string has to have specific physical properties. The most important properties are thickness, evenness of thickness across the length, and stiffness. Piano strings have the problem of being too stiff, and the result is flat harmonics in portions of the spectrum which are important for determining the pitch of the tone (according to Ritsma, the 3rd through fifth harmonics when the fundamental is between 100-400Hz; yes Virginia, pitch perception is not just a matter of the frequency of the fundamental). As Curt says, those flat harmonics make the pitch sound flat, so piano tuners adjust the tuning to make those strings sharper. They are sharp in fundamental frequency (which is an objective phenomenon) but *not* in pitch, which is a psychoacoustic phenomenon. A related matter: guitars with old, worn-out strings sound perpetually out of tune, no matter how hard you try tuning them. Reason: the thickness of the string is not even across its length anymore, so the harmonics have gone sour. - Greg **************************************************************** * Greg Sandell (sandell@ils.nwu.edu) Evanston, IL USA * * Institute for the Learning Sciences, Northwestern University * ****************************************************************
maverick@fir.Berkeley.EDU (Vance Maverick) (12/22/90)
In article <505@anaxagoras.ils.nwu.edu>, sandell@ils.nwu.edu (Greg Sandell) writes: > If the sounds involved were > merely sine tones, or an impoverished harmonic sound (like a square > wave), I would think that the formulaic tuning would sound appropriate. My experience with purely harmonic sounds (say a wavetable instrument in the NeXT MusicKit) suggests that those are the signals for which we are most sensitive to "just" intonation. That is, equal temperament sounds roughest when the harmonics in single tones are clean enough that the proverbial beating between them is obtrusive; for me, then, harmonic sounds lead away from a fixed tuning altogether.
mpogue@vis01.dg.com (Mike Pogue) (12/22/90)
Could you say a few more words about "stretch tuning", and why it sounds better than exact tuning? Do most synthesizers take this tuning into account? -- Mike Pogue Data General Corp. Speaking for myself, not my company.... Westboro, MA.
nsv+@andrew.cmu.edu (Ned S. VanderVen) (12/22/90)
As several recent posts have noted, the stiffness of real strings results in inharmonicity of the series of normal mode frequencies (characteristic frequencies of vibration), and to compensate for this the octaves must be "stretched." But all the explanations posted so far have described the phenomenon incorrectly. First of all, the stiffness results in a series of normal mode frequencies that are sharp, not flat, compared with the harmonic series, and these departures from the harmonic series become progressively sharper for the higher modes of the string. Secondly, a note is not "stretched" in order that its perceived pitch sound "correct" when played alone. Octaves are stretch-tuned in order that the two notes will be relatively free of beats when sounded together. To illustrate how this works, suppose that because of inharmonicity a string of fundamental frequency f has a second-mode frequency of 2.02 f rather than the harmonic value of exactly 2 f. The string sounding an octave higher should be tuned so that its fundamental is equal to the second mode frequency of the lower note in order that no beats be heard when the notes are sounded together. In this example the octave would have to be tuned to a fundamental frequency of 2.02 f to reduce the beat frequency to zero. This sharpening of the octave is the phenomenon of "stretching." The situation is actually more complicated than this because the fourth mode of the lower string will beat with the second mode of the upper string, and no tuning will eliminate all the beats. In practice the tuner will adjust for the most consonant interval, and stretching inevitably results. The inharmonicity depends on several factors: the diameter, length, tension, and elastic modulus of the string. It increases with increasing diameter and elastic modulus, but decreases with increasing length and tension. In pianos the inharmonicity is least in the middle of the keyboard, and that is where the octaves are stretched least. On concert grands the bass strings are long enough that the inharmonicity is not much greater than in the middle of the keyboard, but on a small upright, whose bass strings are too short, the inharmonicity can be severe, and the lower octaves are widely stretched. This, incidentally, provides a test of whether the stretching is due to inharmonicities in the strings or to non-linearities in the hearing mechanism. If it were due primarily to the hearing mechanism we would expect the stretching on all pianos to be about the same. In practice the stretching varies with the instrument, and experiments by Backus have shown that string inharmonicity accounts for nearly all of it.