briang@bari.Sun.COM (Brian Gordon) (12/21/89)
A couple of weeks ago, there was a comment about the difference in "feel" between something played in, say, the key of C (on a piano) and the same thing transposed, say, to C#. I responded with a comment that this was the wonderful world of the equal tempered scale, or "deliberate mistuning". I got a couple of pieces of e-mail asking for more details. After all, if the piano is "equally out of tune in all keys", the intervals should be equally good/bad in all keys, and could not enter into perceived differences. "No sweat", thinks I, "I'll look that up and post a nice scholarly explanation." Well, I haven't found it, and have come up with a couple of experiments that might help. Observation: A simple tune, which sounds comfortable on a well tuned piano when played in the key of C, will sound "brighter" when played on the same piano when transposed and played in the key of C#. That could be because there are relative differences (the intervals of a piano's C# scale are different from those of a C scale), or absolute differences (we react differencly to the higher frequencies than to the lower ones). Experiment: Given recording equipment that can be sped up/slowed down on playback, record the C# version. Then, on playback, slow it down enough to be heard as the C version. Does it "sound like" the C version, or does it still sound "brighter", just lower? Anyone with the gear have the interest? Or, even better, is this a well known experiment with well know results that can be read? Notice that piano tuners already cheat by, for example, "stretching the octaves". The further you get from middle C, the greater a ratio it takes to make an octave "sound right" -- the theoretical 2/1 doesn't work. A high octave which is perfectly tuned by freguency (e.g. makes an electronic tuner happy) sounds flat until the upper note is tweaked up a bit. I seem to recall stories that, in its heyday, Toscanini's NBC Symphony would always record things slightly too fast and pitcedh up 1/2, so that, when slowed down to get into the right key (and speed), they would still be "extra bright". If that is true (and it worked) that would support version #1 above. If version #1 is correct, the question then is, how do the differences get there? Is it designed into the tempered scale, or is it a distortion, like stretched octaves, introduced by technicians. If the latter, how and where? Any piano tuners and/or theoreticians want to hazzard a guess? +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ | Brian G. Gordon briang@Corp.Sun.COM (if you trust exotic mailers) | | ...!sun!briangordon (if you route it yourself) | +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
rbp@well.UUCP (Bob Pasker) (12/21/89)
There's a problem with talking about C vs. C# because C# is "naturally" a
minor key, i.e. C#-minor is the relative minor of the key A-major and
A-major is a minor sixth above the C-major. This means that they are
relatively distant from each other. We'll see what that means in a minute.
The method of well-tempered tuning gives keys which are "distant" to the
tuned-key (usually C) "better" sound. Originally, keyboards (harpsichords,
fortepianos and pianos) that were not well-tempered only sounded "good" in a
few keys, those "close" to the key which they were _usually_ tuned for:
C-major.
In order of distance to the key of C-major we have:
/ sharps: C, G, D, A, E, B, F#/Gb \
closest -< > - furthest
\ flats: F, Bb, Eb, Ab, Db, Gb, Cb /
So, on non-well-tempered keyboards, you could probably play in the keys of:
C, G, F, Bb and sometimes D and Eb without too much dissonance.
The minor keys also sound fine if their relative major key was close to the
tuning:
Major Relative Minor
C A-minor
G E-minor
D B-minor
F D-minor
Bb G-minor
Eb C-minor
With well-tempered tuning, a harpsichordist coud play in distant keys, like
F#, and Cb without "too much" dissonance.
The dissonance in these distant keys is introduced because of the
requirement in western tuning that the dominant (or the "fifth") be 2/3rds
the frequency of the tonic (i.e. A must be 2/3rds the frequency of C) and
the octave above must be 1/2 the frequency (C' must be 1/2 the frequency of
C). It is easy to see how the effects of the mathematics of this are
multiplied (pun intended!) when the tuning gets further away from the key
which the piano is tuned for, since each successive key is a fifth away from
the previous one.
For example:
(note: the "prime" symbol (') indicates the number of octaves above middle
C. D' is the D note in the first octave above the octive containing middle
C.)
Suppose C is of frequency x
then G is of frequency 2x/3 since it's a fifth above C.
and D' is of frequency 4x/9 similarly
A' is of frequency 8x/27 similarly
E'' is of frequency 16x/81 similarly
B'' is of frequency 32x/243 similarly
F''' is of frequency 64x/729 similarly
C'''' is of frequency 128x/2187 similarly
therefore
C''' must be of frequency 256x/2187, twice C'''' (128x/2187)
and C'' must be of frequency 512x/2187
and C' must be of frequency 1024x/2187
and C must be of frequency 2048x/2187
but yet we know that C is x, not 2048x/2187 (.93644262x)
So, if C is x, C'''' should be 4x on an instrument tuned in the key of C not
some approximation. By adjusting the frequency of the notes so they are not
exactly what the harmonic progression wants, but close enough, you get a
keyboard that is out of tune in all keys but sounds "good" in all of them.
Hope this clears things up.
--
- bob
;-----------------------------------------------------------------
; Bob Pasker | rbp@well.sf.ca.us
; San Francisco, CA | +1 415-695-8741djones@megatest.UUCP (Dave Jones) (12/21/89)
From article <15132@well.UUCP>, by rbp@well.UUCP (Bob Pasker): > > The dissonance in these distant keys is introduced because of the > requirement in western tuning that the dominant (or the "fifth") be 2/3rds > the frequency of the tonic (i.e. A must be 2/3rds the frequency of C) and > the octave above must be 1/2 the frequency (C' must be 1/2 the frequency of > C). Bob, Bob, Bob. In Western, Eastern, and every other kind of music, the dominant is 3/2 the frequency of the tonic, not 2/3. "A" is not the dominant of "C", nor is "C" the dominant of "A". And C' is twice the frequency of of C, not half. [Taking a deep breath...] The slight dissonance of well-tempered chords is not due to the approximation of the fifth, which is virtually perfect: 1.4983 as opposed to 3/2. It is mostly due to the bad approximation of the the major third, 1.2599 -- it should be 5/4 -- and of the minor third, 1.1892 -- it should be either 7/6 or 6/5, depending on its function. Other intervals are also off somewhat, but those are the ones that matter. In a major-7 chord, for example, you mostly hear the seven in relation to the three and the five, not to the root. The seven, having a ratio of 15/8 of the root, combines with the root to produce only "buzz" and subsonic undertones. It combines with the third and the fifth to produce resonance.
dts@quad.uucp (David T. Sandberg) (12/21/89)
In article <15132@well.UUCP> rbp@well.UUCP (Bob Pasker) writes: >There's a problem with talking about C vs. C# because C# is "naturally" a >minor key, i.e. C#-minor is the relative minor of the key A-major and >A-major is a minor sixth above the C-major. Just a second... did I suddenly lose my mind? I thought that F#-minor was the relative minor of A-major (C#-minor includes a D#), and also that A-major is a _major_ sixth above the C-major. In any case, the rest of the article was quite thought-provoking. -- David Sandberg dts@quad.uucp or ..uunet!rosevax!sialis!quad!dts "What's the difference between 12-bit and 16-bit? A lot more than 4 bits!"
scott@bbxsda.UUCP (Scott Amspoker) (12/22/89)
In article <15132@well.UUCP> rbp@well.UUCP (Bob Pasker) writes: >[good explanation of equal tuning deleted] Another way to appreciate it is to get an electronic instrument that supports alternate tunings (such as a Kurzweil PX1000 or E-mu Proteus). You don't have to move far from the root key to hear seriously offensive notes. -- Scott Amspoker Basis International, Albuquerque, NM (505) 345-5232 unmvax.cs.unm.edu!bbx!bbxsda!scott
michael@xanadu.com (Michael McClary) (12/22/89)
In article <129476@sun.Eng.Sun.COM> briang@bari.Sun.COM (Brian Gordon) writes: >A couple of weeks ago, there was a comment about the difference in "feel" >between something played in, say, the key of C (on a piano) and the same >thing transposed, say, to C#. I responded with a comment that this was the >wonderful world of the equal tempered scale, or "deliberate mistuning". >[] >If version #1 is correct, the question then is, how do the differences get >there? Is it designed into the tempered scale, or is it a distortion, like >stretched octaves, introduced by technicians. If the latter, how and where? > >Any piano tuners and/or theoreticians want to hazzard a guess? My brother the music major pointed out that he had a HELL of a time tuning pianos, because he kept drifting from equal-interval toward perfect-fifth (which sounds better when you're only playing two notes). Perhaps the paino you were playing on wasn't really well-tempered?
symon@lhotse.cs.unc.edu (James Symon) (12/23/89)
In article <11435@goofy.megatest.UUCP>, djones@megatest.UUCP (Dave Jones) writes: > . . . > In Western, Eastern, and every other kind of music, the dominant is 3/2 > the frequency of the tonic, not 2/3. "A" is not the dominant of "C", nor is > "C" the dominant of "A". And C' is twice the frequency of of C, not half. > Thank you Dave Jones and others for pointing out what a mess that original article was before some poor soul took it seriously. Of course, we all know what it's like to fire off a posting only to read it later and groan in embarassment over the careless errors. However, beyond the carelessness, assuming tempered tuning and that we didn't play the transposed version right after the original, what difference does it make how far in the circle of fifths one is from the other? I leave open the question of latent perfect pitch detecting such small differences. Certainly composers have felt differently about different keys. There is one aspect of this phenomenon (toons transposed taste tangy) that I haven't seen mentioned. I may have missed it. Even if you are given that your piano is well-tuned to be properly well-tempered (so no mathematical difference between the keys of C and C#), and even if it seems doubtful that most of us have enough latent perfect pitch to react differently to C and C# versions played far apart in time (I'm not sure about this), it may be that on a typical piano some keys get used much more than others. ("I don't want to try that piece, too many sharps.") This may lead to a difference in wear in the key actions and the way they strike the strings. Even more noticeable would be differential aging of the strings leading to a difference in the overtone structure. I can imagine several ways that differences in use could cause differences in the strings. Jim Symon | symon@cs.unc.edu Computer Science Dept | {uunet, decvax}!mcnc!unc!symon Chapel Hill, NC 27599-3175 | (919) w:962-1893 h:968-1024 ***Don't use "r" or my header line address***
djones@megatest.UUCP (Dave Jones) (12/23/89)
From article <406@quad.uucp>, by dts@quad.uucp (David T. Sandberg): > In article <15132@well.UUCP> rbp@well.UUCP (Bob Pasker) writes: >>There's a problem with talking about C vs. C# because C# is "naturally" a >>minor key, i.e. C#-minor is the relative minor of the key A-major and >>A-major is a minor sixth above the C-major. > > Just a second... did I suddenly lose my mind? Perhaps. Don't worry about it. Could happen to anybody. (Never happened to me, personally, though.) > I thought that > F#-minor was the relative minor of A-major Correct. > (C#-minor includes a D#), and also that A-major is a _major_ > sixth above the C-major. > Right again. Your mind seems to be functioning quite normally. Of course, C# is the relative minor of E, not A. There may be something in that article that was correct, but if so, I didn't find it. > In any case, the rest of the article was quite thought-provoking. I'll say!
djones@megatest.UUCP (Dave Jones) (12/23/89)
From article <11298@thorin.cs.unc.edu>, by symon@lhotse.cs.unc.edu (James Symon): > > ... it seems doubtful that most of us have enough latent perfect pitch to > react differently to C and C# versions played far apart in time (I'm > not sure about this) ... Most of us probably would not notice a difference. But my previous roommate could have. His sense of pitch was eerie. One day he burst through the front door, still reeling from a self-induced endorphin high. (Racketball junkie.) Startled from a studious contemplation, I dinged a drinking glass against a bottle. It rang with a bell-like clarity. In his pitiful euphoria, he called out, "A-flat!" Realizing that a small embarassment might save him a larger one on some later occasion, I crawled confidently to the piano and struck an A-flat. "Hah!" I said. "The glass is fully four cents sharp."
djones@megatest.UUCP (Dave Jones) (12/23/89)
Just in case we haven't thrashed this poor unfortunate posting enough, notice that F# and Cb, (A.K.A. "B"), are adjacent in the cycle of fifths, not "distant". From article <15132@well.UUCP>, by rbp@well.UUCP (Bob Pasker): > ... > With well-tempered tuning, a harpsichordist coud play in distant keys, like > F#, and Cb without "too much" dissonance. >