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.
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?
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. >