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Piano Impossible to Tune?

DonH56

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Couldn't find my old program, but it was easy enough to recreate enough for the fundamentals. Disclaimer: I am NOT a music major! I do play a bit, mostly amateur gigs now.

First a bit of reference from Wikipedia:

upload_2016-3-18_10-21-39.png

Now, A4 on a piano (A above middle C) is defined as 440.0 Hz and is a standard tuning pitch. Notes in a Western scale are spaced 2^(n/12) apart going up from the root. To make things interesting, let's look at a C major chord starting at middle C (C4), so the notes are C4, E4, G4 (and C4 if we want the top note). A bit of math provides the pitches (frequencies) of each note using equal temperament:

upload_2016-3-18_10-20-22.png

Yes, since C5 is an octave above C4, I cheated on the last one and just multiplied C4 by two.

Now, we prefer to hear tones spaced by integer ratios or "even" intervals. If you look at ratios and such, eventually you find a base frequency from which all the notes are integer multiples. Being lazy, and already knowing the answer, I am going to do a bit of hand waving. Look at the multiples from C4 for the E4 and G4 notes in our chord:

upload_2016-3-18_10-25-0.png

Not integers, or even numbers that lead to integer ratios. Now the hand waving; rather than find a common factor (frequency), I know 1.25 and 1.5 lead to common multiples with integer ratios (5/4 and 6/4). If I plug those in, the new (prime, "p") frequencies are:

upload_2016-3-18_10-29-30.png

So the desired pitch, that sounds good to us, is an E that's a little low and a G that's a little high. This is "just" intonation. Musicians playing instruments that allow us to adjust pitch will alter those notes in the chord to make them sound prettier. Can't do that with a piano, alas.

Now, for musicians who are not mathematicians (or even a lowly engineer such as myself), what we see on a tuner are "cents". Remember there are twelve notes in a scale, logarithmicly spaced, and we define 100 cents between each note. The equation is:

# cents = 1200 * log2(P1/P2) where log2 is log to base 2 and P1 and P2 are our pitches (notes)​

Plugging in our notes/pitches:

upload_2016-3-18_10-35-57.png

Note log(x)/log(n) = the log of x to base n

If you read the Wikipedia quote above, or look up a major chord in a music theory book, sure enough the third in a major chord needs to be about 14 cents low, and the fifth about 2 cents high, to create a pleasing sound. The theory and math work the same for other chords, though actual numbers will vary, natch.

Ain't it great when the theory and the reality actually meet?

HTH - Don
 

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amirm

amirm

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So the desired pitch, that sounds good to us, is an E that's a little low and a G that's a little high. This is "just" intonation. Musicians playing instruments that allow us to adjust pitch will alter those notes in the chord to make them sound prettier. Can't do that with a piano, alas.
Thanks for the detailed explanation. I have to read it a few times to understand it given my total lack of experience in this area :). But did want to ask about this. Can't a piano be tuned differently just like other instruments you mention? Or do you mean due to lack of difficulty relative to something that has its tuning elements within reach all the time?
 

DonH56

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The problem is that a piano cannot be tuned "on the fly". The E in a C major chord needs to be about 14 cents low, but if the next chord is an E major chord, that same E needs to be dead on pitch. In a major chord the third needs to be 14 cents low, and the fifth a couple of cents high, but the notes in the chord depend upon what chord is being played. Brass players can adjust their embouchure ("lips") and slides to adjust pitch; reeds can adjust embouchure and tension on the reed; strings can adjust finger positions. The piano plays the pitch of the strings the hammer strikes, no easy way to change their length. Note most pianos are tuned so that the (two or three) strings struck by the hammer are tuned a little "off" to help this issue and to provide a "fuller" sound than that of just a single string.

C maj = C, E, G, C -- E (third) needs to be low, G needs to be high
E maj = E, G#, B, E -- E needs to be on pitch, G# low, B high

Amir, is there a way to attach a "wav" file? I created a couple of chord files so you could hear the difference, but VB said it is not an allowed file type. I am able to do this on other fora, just curious.
 
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