Thursday, December 7, 2006

thought experiments!
There is a problem that constantly comes up in tuning that has always struck me as interesting. This problem is the comma.

Take the note A4. Normally tuned, A4 has a frequency of 440 hz - that is to say, there are 440 "beats" of the soundwave through the air in the space of a second. The major chord A based off of this tone will be A4, C#5, E5. For now let's look at the C#. The single interval of a major third, in this case A4-C#5, tunes most naturally into the frequency ratio 5:4 - that is to say, for every 5 "beats" that the higher tone (the C#) makes in the air every second, the lower tone (the A) will only make 4. Tuned in this way, then, one calculates the frequency of the tone C#5 by taking the A4 (440 hz) and multiplying by 5/4. This puts the C#5 at 550 hz.

But there are other ways of tuning it. Also in the A major chord is the note E5. E5 here is tuned as an interval of a perfect fifth up from A4. A perfect fifth has the frequency ratio 3:2 (three beats for the E5 for every two of the A4), putting the E5 at 660 hz. However, we don't need to stop here - we can also build notes off of the E! The note a perfect fifth over E5, B5, will have the frequency 990 hz. For purposes of simplicity, let's drop this note down an octave - to B4. This is done simply by halving the frequency (990 * 1/2), resulting in B4 having a tone at 495 hz. B4 too has a note a fifth above it, which is F#5 (495 * 3/2). The note a fifth above F#5, at 742.5 hz, is C#6, which we can drop an octave to C#5. We already calculated the C#5 above in the previous paragraph, but here we shall do so again. It will be: 742.5 * 3/2 * 1/2. The frequency that comes out is... 556.875 hz.

What just happened? Above we calculated C#5 at 550 hz, and here it jumps up to nearly 557. Earlier we had calculated it as a major third, 5:4, directly from A4; here we arrive at it from four perfect fifths dropped two octaves ((3/2)^4 * (1/2)^2), which turns out to be a ratio of 81:64 above A4. These two different kinds of C#5 have a ratio to one another of 81:80 (the technical name for this ratio is the "syntonic comma"), a difference which is normally imperceptible to the human ear (we just hear them as the same tome). However, play the 556.875 C# as a major third against A, and it will sound somewhat uneven and unsteady where the 550 C# sounded pure and consonant. Similarly, play 550 as a perfect fifth above F# and it will sound ugly and jarring, while the 556.875 again sounds pure and consonant. Looking at a system of just four tones (the full traditional scale is 12), we have found basic foundational problems of incompatibility. Some tones just do not work with others in an acceptable way. The C# in this situation has to either pick one frequency and be content with some intervals just sounding bad with it present; or it can be split so that there are always two different C#5s, never the one meeting the other; or the note can be "tempered" to somewhere in-between these two extremes as a compromise (in modern temperament, the C#5 is tuned to 554.36526... hz). None of these are perfect solutions, because by the basic fact of the numbers themselves the perfect solution is impossible.

It occurs to me that there may be similar kinds of commas in philosophy. In Kant's first Critique there's the section on the Antinomy of Pure Reason, which I still don't understand in anything like an adequate way. If the systematic nature of metaphysics necessarily gives rise to commas, then a new task in philosophy may be: to point out these commas in a clear way so that a more well-informed kind of metaphysical "tuning" can take place.
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