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Sonic Foot

September 17th, 2009
music  [html]
At an open house our new neighbors held last week, there was a large organ pipe in the front hall. This reminded me of previous musing on why the note 'C', which we think of as pretty central, happens to come out to be almost exactly one foot long. As there were several people at this open house who knew a good bit about music, I started trying to see if any of them knew how this came to be. They brought up three good points in the direction of coincidince:
  1. While we think of 'C' as central, to many instruments it is not relevant at all. Brass instruments like flat keys and string instruments like sharp keys. Some note is going to be close to 12 inches, so it could easily just be coincidence that it is 'C'.
  2. Pitch has changed over the years, generally getting sharper. So if with 'A' at 440Hz 'C' is close to (power of 2 multiples of) 1 foot, then with 'A' at, say, 415Hz, 'C' would have come from a tube almost 13 inches long. There are examples of 'A' as high as 500Hz and as low as 390Hz.
  3. Were people even working in feet back then?
So how is 'C' central? One important way is that in [just intonation] (instead of modern [equal temperment]), 'C' was often the key where the ratios were perfect. That is, in the key of 'C' you would have the fifth be 3/2 times the length of the tonic and the fourth would be 4/3 times. This works out nicely if your 'C' is 12 inches, as then the fifth, 'G' is 18 inches and the fourth is 16 inches. You can, of course, do this in any key, but if you do it in C then they all come out in inches.

Looking for historical detail, I find:

The first mention of temperament is found in 1496 in the treatise Practica musica by the Italian theorist Franchino Gafori, who stated that organists flatten fifths by a small, indefinite amount. This practice was formalised in what is called the mesotonic or meantone (also written mean-tone) scale. It was always particularly favoured by organists and explains why organ music from the period the early sixteenth to the nineteenth century was written in a relatively small number of keys, those that this scale favoured. Arnolt Schlick's Spiegel der Orgelmacher und Organisten (1511) described both the practice of and formulae for mean-tone tuning which makes it clear that it was already in use. Pietro Aron produced a more thorough analysis in Toscanello in Musica (1523), which sufficed for all practical purposes. The earliest complete description was published by Francisco de Salinas in De Musica libri septem (1577).

Based on C, the method relied on using the first five notes from the circle of fifths from C, namely C, G, D, A, E and setting a pure third between C-E by narrowing the fifths by a small amount - from a ratio of (3:2) to a ratio of (2.99:2). D, the note between C and E was set so that the ratio between D and C was identical to that between E and D, so placing D in the mean position between C and E, hence the scale's name. What happened after this to complete the chromatic scale introduced a number of variants which only the more studious of our readers are likely to pursue. Suffice it to point out that the results generally work well in the keys C, G, D, F and B flat but outside these serious problems arise and composers writing for this system avoided keys more distant from C. -- www.dolmetsch.com/musictheory27.htm#mean

The same source claims a few paragraphs later that "in England, it was not until 1842 that the first organ, that of St. Nicholas in Newcastle-upon-Tyne, was tuned to equal temperament."

I wouldn't say this is resolved yet, but I do think that our choice for the pitch 'C' is probably because that pitch is what you get from a 12" (or 2', 4', 8', 16', ...) pipe.

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