Bear with me; this will be a bit long. From a review of the recent book How Equal Temperament Ruined Harmony (and Why You Should Care), by Ross W. Duffin.

Pluck a cello string and get a musical tone. For a second, higher tone the cellist presses the string against the neck of the cello, effectively making the string shorter (since only the part below his pressing finger vibrates). Shorter string, higher tone.

Do the two tones harmonize, or are they dissonant? Pythagoras discovered that the most pleasing sound-intervals come when the original and shortened string lengths are in a simple ratio. For example, a 2:1 ratio (the original string is twice the length of the shortened one) produces tones an octave apart. With a 3:2 ratio, the tones are apart by an interval that musicians call a 'fifth'.

(We now know that the higher tone is a faster vibration: halve the string length and you double its vibration. From here on out it will be simpler to refer to ratios between rates-of-vibration rather than ratios of string-lengths.)

Thus, an octave represents a doubling (100% increase) in the rate of vibration, and a fifth represents a 3:2 increase (50%) in the speed of vibration.

Now look at a piano, starting at the lowest C. Go up a fifth and you reach G; go up another fifth and you reach D (an octave+ above the original C). Continue thus and you go through a cycle (C-G-D-A-E-B-F#-C#-D#-A#-F-C); after a dozen fifths you return to C, seven octaves above the original C. In other words, on the piano 12 fifths = 7 octaves.

The difficulty is that mathematically, 12 fifths does not equal 7 octaves. A fifth is a 50% increase; an octave is a 100% increase – and twelve of the former is very slightly more than seven of the latter (the 531441/524288 more, if you care). That gap is called the Pythagorean comma.

So if you want octaves to be precisely an octave apart, and thus harmonize properly, you have to fudge the intervals between the intermediate notes.

The question becomes, "Which interval(s) shall I fudge, and how much? Which notes should be slightly 'out of tune' from their perfect-sounding values?" The choice made is called the temperament. "It means redefining musical intervals so as to avoid the comma problem, smoothing out its harshness by distributing that unruly remainder somehow throughout the scale."

That's the end of the vocabulary aspects of this post, but let me finish with the musical ones. The author's point is that in the past many temperaments were used – all different from that which prevails today. As a result, "Most of us have never heard Bach's music as he himself heard it."

Today's temperament spreads the comma evenly between all pairs of adjacent notes (that is, their frequencies always differ by a ratio of the 12th root of 2). As a result, a piece will sound essentially the same when it is transposed into another key. But past musicians used tunings where adjacent notes did not differ equally: some differed more, and some less. As a result, a piece would sound different in a different key. Now you see why compositions would have such titles as "Fantasy in D Minor": the choice of key mattered. "Each musical key has a distinct, individual character because of its particular distribution of commas."

Very interesting. Whenever my composer friend, Erling, tries to explain the intricacies of just intonation and equal temperament, my head begins to ache. This use of comma made me revisit the words meaning in Greek κομμα komma: 'stamp or impression of a coin', 'coinage', 'that which is cut off, piece' (reflecting its origin from the verb κοπτω koptō 'to strike, smite, knock down; cut off'), 'refuse of corn (in threshing), chaff', 'short clause (of a sentence)', and 'contusion'.

quote:

As a result, "Most of us have never heard Bach's music as he himself heard it."

Well, some of it, anyway. I think The Well-Tempered Clavier was composed for equal-temperament tuning.

Two more extracts from the book review. They have nothing to do with words, but may interest those more familiar than I with the musical subject.

Mr. Duffin's Web site, one discovers, gives Bach chorales and fugues electronically synthesized in different temperaments. [Note: I believe that site is here, but haven't found the audio on it.]

He mentions in passing the scholar Bradley Lehman's recent discovery that Bach encoded his own favored temerament in the apparently ornamental doodles and knoted squiggles he put on the title page [of a piece]. [Note: you can find info by googling up Bradley Lehman; it's beyond my understanding.]