Musicians wrestle with equal-interval measurement. The twelve tones in the chromatic scale are shown in column 1 of the Table. "Just intonation" defines the consonance we hear when the two notes sounded together bear a simple numeric ratio to each other. In column 2 of the Table are the well-known ratios from Pythagoras. They are all derived from the ratios of the products of the prime numbers 2, 3, and 5. Unison has a frequency ratio of 1:1 (C:C, i.e., middle C on the piano to middle C) and the octave is 1:2 (C:c, i.e., middle C to the C eight white keys to the right). All tuning systems accept these fundamental ratios. Next comes the perfect fifth (C-G) with the ratio 2:3 and the perfect fourth (C-F) with the ratio 3:4.
Tone | Concordant Frequency Ratio | Exact Ratios |
c | 8 = 1:2 | 8.00 |
B | 7.55 | |
A# | 7.13 | |
A | 6.73 | |
G# | 6.35 | |
G | 6 = 2:3 | 5.99 |
F# | 5.66 | |
F | 5.33 = 3:4 | 5.34 |
E | 5 | 5.04 |
D# | 4.76 | |
D | 4.50 = 8:9 | 4.49 |
C# | 4.24 | |
C | 4 = 1:1 | 4.00 |
Column 3 shows the steps of the chromatic scale indicated by equal frequency ratio divisions using the multiplicative constant of (2)1/12 = 1.0595. This table shows how the Pythagorean system, derived from studies of the monochord, corresponds to the equal interval system. This also shows why "music" was part of the quantitative "quadrivium", the four liberal arts required in medieval times to advance from B.A. to M.A., the other three being arithmetic, geometry and astronomy. The "trivial" linguistic arts, the "trivium", were grammar, rhetoric and logic.
A natural major third is 5:4 and consonant. In exact ratio tuning, E with C, although a major third, is 5.04 to 4 and very discordant, . An exact cycle of four pure fifths, C-G-d-a-e' produces a major third out of tune when compared to a true major third of e' to c' of 5:4. This discord was known as the "comma of Didymus". Historians of musical theory consider this discord the reason why early medieval music extolled the tone, fourth, fifth, and seventh as "concordant" and treated the major third as "discordant", a wolf chord.
Until about the middle of the eighteenth century, ninety-five per cent of all pipe organs were tuned in mean-tone temperament. Finn Viderø has made several recordings on just such an organ built in 1616 in the castle at Frederiksborg, Denmark. I have heard the organ in recital there, and it produces a truly beautiful sound.
The commonest mean-tone system uses eight major thirds (C-E, E-G#, Bb-D, D-F#, E-G#, Bb-D, D-F#, F-A, A-C#, Eb-G, G-B). These are exactly in tune and many common chords can be produced. Musicians have long thought these intervals were more pleasant to hear than those in equal temperament where no major third, or any other interval, is in natural tune save the octave. However, mean-tone temperament makes only about a dozen keys available, and the rest don't sound well.
In Scales: Music and Measurement (RMT 15:3, p. 838) four interesting comparisons were made between musical tunings and measurement theory:
1. "… mathematical perfections were claimed for 'Pythagorean' tunings, as they are now for some IRT models."
"Just intonation" is both mathematically precise and musically satisfying, but only in certain keys. It does not have objectivity or generality, i.e. it cannot survive a transposition to other keys. The "equal temperament" scheme overcame these drawbacks.
2. "… special practical virtues were perceived in 'just meantone' tunings, as they are now in raw-score-weighting schemes."
Certainly there are unique advantages for specific keys, but no generality is possible. Without generality, our music is restricted.
3. "Pythagorean tuning was simple, and musically effective. Its limitation was that only 11 of the 12 notes of an octave could be in tune simultaneously. Yet it was so easy and familiar, just as raw scores are today, … ."
We usually begin with the system first discovered and advance. Oppenheimer (1955) said, "all sciences, arise as refinements, corrections, and adaptations of common sense. … these are traits that any science must have before it pretends to be one. One is the quest for objectivity. I mean not in a metaphysical sense; but in a very practical sense … ." (p. 128)
4. "… in a remarkable parallel to the current proliferation of psychometric models, 'the history of tuning is saturated with clever and original theories that have no practical application.'"
A host of alternate tunings have been proposed, and many have a long historical lineage, but the major question has always remained, "What generality exists?" Without generality there can be no application.
Mark H. Stone
Oppenheimer, R. (1955). Analogy in science. Presented at the 63rd Annual Meeting of the American Psychological Association, San Francisco, CA, September 4, 1955.
Musical temperament. Stone, MH. … 16:2 p.873
Musical temperament. Stone, MH. … Rasch Measurement Transactions, 2002, 16:2 p.873
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