Musical Offerings, Fall 2018

Musical Offerings ⦁ 2018 ⦁ Volume 9 ⦁ Number 2 63 are also convenient because of mathematical logarithm rules. For instance, what is the distance between A4 and E5? We must multiply frequency ratios: A4 to A5 to E5. 2/1 x 3/2 = 3/1. However, cent values can be added together. Going from A4 to A5 to E5, add an octave and a fifth. 1200 + 702 = 1902. Thus, the ability to add cent values is quite helpful. The greater the number of cents, the greater the distance between two notes. With this understanding, a problem arises. If a person begins on the lowest C of the piano (C1) and ascends 7 perfect octaves, he/she will land on C8. Similarly, that person could also begin on C1 and ascend 12 perfect fifths, landing on C8. However, when doing this process mathematically, the resulting frequencies are different. As shown in Figure 1, an octave is equivalent to 1200 cents. Seven octaves then correspond to a frequency change of 7 × 1200 = 8400 cents. However, marching up twelve perfect fifths, there is a corresponding frequency change of 12 × 702 = 8424 cents. Thus, there is a 24-cent difference between this chain of octaves and fifths. The 24-cent difference can also be described mathematically by the frequency ratio 3 12 /2 19 . This difference has been known since the time of Pythagoras in 550 B.C . 5 Because of this 24-cent difference, known as the Pythagorean comma, the musical “circle” cannot be completed. Note that a 24-cent difference corresponds to approximately 1/3 the difference of a semitone. While it may not seem like much, this difference causes a terrible tuning issue. As Stuart Isacoff put it, “In order for the twelve pitches generated through the proportion 3:2 to complete a path from ‘do’ to ‘do,’ the circle has somehow to be adjusted or ‘rounded off.’ ” 6 The problem of the Pythagorean comma is solved using some type of systematic adjustment. The adjustments can be divided into two general categories: tuning and temperament. According to J. Murray Barbour, a tuning system is one “in which all intervals may be expressed as the ratio of two integers. ” 7 Conversely, Barbour says that “a temperament is a modification of tuning which needs radical numbers to express the ratios of some or all of its intervals.” Radical numbers are those such as √2, π, or 5 1/4 . They cannot be written as the ratio of two integers. Pythagorean intonation and 5 Fauvel, Flood, and Wilson, Music and Mathematics , 18. 6 Stuart Isacoff, Temperament: How Music Became a Battleground for the Great Minds of Western Civilization (New York: Vintage, 2003), 65. 7 J. Murray Barbour, Tuning and Temperament: A Historical Survey (East Lansing: Michigan State College Press, 1953), 5.

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