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The Remarkable Mathematical Coincidence of the Tempered Scale

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The Remarkable Mathematical Coincidence of the Tempered Scale The Well Tempered Pythagorean: The Remarkable Relation Between Western and Natural Harmonic Music Robert J. Marks II University of Washington CIA Lab Department of Electrical Engineering Seattle, WA [email protected] Abstract Introduction True natural harmony, popularized by sixth Newtonian physics applied to the vibrating century BC Greek philosopher Pythagoras, strings of violins, violas and guitars; and the required simultaneously sounded frequency to vibrating air columns of bugles, clarinets, have ratios equal to the ratios of small whole trombones and pipe organs, show that available numbers. The tempered scale of western music, tones of a string or an air column of fixed length on the other hand, requires all semitone are related by integer multiples (harmonics) of a frequencies to have a ratio of the awkward fundamental frequency. Tonal color is largely number, 12 2 . The number 12 2 and, except for crafted by choice of the strength of the the octave, all western music intervals, can’t be components of the harmonics in a tone. exactly expressed as the ratio of small numbers. The physics of vibrating strings and air Indeed, the frequency ratios can’t be exactly columns can be used to generate a partial expressed as the ratio of any whole numbers. differential equation dubbed the wave equation. They are irrational. Nevertheless, the tempered When subjected to boundary conditions (e.g. a scale can be used to generate natural string is constrained not to move at the bridge Pythagorean ratios to an accuracy often audibly boundary of a guitar), the wave equation gives indistinguishable to a trained listener. The rise to a harmonic or Fourier series solution of tempered scale, in addition, allows freedom of the wave equation corresponding to integer key changes (modulation) in musical works multiples of the lowest frequency allowed by the difficult to instrumentally achieve in the boundary constraints.1 Pythagorean system of harmony. The The Mth harmonic of a reference tone, geometrically spaced frequency intervals of the or root, is simply M+1 times the frequency of the tempered scale also make perfect use of the reference frequency and can be expressed as the logarithmic frequency perception of the human ratio 1:M+1. For a vibrating string or air ear. The ability of the tempered scale to generate column, the root can be taken as the lowest near perfect Pythagorean harmonies can be frequency allowed by the physics of the imposed demonstrated in a number of ways. A deeper boundary constraints. Simultaneous sounding of question asks why this remarkable relationship notes corresponding to the root and its first five exists. There is, remarkably, no fundamental harmonics constitute a natural major chord. mathematical or physical truth explaining the Clearly, then, notes with frequency ratios of observed properties. The philosophical answer, 1:M+1 form pleasing harmony when M is small. we propose, is either a wonderful cosmic More generally, Pythagorean harmony coincidence - or the clever design of a Creator claims two audio tones will harmonize when the desiring our appreciation of both the intricate and ratio of their frequencies are equal to a ratio of beautiful harmonies of western music and the small whole numbers (i.e. positive integers). flexibility to richly manipulate them in almost This follows directly from the pleasing harmony innumerable ways. 1 Mathematical details of the derivation of the wave equation for a vibrating string and its harmonic (or Fourier series) solution is in Appendix A. of harmonics. For example, the second harmonic advantages in comparison to a strict Pythagorean (1:3) and the third (1:4) harmonic harmonize. scale. These two notes have a relative ratio2 of 3:4 and, indeed, form a the interval of a natural perfect Modulation. The tempered scale allows fourth. This pleasing musical interval is, as modulation among different musical keys. required by Pythagoras, the ratio of two small The tonic or reference note can be changed whole numbers. since, in the tempered scale, no note is Western music, on the other hand, is not favored above another. Using the tempered based on the ratio of small whole numbers. It is, scale, for example, one can change melody rather, built around twelfth root of two = and harmony from the key of C to the key of 1 G using the same set of notes. This cannot 12 2 = 212 = 1.05946... The number 12 comes be done in the Pythagorean system. This from the division of octave into 12 equally remarkable property of the tempered scale 12 was celebrated by J. S. Bach in The Well spaced chromatic steps. By definition, 2 , Tempered Clavier wherein all twenty four when multiplied by itself 12 times, is equal to 3 major and minor scales were used in a single two. The ratio of the frequency of any two work. 4 12 adjacent semitones , such as C# to C, is 2 . Dynamic Range Perception. The human Since a perfect fourth is five semitones, the ratio sense of frequency perception is 12 5 approximately logarithmic. This allows a of frequencies is (2 ). This seems a far cry larger dynamic range of perception.7 The from the Pythagorean frequency ratio of 3:4. tempered scale logarithmically divides each When, however, the numbers are evaluated, we octave nicely into twelve equal logarithmic find intervals. 4 5 Why does the flexible tempered scale so = 1.33333 ≈ ()12 2 = 1.33484 . (1) 3 well mimic the structured whole number Pythagorean frequency ratios? The answer, as The difference between these notes is a we illustrate, is either a marvelous coincidence miniscule audibly indistinguishable5 1.955% of a of nature or clever design of the Creator of semitone, or 1.955 cents.6 music. There is no foundational mathematical or The near numerical equivalence in physical reason the relationship between Equation (1) illustrates an more general Pythagorean and tempered western music should awesome property of the tempered scale of exist. But is does. The rich flexibility of the western music. Using the awkward number, tempered scale and the wonderful and bountiful 12 archives of western music are testimony to this 2 , frequencies can be generated whose ratios mysterious relationship. are nearly identical to ratios of whole numbers. The tempered scale also offers important Pythagorean Harmony Harmony in western music is based on harmonics - also called overtones. According to 2 Equal to the ratio of one fifth to one fourth. Pythagoras, tones are harmonious when their 3 Another illustration is compound interest. A frequencies are related by ratios of small whole one time investment made at an annual rate of numbers. The interval of an octave, or interest of 12 2 - 1 = 5.946… % annual interest diapason, is characterized by ratios of 1:2, 2:1 will double in 12 years. 4 A piano tuner will tell you, as is the case with 7 Human perception of sound intensity is also most quantifiable human characteristics, the ratio roughly logarithmic. If were not, the sound of is not exact. Lower notes on a piano, for 100 violins would be perceived as 100 times as example, are tuned a bit lower than dictated by loud as that of one. Thankfully this is not the the 12 2 ratio. case and we can enjoy in our comfortable 5 The method to compute the semitones or cents hearing range a violin solo or an orchestra of between two frequencies is discussed in violins. The increase in volume from 1 to 10 Appendix B. violins results roughly in the same incremental 6 A semitone is divided into 100 cents. volume increase as 10 to 100 violins. and 2:4. If any frequency is multiplied by 2N 13. 14:1 is an octave above the sixth where N is an integer, the resulting frequency is harmonic. (Bb7) related to the original frequency by N octaves. 14. 15:1 is both the second harmonic of For example, A above middle C is currently, by fourth harmonic and the fourth universal agreement, is 440 Hz8. Then 440 Hz ÷ harmonic of the second harmonic. (B7) 2 = 220 Hz is A below middle C and 440×23 = 15. 16:1 is an octave above the seventh 440×8 = 3520 Hz is the frequency of A a total of harmonic (C8). N=3 octaves above A above middle C. The ratio of 2:3 is the perfect fifth or diapente interval while 3:4 is the perfect fourth or diatesseron interval. Numbers that can be expressed as ratio of integers are rational numbers. Pythagoras claims harmony occurs between notes when the ratio of their frequencies are small whole numbers. Figure 1: Tempered notes closest to eight Harmonics result naturally from the harmonics when the root is middle C (a.k.a. physics of vibrating strings and air columns. C4). The first four harmonics are numbered. The rules for the first few harmonics follow. Harmonics 2 through 4 (G4,C5,E5,G5) are The intervals cited are Pythagorean (or natural) those used in bugle melodies. since there relations are determined by ratios of small numbers. After each entry is the note corresponding to a root of middle C, denoted C4. There exist variations of harmonic The closest tempered notes are shown in Figure frequencies from the corresponding tempered Error! Reference source not found. when notes. This variation is shown in Table 1. The middle C is the root. fourth column, headed Ratio, contains the 0. 1:1 defines the reference note or root. normalized frequencies of the harmonics C4. normalized (divided) by the frequency of the 1. 2:1, with twice the frequency, is an root. The next column, labeled Temper, octave above the root.
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