THE ILL-TEMPERED MATHEMATICIAN. John R. Silvester Department of Mathematics King s College London

Transcription

1 THE ILL-TEMPERED MATHEMATICIAN John R. Silvester Department of Mathematics King s College London 1

2 From Percy Scholes The Oxford Companion to Music: Temperament means an adjustment in tuning in order to get rid of gross inaccuracy in the intervals between certain notes an adjustment by the distribution of this inaccuracy over the intervals in general (or some of them), so that small disturbance of the ear results. Any close discussion of this subject inevitably becomes highly mathematical, and mathematical treatment is aside from the scope and purpose of this book. (Excuses, excuses!)

3 A vibrating body (such as a piano string, or the column of air in an organ pipe) has a certain natural or fundamental frequency. But it will also readily vibrate at whole multiples of this frequency. The note which is n times the fundamental frequency is its n th harmonic, and the intervals between these harmonics, for small n, are the basis of music. The most important interval aside from the ratio of 1 : 1, or unison, which is really no interval at all is the octave, given by a frequency ratio 1 :. Notes an octave apart sound almost the same, and indeed are given the same name A, B, C,... 3

4 The next most important interval is the fifth (strictly, the perfect fifth), given by the ratio : 3. And then the major third, 4 : 5. (The (perfect) fourth, 3 : 4, is really just a fifth turned upside down, since : 4 is an octave.) Musicians count the notes at both ends of an interval. So an octave (or 8 ve ) means counting along eight notes, as C C in the C major scale: C D E F G A B C. And a fifth means counting along five notes, as C G: C D E F G. Put the two together and you get a twelfth, because = 1. (OK?) However, there are black notes as well as white, and the intervals in that C major scale are not all the same: there are no black notes between E and F, nor between B and C, so these intervals are half the size of the others: semitones, whereas the others are tones. 4

5 In fact, an octave is equal to 1 semitones: C C D D E F F G G A A B C or, if you prefer, C D D E E F G G A A B B C and here the intervals from each note to the next are all the same, at least on an equal-tempered instrument (of which more anon). And a fifth is 7 semitones: C C D D E F F G. But why 1 notes to an octave? Why not some other number? 5

6 Suppose we try to construct a scale using only octaves and fifths. That means we start with a particular frequency, and allow ourselves to multiply or divide it by (multiply to go up an octave, divide to go down) or by 3 (to go up or down by a fifth). Will we ever arrive at a note using only octaves and at the same note using only fifths? For this to happen, we d need n = ( ) 3 m, or n+m = 3 m, for some integers n and m. Unfortunately this is impossible unless n = m = 0; but can we perhaps find an approximate solution? 6

7 If n ( 3 ) m, then (taking logs) nln() mln ( 3 ), or n m ln(3/) ln(). So we need a rational approximation to x = ln(3/) ln() continued fraction, 1 x = = ln(3) ln() 1. As a and the convergents are 1, 1, 3 5, 7 1, 4 41,... So these rationals are increasingly good approximations to x. We shall choose 1 7 as our approximation: in fact x and

8 The fact that 7 1 x says that 7 ( 3 ) 1, that is, going up 7 octaves is roughly the same as going up by a fifth 1 times; alternatively, if we divide an octave into 1 equal intervals, then 7 of them is roughly the same as an interval of a fifth. So that s why we have 1 semitones in an octave, and 7 in a fifth. If we had chosen 5 3 instead, then we would have to divide the octave into 5 intervals instead of 1, which would give us a pentatonic scale. This is in fact used in bagpipes, and in some eastern instruments, e.g., in Japan. The fifths are then less accurate than in our scale, because 3 5 isn t as good an approximation to x as is 1 7. If we want more accurate fifths than ours, we have to go to the next approximation, 4 41, which would involve dividing each octave into 41 parts. This is too many for most people, though it has been tried. 8

9 Here is a visual way of seeing that x 7 1 : x 1 9

10 Temperaments (1) The Pythagorean scale. Strategy: get all the fifths exactly right well, all but one of them (which is the best we can do). Hide the bad one away where (we hope) it won t matter too much: G D (=E ). So: E B F C G D A E B F C G ( ) 3 3 ( ) ( ) 3 ( ) 3 3 ( ) 3 4 ( ) 3 5 ( ) 3 6 ( ) 3 7 ( ) 3 8 Now re-order and multiply or divide by appropriate powers of to put everything in the same octave: C C D E E F F G G A B B C

11 How good is this? Small differences in pitch are measured in cents; a cent is 1% of a semitone, an interval of 1 : 1/100, so 100 cents is one (equal-tempered) semitone, an interval of 1 : 1/1. The next table shows the errors in the fifths and thirds, and the sizes of the semitones, for the Pythagorean scale: C C D E E F F G G A B B Fifths: Thirds: Semis: As you can see, all the fifths are perfect(!) except at G, but most of the thirds are pretty awful. And the semitones come in two sizes! 11

12 () Just intonation. Here we attempt to get fifths and thirds exactly right in as many keys as we can. We set up a 3 5 table, with C = 1 in the middle, with rows corresponding to thirds and columns to fifths. Since a third is 4 : 5 and a fifth is : 3, going up a third involves an extra factor of 5, and going up a fifth involves an extra factor of 3. (Confused?) The powers of are there to make everything end up in the same octave: D = 10 9 A = 5 3 E = 5 4 B = 15 8 F = 45 3 B = 16 9 F = 4 3 C = 1 G = 3 D = 9 8 F = C = G = 8 5 E = 6 5 B = 9 5 Notice that, in the first and last columns, we have values for each of D, B and F : 3 = 8 possibilities. We must choose! We ll take all of the last column, and ignore the first. 1

13 Re-order: C C D E E F F G G A B B C Here are the errors, and the semitones, for just intonation: C C D E E F F G G A B B Fifths: Thirds: Semis: So we now have six good keys (C, C, E, F, G, G ) but the rest are pretty bad; and we have four sizes of semitone. 13

14 (3) Mean tone temperament. Here we are going to concentrate on getting the thirds right. There are 4 semitones in a (major) third, for example C E goes C C D E E; and 1/4 = 3, so three major thirds can be stacked to give an octave, for example C E G C. However, these can t all be exact, for , which is near to but not equal to. Our 1 notes can be split into four disjoint stacks: C E G C, G B E G, D F B D, and A C F A. We shall make two of the thirds in each stack exactly right, and not worry about the other. Notice that the stacks (as written) start with C, G, D, A (going up in fifths), and another fifth (a fourth fifth!) takes us to E, back in the first stack and a third up from C well, actually two octaves and a third (a seventeenth!) up from the initial C. (Two octaves is a fifteenth: = 15, and then = 17. OK?) 14

15 Now C E, seventeen notes up the scale, takes us to the 5 th harmonic of C, that is, a frequency ratio of 1 : 5. Instead of attempting to divide this interval into four exact fifths (which is impossible, since ( ) 3 4 = = ), we split it into four equal intervals (ratios) of 1 : 5 1/ So: F 5 1/4 A 53/4 C 5 7/4 16 B 4 5 1/ D 51/ F 5 3/ 8 E 4 5 3/4 G 5 1/4 B 55/4 4 C 1 E 4 5 G 5 16 Here going along a row multiplies by 5, and going up a column (or from the top of one column to the bottom of the next) multiples by 5 1/4, with appropriate adjustments for octave. 15

16 Re-order: C C D E E F F G G A B B C 1 57/ / / 5 3/ / / / /4 5 1/ 4 Here are the errors and semitones, for mean tone temperament: C C D E E F F G G A B B Fifths: Thirds: Semis: If we regard 5 cents as not too bad, then we have eight good keys, the best yet; and we are back with only two sizes of semitones. 16

17 Of course we had to choose where to divide the stack C E G C, and our choice made the key of G (A ) pretty awful. Some organs got around this by having knobs that swapped all the G s to A s (and similarly for the other bad notes), so you set these knobs according to the key of the piece you wanted to play. This meant the organ had to have a lot of extra pipes and mechanism. The organ at the Temple Church, just up the road from KCL, had some black keys split: the front half of G /A played G, and the back half (which was raised) played A ; and similarly for D /E. Sir Charles Wheatstone, who was at KCL, must have known this organ; in 189 he invented the English concertina, which was tuned in mean tone and had seperate buttons for G and A, and for D and E. Modern versions are tuned in equal temperament, but retain duplicate buttons for these notes. 17

18 Historically, many other temperaments have been tried; for example, Young temperament (invented in England in 1799 by Thomas Young) is still used by some harpsichord enthusiasts. In Germany in 1691, Andreas Werckmeister published several temperaments, and many people think Bach s music should be played using Werckmeister III. Bach certainly had a fairly good way of tuning his harpsichord, and in 17 wrote Das Wohltemperierte Klavier (The Well-tempered Clavier), a set of 4 preludes and fugues designed to show that it was possible to play in all twelve major and minor keys. Later, he did it again, just to show it wasn t a fluke, and the whole collection is known to aficionados as The 48. Recently, Dr Bradley Lehman has claimed that the squiggles Bach drew as decoration around the title page of the 48 are in fact a coded set of instructions on how to tune your harpsichord, so we now have the Bradley Lehman temperament. 18

19 (4) Bradley Lehman temperament. Going up 7 octaves multiplies frequency by 7 = 18, whereas going up by 1 fifths multiplies it by ( 3 ) The ratio of these, which is 19 : 3 1, or about 1 : 1.01, is called a Pythagorean comma. So to make everything fit together, some or all the fifths have to be flattened (shrunk) a little, with the total shrinkage equal to one comma. Bach s method, according to Bradley Lehman s interpretation of those squiggles, was to shrink the fifths F C G D A E by one sixth of a comma each, leave the fifths E B F C pure (i.e., exactly : 3), and shrink the fifths C G E B by one twelfth of a comma each. So the total shrinkage is now (5 1 6 )+(3 1 1 ) = comma, so the final fifth, B F, is stretched by one twelfth of a comma. 19

20 Now an interval of a fifth shrunk by one sixth of a comma is 1 : 3 x, where x 6 = 19 19/6 31, or x = 3. Taking C as 1 we obtain F C G D A E 3 13/6 1 13/6 3 13/3 13/ 6/ The next three are easy, as we just multiply by 3 : B F C 3/3 0/3 17/ For the rest, we multiply by 3 y, where y1 = 19 19/1 31, or y = 3 : G E B 5/4 3 41/6 3 89/1 3 0

21 Now re-order and adjust by appropriate powers of to put everything in the same octave: C C D E E F F G G A B B C 1 5/3 3 10/3 11/ /3 3 11/3 13/ / /4 3 11/ 9/ /3 3 3 Here are the errors and semitones, for Bradley Lehman temperament: C C D E E F F G G A B B Fifths: Thirds: Semis: Except perhaps for C and F, the keys are all subtly different, a fact which (it is claimed) Bach exploited wonderfully in his writing. 1

22 (5) Equal temperament. Here we simply make all semitones exactly the same, 1 : 1/1. So: C C D E E F F G G A B B C Here are the errors, and the semitones: C C D E E F F G G A B B Fifths: Thirds: Semis: The fifths are pretty good, but the thirds much less so though at least there are none that are really awful. (Critics say it is too bland, and don t like the fact that all keys sound the same.)

23 Tuning a keyboard instrument. Since 1939, concert pitch has been standardised at A = 440 Hz (the A above middle C); so the D below this is 440 7/ Hz. The third harmonic of this is three times the frequency, or Hz, whereas the second harmonic of the A is 440 = 880 Hz. So these two harmonics will beat at = Hz, or about one beat per second. A similar calculation is done for each note of an octave around the middle of the instrument, and this temperament octave is then tuned in fifths and fourths, listening to the beats and timing them with a metronome. 3

24 One can then tune the rest of the instrument by setting up exact octaves from the temperament octave; but unfortunately on the piano there is a problem. Because of the stiffness of piano strings, they suffer from inharmonicity, that is, the harmonics are not quite in tune: the n th harmonic is a bit more than n times the frequency of the fundamental, with error varying approximately as the square of n. The effect is worse as the strings get shorter, towards the top of the piano; and it is also worse towards the bass, as the strings get thicker. (You need thicker strings in the bass, or the piano would have to be much bigger.) So a good tuner stretches the octaves, by listening to the beats between the harmonics of octaves, double octaves, triple octaves, etc. The stretching over the whole range of a concert grand can be of the order of 30 cents. A side-effect is that this puts the fifths more or less back in tune, cancelling the inaccuracy of equal temperament! (But it makes the thirds worse.) 4

25 Organ pitch. The pitch of organ stops is given in feet: an 8ft stop sounds as on a piano; a 4ft stop is an octave higher, and so on. Why? The organ keyboard is usually 5 octaves long, from the C two octaves below middle C to the C three octaves above. In Bach s time, pitch was somewhat higher than now, with A at 480 Hz, instead of 440. So middle C was = 88 Hz, and the bottom C on the keyboard was = 7 Hz. The speed of sound is 116 feet per second, so the wavelength of a 7 Hz note is feet. An open pipe to make this note has half this length, 7.8 feet. By the time you have added the (non-sounding) part between the mouth and the foot of the pipe, you are looking at an 8 foot pipe. 5

26 The Doppler effect. Sound waves are a succession of pulses of air-pressure, a note of frequency n Hz consisting of one pulse every 1 n seconds. These travel through air at a constant speed V, the speed of sound, about 768 mph in dry air at 0 C. If pulse 1 happens at time t = 0, then pulse happens at t = 1 n, by which time pulse 1 has travelled a distance V 1 n = V n. Now suppose the source of the sound is travelling with speed v towards the listener. Pulse 1 happens at t = 0; and pulse at t = n 1, as before. By then, pulse 1 has travelled a distance V n and the source has travelled a distance n v towards the listener. So the distance between the travelling pulses is V n n v = V v n = d, say. 6

27 These pulses are still travelling at speed V, so the time-interval between their arrival at the listener is V d, and thus the frequency of the note heard is the reciprocal of this, V d = V v V n, which is higher than the original note. If the source now passes the listener, then as it retreats we do the same calculation with v in place of v, so the note heard will have frequency V V+vn, which is lower than the original note. The interval between the two notes heard by the listener is thus say; solving V+v V v V V v n : V V +v n = V +v V v : 1 = x : 1, = x for v, we get v = x 1 x+1 V. 7

28 Suppose the listener hears the note drop by a minor third, so x = 5 6. Then v = = +1V 6+5 V = V 11 = m.p.h. Similar calculations yield the following table: semitone 5 mph perfect fourth 110 mph tone 45 mph perfect fifth 154 mph minor third 70 mph octave 56 mph major third 85 mph two octaves 461 mph (So traffic police might find music lessons helpful!) 8