Taming the pythagorean wolf

wolf oe Taming he the Pythagorean Pythag LDH. D.H. Griffel School of Mathematics, University of Bristol, Bristol BS8 1TW, UK Introduction Why does a conference on "Science and Imagination" include a paper on musical scales? It is an obscure subject, of interest mainly to musicians, and a small minority of musicians at that. Yet it was once a central problem in science, attracting the attention of Kepler, Galileo, Stevin, Descartes, Huygens, Hooke, Newton, Leibniz, Young, Helmholtz, and many others. The problem of musical scales is not straightforward. It resembles many difficult problems in the applied and environmental sciences, in that it involves applying quantitative criteria in areas which are difficult to quantify completely. The problem of tuning illustrates (in a context free of difficult value judgements) how wrong one can be. Musical notes and intervals The basic physical facts are sketched here very briefly. For more information see refs 1 or 2. A musical note is a vibration: higher notes correspond to faster vibrations, Most Western music uses only a few of the infinitely many possible notes; they are given names based on the letters A, ....G. If one note has vibration frequency twice that of another, we hear them as essentially the same note, in different registers. The two notes are given the same name. Thus the notes of frequency 220 and 440 hertz are both called A (sometimes a and a’ respectively). Two notes of frequencies f and 2f are said to be an octave apart. It is frequency ratios that determine the musical effect of a pair of notes sounded together. An interval is a pair of notes with a given frequency ratio: all intervals with the same ratio sound alike, regardless of the absolute frequencies of the notes. European music splits an octave into 12 parts, each of which is called a semitone. Thus there are 12 different notes between one A and the next A. They are laid out on a keyboard as-shown in Figure 1. Some of them are considered more important, and have white keys and simple names A, B, ...,G. The others have black keys, and names which relate them . to the nearest white note. Thus F” means the black note a semitone above F, and B° means the black note a semitone below B. The principal intervals and the circle of 5ths The fundamental interval is the octave, defined above. The next most important are shown in Table 1. Disregard the last column for the present, and think of the intervals as defined by their size in semitones. Speculations in Science and Technology, Vol. 11, No. 4, Page 273

D. H. Griffel B|c ve Made — E \ i yn eN c C|DIE /F|6JA Figure 1 The keyboard. / 3? \ N E_ (d)- C Ab Figure 2 The circle of Sths. B, F”, c, Gc’, Eb, Bb, Thus a 5th up from C is G (see Figure 1). It is five white notes up from C, hence the name "Sth", But a 5th up from B is the black note F* » Seven semitones away from B. Similarly, a 3rd up from C is E, a step of three white notes. Again, not every step of three white notes is a 3rd: the defining property of an interval is its frequency ratio, given here in terms of semitones. Going up in steps of a 5th from A gives A, E, F,C,G,D, A, … The sequence runs through all the notes and then return s to its starting point, This is often called the circle of Sths; see Figure 2. Taming the Pythagorean wolf 4th Octave Sth Interval 4 5 12 7 Size in semitones 5/4 4/3 2/1 3/2 Frequency ratio Table 1 Principal intervals “ Figure 3 Pythagoras investigating harmonic ratios, according to Gafurius (~ 1500 AD).

D. H. Griffel Pure intervals Some intervals andd chords sound pleasant, some do not. Experiment i s (going back to Pyego = see Figure 3) reveal a striking and beautiful fact: intervals sound pleasant , a Cir irequency ratios are simple rational number s, as shown in Table 1. See ref. 3 or the development of an understanding of this; but it is the fact, not the underl i mechanism, that concerns us here. oye . When h an interva : l is slightl | y out of ) tune, , an un pleasant quavering i sound is heard, call beats . As the interval is brought into tune, the beats get slower and finally disappear when the interva l is pure. See refs 1 or 2 for further information. The fundamental paradox Putting intervals together means multipl plying yi their i frequency ratios. Proof: if 2 uw frequencies x, Y, Zs then the intervals XY and YZ have ratios ir and 3h. com iningimerva ls XY andYZ gives XZ, with ratio z/x = (y/x)(2/y). For example ultiplying the ratios 3/2 and 4/3, giving 2. This agreesgoing wi the factane rather that a Sth plus5 a 4th gives an octave. ane ' . IÍnverting an interva interval l (that (th t al is,isis,, going soi down] p by the given number of semitones) corresponds to inverting the frequency We> are now in a position 10 see the fundam i ental diffiiculty in tuning i a ke: board A nfGoing EP a 5th and then down a 4th is equivalent to going up a tone (two semitones) 1 fol ows from theabove that the frequency ratio of a tone is (3/2)/( 4/3) = 9/8. Now, six en ve, which must have have freque 1 ncy ratio 2 - out of t une octaves | are The y quite intolerable. But combining six tones gives a frequency ratio (9/8)° = 2.027. It is impossible for all 4ths and Sths to be perfectly in prt - - — tune Taming the Pythagorean wolf 277 The spiral of Sths The difficulty can be viewed in terms of the circle of Sths. Twelve Sths add up to seven octaves. But 12 pure Sths give GA)? = 129.7, whereas seven octaves give 2’ = 128. Thus going round a cycle of pure Sths, starting from C, say, brings one back to a note different from the starting point. It is in fact B°, which is slightly higher that C when Sths are tuned pure. The circle of Sths has becomea spiral: see Figure 4. The Pythagorean wolf The spiral of Sths does not fit into a keyboard with only 12 notes to the octave, Some kind of compromise is needed. One solution is to make 11 of the Sths pure, and distort the last one so as to close the circle of Sths. The last 5th is then flat by 1.3 percent. It beats about eight times a second (in the tenor register), and sounds really unpleasant. It is called a “wolf 5th”, because it howls instead of singing sweetly. What can one do about this? One answer is to hide the wolf in the forest - among the black notes in Figure 1. Mediaeval and Renaissance music uses only a restricted range of harmonies, so if the wolf Sth is between G° and EP, it will hardly ever be heard. However, there are other problems. Pythagorean 3rds We showed above that when 5ths are pure, a tone has ratio 9/8. Therefore a 3rd (equal to two tones) has ratio (9/8)? = 1.266. But this is very out of tune: see Table 1. Thus pure Sths imply very sour 3rds: they are called Pythagorean 3rds. The lone wolf Sth could easily be hidden. But with most of the Sths pure, nearly all the 3rds are badly out of tune, This is a serious problem. Early mediaeval music was built largely on 4ths and Sths, and did not use 3rds. Therefore the Pythagorean system was satisfactory. But the development of music using a wider range of harmonies required a system in which 3rds and common chords are tolerably in tune. | Equal temperament One obvious approach to the problem of tuning is to make all the semitones equal. Since 12 semitones make an octave, each one will then have a ratio of 2°”. It follows that all the Sths will have frequency ratio 2”! = 1.498. The Sths are said to be “tempered”, that is, deliberately made slightly impure. A tuning system with tempered intervals is called a “temperament”. The system described here is called "equal temperament” because all Sths are tempered by the same amount, from 1.5 to 1.498. The Sths are only slightly tempered; 1.498 is almost 3/2. But 3rds in equal temperament have frequency ratio 2/2 2 1.260, quite badly out of tune. They beat about eight times a second in the tenor register, Figure 4 The spiral ofSths. On a modern piano the beats are not very noticeable, and we have all grown accustomed to the sound of equal temperament. But on earlier instruments, with a more incisive tone, the beats are much more prominent. So although equal temperament wa: invented in the sixteenth century, it was not generally used for keyboard instruments unti the nineteenth century (when the clear tone of the harpsichord had given way to th blander sound of the piano).

D. H. Griffel Optimisation Mean tone There is no complete solution to the temperament problem. There have been many compromise solutions (see ref. 4). One of the earliest, called “mean tone tuning", tempers the Sths so as to make the 3rds pure. Because four Sths make a 3rd (see Figure 2), the error in the Pythagorean 3rd is divided among four Sths in this system; so each of them needs only a slight tempering, and in practice sounds very nearly pure. This is a good scheme for much Renaissance and early Baroque music. But it has limitations. All the white notes can be tuned in this way to give pure 3rds and nearly pure Sths. The black notes can be tuned to give pure 3rds with the white notes. But there are ambiguities. The note marked G*/A? in Figure 1 can be tuned to give a pure 3rd with E, or a pure 3rd with C, but not both. If it is a pure 3rd with E, it is called G°, and if it is a pure 3rd with C it is called A>. These two notes are very Clearly different; the interval cic in mean tone has ratio 1.28. This is so far from 5/4 that G°C in mean tone sounds quite intolerable. 279 Taming the Pythagorean wolf . Renaissance and early Baroque music used a limited range of harmonies. Since G° and A” were not often needed in the.same piece, one could tune to whichever was needed. But as music developed, more flexible temperaments were needed. | vont " forfor the various chords; a i is of ref. 5 i gives a pattern of " “degrees of avoidance” One can “ee prominently. use not does Bach which chord oh depes is assigned to a mal cho va me 2 impurities look for a temperament for which the pattern of descr be may result The f degrees of avoidance. clavier. It is similar to the eighteenth torawerament for laying Bach’s well-tempered tury unequal temperaments described above. | temperaments, sce refs 6 and 7. CFO more mathematical approaches to optimising Other kinds of scale? pand on the i of attack on the temperament pro blem is to aband mpletely different line g Me dividin on based s system ed temper division of the octave into 12 parts. Equally they But 8). ref. (see es centuri l severa for d studie octave into 19, 31, or 53 parts have been are still very much a minority taste. Concluding remarks In mean tone, many chords are very nearly pure, but others are practically unusable, Many alternative schemes were devised in the eighteenth century, which even out the inequalities to various extents. In.a typical example (Vallotti’s temperament), half of the Sths are pure and half are slightly impure; five of the 3rds are better than equal temperament and five are worse. The most often used chords are better than in equal temperament, and even the worst are much better than the Pythagorean or mean tone wolves. Unequal temperaments of this kind have two advantages. Many of the commonest chords are better in tune than in equal temperament. And it follows that chords on different keynotes have different sounds. Therefore music in different keys will sound different. Moving from one key to another is an essential part of the language of classical music. An unequal temperament gives a rougher or sharper sound when thé music modulates into remote keys, and the return to the home key near the end of a movement will generally bring a smoother tone-colour. This element of the music is lost in equal temperament. Bach’s "well-tempered clavier" In 1722, J.S. Bach completed a set of preludes and fugues in all the 24 possible keys. He called it “The well-tempered keyboard”, the point being that it needs a temperament in which all chords are acceptably in tune. It used to be thought that "well- tempered" means equally tempered. But looking carefully at the music refutes this. Bach is rather cautious in using chords which are noticeably impure in typical eighteenth century temperaments. For example, they may be sounded so briefly that there is no time to hear how out of tune they are. Or they may be sounded in one hand while the other hand plays faster-moving notes which distract attention from the impure chord. See ref. 5 for a detailed study, concluding that there is definite evidence that Bach did not intend equal temperament. en tend to be i al experiience. Instruments ory should be viewed in the light of practic it responds 1 t; concer a hout throug tune in tly perfec stay able À harpsichord will not is little point 1 the concert room. So there the warm and perhaps moist atmosphere of | | ion. precis temperaments with extreme ing ations too seriously.Theae there is another reason for not taking the calcul ed as ary evil, | necess a are above are based on the idea that impure chords eig e a sing = discus when noted, We cation. implifi imp overs i is s an as possibi le. But this rf mu i g between g00od and bad keyss can add to the we century systems, that movin riate place can ha approp an in hord i from contemporaryy 1 music is all about emotion (as is clear ue music Baroqe effect.oar nalMowe piece. aemotio :0 t language of| the par is t ramen tempe al Unequ ic itself). as well as from the music i tings ance misses the musical to reducing disson this music A strictly mathematical approach tative optimisation techniques eh Poin many fields, a straight application of quanti s of the problem arc unquantifia e. to go wrong because some important features ation. imagin but ise, expert needed is not only mathematical and scientific References © 1. Helmholtz, H., On the Sensations of Tone, (Transl.). Longman, London (1885), powerae, KN (1954). . of Music. Springer, New Roederer, I ., Introduction to the Physics and aoe), | ND Eighteenth century systems LF. u Quantifying Music. Reidel, Dordrecht (1984). of Music| and Musicians. Macmillan, London . em. j ts" in The New Grove Dictionary empered Clavier,” Early Music, 7,23 Temperament: Intemal Evidence from the Well-t I. “Bach's | 9). 19, 554l Tempera “Optima r Harmon tical Review, y", ment”, 6Ÿ Goline Parker, in DB.AA, "Numbe MathemaSIAM Intelligencer, 8, 562 18-21(1977). (1986). 5 Ba ia, 16-A, 217-225 8. Gamer, C. and Wilson, RI, "Musical Block Designs”, Ars Combinator