Music, consonant and dissonant

Chapter 5 Music, consonant and dissonant This chapter covers the elementary aspects of Physics of Music related to the concepts of consonance and dissonance, and the related concepts of Music Theory. Everything is treated in the simplest possible way: many books introduce all kinds of complications right away, and the result is often an unfortunate confusion. For various reasons, many scholars as well scholars have been drawn into what seems like never-ending considerations of tuning and temperament. A small but vocal group argues against the Equal Temperament, and the impact is sometimes quite worrying. At a recent dinner in a restaurant, I was explaining to my wife some subtle aspects of the circle of 5ths, and I draw it on a napkin. The waitress noticed it and said Oh, I know this. We are just reading at school the wonderful book How Equal Temperament ruined Music. I pointed out that the title is How Equal Temperament ruined Harmony, and that I think even that is a very misleading and in fact harmful claim. The waitress was not impressed: That is not what our teacher says. In an attempt to clarify the issues involved, our treatment of the tuning and temperament is more thorough than many other parts of the Text. 5.1 Basic Music concepts Historical development of Western music resulted in a complex structure, and most of the complexity is connected to harmony (the other main attributes of music are melody and rhythm). It is when we consider harmony that we encounter the fundamental concepts of consonance and dissonance, intervals and chords, and tuning and temperament - as well as physics, biology, mathematics and music thery.. 5.1.1 Physics underpinning As argued in Chapter XXX, the basic physics of music reflects the fact that the frequency spectrum of tones produced by simple linear systems, such as a vibrating string or vibrating 97

98 CHAPTER 5. MUSIC, CONSONANT AND DISSONANT column of air in a pipe, is often strictly harmonic a This means that the individual overtones are equidistant in frequency: f n = n.f 1 n = 1, 2, 3,.. where f 1 is the lowest frequency (the fundamental ). This is true in any culture (as well as in any alien world :-) We shall also see the math underpinning of tuning and temperament, as soon as we start explaining the concepts. 5.1.2 Biology underpinning a) As we have seen in the previous chapter, human perception of frequencies is logarithmic: a given ratio of frequencies corresponds to the same distance (in millimeters) between the places of the maximum response of hair cells along the cochlea This means that in perceiving the intervals (or chords), what is important is not just the differences of frequencies of the tones but also (and mainly b ) their ratios. b) As we know, the waveforms corresponding to a periodic waveform are periodic. This is contributing to our perception of the intervals contained in the harmonic tone as being consonant. 5.1.3 Biological and cultural underpinnings And last but not least: a baby, newborn or even before, is constantly exposed to the periodic sounds, even as elementary as utterances of definite pitch ( Oh, what a beautiful baby etc). This is true even for non-musical babies and non-musical parents. And as we will see, the intervals considered consonant are contained in any periodic sound as the ratios of the harmonics. As a consequence, our concepts of consonance and dissonance develop pretty soon, and are - in their main features - universally shared. 5.1.4 A semi-historical overview of harmony It all started with Pythagoras, again c. He is credited with recognizing that intervals are consonant if the ratio of the two tones are in the ratio of small integers. The simplest ratio of frequencies occurring in the harmonic series is 2:1. The resulting tones, with frequencies 1,2,4,8,16,... are perceived so similarly that they may be called the same tone but higher a In many cases, the underlying mechanism is the production of a periodic waveform. This is particularly relevant in cases when the instrument itself has the passive spectrum that is slightly inharmomic. b The differences of frequencies are also important: they determine the beat rates and thereby the perception of the quality of consonances and dissonances/tuning and temperament c This is a cartoon account. Very little is actually known about the person of Pythagoras, and he has not left any writings :-( But the Pythagorean tuning is a generally used term - as is his triangle :-)

5.1. BASIC MUSIC CONCEPTS 99 up. The interval of 2:1 came to be called an octave (the motivation for the name octave will be provided shortly). The task now is to provide a meaningful subdivision of the octave. The first step results from the next simplest ratio 3:2. When you put six such intervals on top of each other, you will have a series of tones with frequencies 1, 3/2, (3/2) 2, (3/2) 3, (3/2) 4, (3/2) 5, (3/2) 6 But many of these tones belong to higher octave that the one we started from. Therefore, every time that happens, we bring the tone down an octave by dividing the frequency by 2. This provides 7 notes in the same octave with frequencies 1, 3/2 = 1.5, (3/2) 2 /2 = 1.125, (3/2) 3 /2 = 1.6875, (3/2) 4 /4 = 1.2656, (3/2) 5 /4 = 1.8984, (3/2) 6 /8 = 1.4238 and when we order these tones by frequency and add the note corresponding to the first note of the next octave we obtain 1, 1.125, 1.2656, 1.4238, 1.5, 1.6875, 1.8984, 2. Then we calculate the widths (in cents) of the intervals between two consecutive tones: 204, 191, 204, 89, 204, 204, 90 These tones eventually became what we now call the white keys on the piano: F, G, A, B, C, D, E, F... where F with frequency=2.0 is the first note of the next octave - see Figure 5.1. Note that these widths come in two categories: those close to 200 cents - these are called tones and those close to 100 cents ( semitones ). In the now prevailing Equal Temperament all tones are defined as exactly 200 cents, and all semitones as 100 cents. By shifting the starting (and ending) tone, one obtains 7 different modes that have lovely Greek names: from F to F is the Lydian mode, from G to G = Mixolydian, A = Aeolian, B = Locrian, C = Ionian, D = Dorian and from E to E is the Phrygian mode. The Ionian mode corresponds to the modern Major scale. It is fun to get the feel of these modes on the piano. The original Greek modes were more subtle, and the subsequent historical development has been exceedingly complex very interesting but largely irrelevant today. Therefore, we now restrict ourselves to a very simple account emphasizing Equal Temperament (but we will discuss the issues of tuning, microtuning and temperament in considerable detail in section XXX).

100 CHAPTER 5. MUSIC, CONSONANT AND DISSONANT 5.1.5 The Musical Keyboard All of the simplest music theory we will need is illustrated by a regular piano keyboard (see Figure 5.2). Each octave (factor of 2 in frequency) is divided into twelve (logarithmically) equal semitones - therefore each semitone corresponds to the same frequency ratio of 2 1/12 i.e. by 100 cents. The resulting chromatic scale can be played by using successive keys that may be white or black as illustrated d The white keys are known (in the German convention) as C, D, E, F, G, A, H and C again we have encountered these tones in previous section. A semitone lower than one of these names is denoted by a flat = b so for example the black key below the D signifies D-flat, or D b. Similarly a semitone higher that one of the names is denoted by a sharp or # so that the black key above G will signify G-sharp or G #. There is one exception: the black key below H is not called H b but B. In the English convention the names are slightly more logical: C, D, E, F, G, A, B and C again, without the German business around the 11th semitone. On the other hand, in the German notation the name Bach can be played as a melody (B.A.C.H. and the great cantor took advantage of this to sign his name in his Unfinished Fugue and then he died :-( Notes: Voice teachers often use the names do, re, mi, fa, sol, la, si and do again. In the Equal Temperament D b = C # etc; more about this below. We shall use the English notation. 5.1.6 Intervals A musical interval consists of two notes, and the name of the interval indicates its width. By convention we start from a C and determine the number of the white keys representing the upper note of the interval. Then the convention is: 1 (the same key) produces a unison : width of 0 semitones. This may seem a little funny on the keyboard, but the unison produced by two (or more) instruments is a genuine musical interval (of zero width). 2 (from C to the next white key, i.e. D) : a second : width of 2 semitones 3 (from C to the third key, i.e. E sounding together with the starting C: a third : 4 semitones etc, i.e.: 4 C-E: a fourth key: 5 semitones 5 C-G: a fifth key: 7 semitones 6 C-A: a sixth : 9 semitones 7 C-B: a seventh : 11 semitones 8 C-C: an octave i.e. the eighth key: 12 semitones d The irregular spacing of the black and white keys developed historically, presumably to assist the player in quickly finding the right key.

5.1. BASIC MUSIC CONCEPTS 101 The 4th, 5th and octave are considered perfect (even when they are not more on this below) The other intervals as defined above are called Major (so e.g. C E is a Major third or 3 M ; similar for 2 M, 6 M and 7 M.) Intervals a semitone narrower than a Major interval are called minor (so e.g. C A b is a minor 6th or 6 m The interval 6 semitones wide (C G b ) is called a tritone (for obvious reasons) All this can be now be summarized as names of the interval as function of its width in semitones: 0 unison 1 2 m 2 2 M 3 3 m 4 3 M 5 perfect 4 th 6 tritone 7 perfect 5 th 8 6 m 9 6 M 10 7 m 11 7 M 12 octave This Table can be used for all intervals, even if they do not begin with a C. Just use the Keyboard Figure, count the number of semitones between the lower and upper tone, and look up the name in the Table. 5.1.7 Chords Three or more tones played together are called chords. There is a large number of combination in musical use, and their judicious choice is the essence of Harmony. For our purposes we will use just the two basic chords: a Major chord : a 3 m on top of 3 M, for example C-E-G a minor chord : a 3 M on top of 3 m, for example A-C-E Note that the Major chord, ideally with its frequency ratios of 4:5:6, is part of a harmonic series (with the fundamental two octaves lower). The minor chord has the same three intervals 3 M, 3 m and 5 th but they are in the wrong order. Am extremely useful way to understand the structure and relationship of various Major and minor chords is the famous circle of fifths displayed on Figure 5.3. In the clockwise direction, each triplet of successive chords represents a subdominant, tonic and dominant of a musical key, and much of simple music (and some not so simple) can be harmonised using

102 CHAPTER 5. MUSIC, CONSONANT AND DISSONANT those three chords. The distance along the circumference measures a musical distance between the chords and corresponding scales: the larger the distance, the more difficult is to modulate between them. For most musicians, especially for those closely involved with harmony (piano/organ players, composers,...) a complete mastering of the Circle is an absolute necessity. I know professional jazz pianists who practice that mastery every single day. 5.2 Consonance and Dissonance In this section, the simplest possible account of the concepts is presented. The discussion is limited to the Western music inclusion of other musical cultures is beyond the scope of this book. Note that even limiting ourselves to the classical 12-tone music, scholars would complain about oversimplification in our treatment. I believe that before considering all kinds of complications, one has to understand the basics and they are not trivial. In addition: if you understand the fundamental concepts, you will be well positioned to study more advanced concepts on your own. 5.2.1 Pythagoras explained The basic fact behind the concept of consonance and dissonance is that the perception of two sinusoidal (i.e. pure ) tones is unpleasant if their frequencies are not too close AND not too far away from each other. If they are closer than approximately 5 Hz, we hear (can count) the individual beats. If they are quite far from each other, we hear the two individual tones. But if they are in the critical region, the rapid beats create a disturbing (harsh, bleating) sound. The critical region is about 2-3 semitones, and it corresponds to the overlapping regions on the cochlea that we discussed in the previous chapter. This effect has been thoroughly investigated experimentally. Figure 5.4 shows the typical result. For complex tones with several partials, we have to investigate all combinations of one partial from the bottom tone with one partial of the top tone. From this immediately follows Pythagoras famous observation: if the ratio of the fundamentals of two tones is a ratio of two small integers, such as 1:1 (a unison ) 2:1 (an octave ) 3:2 (a just 5 th 4:3 (a just 4 th 5:4 (a just Major third 3 M ) 6:5 (a just minor third 3 m ) then the resulting sound is consonant, because all pairs of partials (in fact not all, but most of the important ones see below) are either too close or too far from each other to produce the rough rapid beats i.e. fall into the same critical region.

5.3. TUNING AND TEMPERAMENT 103 This is illustrated in Figure 5.5 Because of the fundamental properties of the octave, for each of the above intervals their complements to the octave sound quite similar to the original interval. This leads to (see again the keyboard slide ): complement of a 5 th will have frequency ratio 2 2 = 4 : 3 - this is the interval of a 4th 3 complement of a Major third (3 M ) will have frequency ratio 2 4 = 8 : 5 - this is the interval 5 of a minor 6 th (6 m ) complement of a minor third (3 m ) will have frequency ratio 2 5 = 5 : 3 - this is the interval 6 of a Major 6 th (6 M ) (and complement of an octave is a unison). To complete all these complements: for dissonant intervals, the complement of 2 m is 7 M, the complement of 2 M is 7 m and the complement of a tritone (frequency ratio of 2) is another tritone. All of this is in Figure 5.6. 5.3 Tuning and Temperament 5.3.1 The difficulty There is a mathematical and fundamental difficulty with the tuning system based on the small-integer ratios as outlined above. We would like the pure consonant intervals, upon addition on top of other pure intervals, to close into one or several octaves, and they almost do: (5/4) 3 = 1.95 i.e. three Major thirds are almost equal one octave.the difference is called a lesser diesis and it is equal to 3986.31log (5/4)3 = -41.06 cents. In plain English: three 2 just Major thirds come 41.06 cents short of the octave. (6/5) 4 = 2.07 i.e. four minor thirds are almost equal one octave. This is called a greater diesis and it is equal to +62.57 cents: four minor thirds are 62.57 cents wider than an octave e. (3/2) 12 = 129.75 2 7 = 128 i.e. 12 fifths almost equal 7 octaves. The difference is called the ditonic or Pythagorean comma and equals = 23.46 cents and the final discrepancy is the difference between a Major third determined from four just 5ths minus two octaves, and the just Major third as given from first principles. This is called a syntonic comma and equals to 21.51 cents. These discrepancies are the difficulty. As we shall see, the problem goes beyond the choice of twelve semitones per octave and beyond the difficulties presented by keyboard e Combining this with the previous result leads to some amusing composing see Figure 5.7

104 CHAPTER 5. MUSIC, CONSONANT AND DISSONANT (and other quantized, fretted )) instruments. In fact, even though an a-capella choir or a string quartet can produce all intervals perfectly pure, they will pay the price: the overall pitch will drift - see Figure 5.7. 5.3.2 Tunings A tuning is any scheme in which as many intervals as possible are kept pure, letting the inevitable discrepancies fall where they may, and the composer has to avoid them. The earliest Tuning is attributed to Pythagoras (again!). It is built by simply setting eleven fifths to the pure (beatless) 702 cents. This can be calculated either by repeatedly multiplying the frequency of the bottom tone by 3/2, or by adding 702 cents to the value of the bottom note expressed in cents (which method is better?). In practice, one tunes a perfect 5th by removing the beats between the 3rd harmonics of the bottom tone and the 2nd harmonics of the top tone, for each of the 11 fifths. As a result the last (12th) fifth is awful (7 1200 11 702 = 678 cents instead of 702) and many 3M thirds are quite bad, too. Another approach is illustrated by the decaus tuning which has six major and six minor chords perfect, at the expense of the 12 remaining ones being unusable. Music using just the perfect chords sounds very smooth, but after a while the key limitation can make it boring. The decaus tuning provides us with a good exercise. Figure 5.8 shows what and how is being done: the original Solomon decaus treatise is quite difficult to present, so following Hall we start at F go up 3 just perfect 5ths to get F, C, G and D, and from each of these notes we go up two just Major thirds. The results are on Figure 5.9. We see that 6 Major chords (C, E, F, G, A and B) and 6 minor chords (c #, e b, e, a b, a and b) are perfect, and as expected we pay a heavy price in the quality of the other 6+6 chords. Many compromises and improvements/ solutions were designed and used in the past. Sometimes keyboards with extra keys were built, providing the required difference between an F# and a Gb etc. An excellent example of a successful design is the recent(!) pipe organ built in Goteborg(Sweden) following the 18th century principles see the keyboard of this instrument in Figure 5.10. Another possibility is to extend the ET to more than 12 equal intervals per octave. After 12, the next suitable division f is 19, then 31, 53,... But all these schemes require considerable expense as well as much skill on the part of the performer. 5.3.3 Temperaments In a circulating temperament the tuner attempts to spread the discrepancies around, so all chords are tolerable in all keys, although none is perfect. The simplest such temperament is the 12-tone Equal Temperament ( 12-tet ), where all fifths are tempered from the just f see a 19-tet design on Figure 5.11

5.3. TUNING AND TEMPERAMENT 105 702 cents to 700 cents, all Major thirds from the just 386 cents to 400 cents, and all minor thirds from 316 cent to 300 cents. Bach wrote his famous 48 12 (prelude+fugue) times 2 (Major and minor) times 2 (he could not help doing the whole marvelous thing all over again) to celebrate the new and exciting possibility to modulate freely to any key he wanted. It is not known which of these well-tempered schemes g he used. It was unlikely the exactly Equal Temperament, due to the difficulty of its use (since all intervals beat, one has to laboriously count the beat rates). However, Bach was a pragmatic, and supremely capable musician. There is testimony that he was able to tune his harpsichord in less than 15 minutes. I don t think he had to count beats to produce something practically identical to ET he was able to do it just by ear! Since more than 100 years there is a flourishing industry of research into the question what exactly was Bach s temperament. The claims are sometimes phantasmagoric and I believe often a waste of time (more about this below). In any case, the practical tuning difficulty disappeared, to a large extent, with the advent of electronic tuners. It should also be noted that if no quantized-pitch instruments are incluved, it is in principle possible to achieve the best possible tuning in real time during the performance (string quartets, a-capella choirs etc.). And a harpsichord can, in principle, be completely retuned during an intermission of a recital. A large pipe organ is a different story, as is any performance that includes equally tempered instruments. 5.3.4 More on the dissonance/consonance of complex tones Figures 5.12 and 5.13 show a detailed investigation of a Major chord consonances and dissonances: Figure 5.12 shows the harmonic spectrum as function of frequency. The reader is invited to multiply the integers by the root frequency f 0 ; for convenience we choose root frequency = 100 Hz so that the harmonics 4, 5 and 6 have frequencies 400, 500 and 600 Hz. As explained in the text, these three frequencies make up a Major chord; we will call its notes C1, E1 and G1. So the implied root of this chord is indeed 100 Hz: the C two octaves below C1. Each of these three notes are the fundamentals of their own Fourier series, and their higher harmonics (C2,C3,C4.., E2, E3,... G2,...G6) are integer multiples of the root frequency as indicated. Sometimes we get an overlap: for example G2 overlaps with C3 because 2*3 = 3*2 etc. (ignore the small splits for the moment). The whole scheme shown covers a little more than 5 octaves as indicated on top. Figure 5.13 shows the same range plotted as function of frequency (for example 2 is the second harmonic of the implied root or 200 Hz or 12 semitones or 1200 cents, 4 is fourth harmonic or 400 Hz or 24 semitones or 2400 cents etc). g Most people say that Bach used a well-temperament. This is a linguistic atrocity; I much prefer to say a well-tempered scheme or a circulating temperament.

106 CHAPTER 5. MUSIC, CONSONANT AND DISSONANT As we know, the first 6 harmonics of C1, E1 and G1 fall close to the centers of a piano keyboard (this is in fact how the piano keyboard was designed in the first place). So there are small discrepancies, and in addition E1 and all its harmonics come short by 14.6 cents because the Major third is 386 in just tuning but 400 cents in equal temperament, and G1 overshoots by 2 cents (700-702 = -2). Note carefully that the harmonics of C1 will include the notes C,E and G; harmonics of E1 will include E,G# and B, harmonics of G1 will have G, B and D. The whole range corresponds again to a little over 5 octaves, but now these look equidistant as expected. Note the piano keyboard shown here is designed to have the same distance between any two consecutive semitones in order to correspond to a scale linear in cents (so it makes sense mathematically but it would be a little difficult to play :-( Back to the top display linear in frequency: the differences between the equal temperament and the just tuning are now increasing as we go up in frequency (because the ratios stay the same!). These Figures shows that the harmony of a simple major chord is in fact quite complex and a fair amount of dissonance may be present, even in just tuning. This effect can be significantly enhanced by playing a nominally consonant chord in a wide spacing. As an example, Figure 5.14 shows a form of an inversion of the F major chord where a C is separated from the F and A by two octaves. The two minor seconds between the harmonics of the C and the fundamentals of the F and A are very audible and, needless to say, very dissonant h. 5.3.5 Tuning and Temperament myths and follies As noted above, innumerably many articles have been written (and are still being written, often by math amateurs or by retired professors) claiming to have solved the tuning problem. As discussed above, this is simply not possible. As for temperaments, there are even more articles claiming to have discovered the best temperament (or the temperament Bach has used ). I think I have good reasons to believe that this is misguided, too. 5.3.6 Back to history Figure 5.15 shows the quality of the major thirds i of the 12 major chords in Equal Temperament, compared with the Pythagoras tuning, two 16xx meantone schemes, 17xx Neidhart and Valotti, and 20xx Lehman. What is plotted is the distance of the major 3rd from its pure ratio 5/4, measured in cents. As a rule of thumb, the values up to 10 sound good, between 10 and 20 reasonable, h J.S.Bach the supreme master of consonance and dissonance had a keen ability (if not theoretical understanding) of using such phenomena to a great effect. i With an exception noted in the text, an inclusion of the 5ths - or equivalently of the minor thirds - does not change our conclusions.

5.3. TUNING AND TEMPERAMENT 107 and to be over 20 is bad. For the Pythagorean, 1/4 comma and 1/6 comma tunings, all values not explicitly plotted are good (showing this would make the Figure too busy...). In ET, all fifths are tempered equally, so all are imperfect but not too bad. And of course all 3M are imperfect, again all equally as indicted on Figure - all are reasonable. You might say it is democracy in action, or mediocrity in action :-) Now we will offer a simplified historical account: At first sight, the ancient Pythagorean tuning has much to offer. Four thirds are good, the rest (all 22 cents off) are at the boundary between reasonable and bad, and all but one fifths are perfect (by construction). However: 22 cents really is quite bad, one of the good thirds actually belongs to the twelfth bad fifth, and to really use a key one would like to have not only the tonic but also the dominant and subdominant chords usable. All this leaves us with a single key of C with a good tonic, subdominant and dominant :-( The meantone tunings use a systematic mathematical prescription: for an 1/n meantone, temper 11 fifths (downwards) each by (syntonic comma)/n and let the 12th fifth (and all other intervals) come where they may. So for our two meantones, the 1/4 will have the wolf 5th = XXX cents and the 1/6 will have XXX cents (HW). It turns out that for the 1/4 meantone, there are 8 thirds that will be completely pure. You pay for it by having the remaining four thirds unusable as indicated on Figure. In the 1/6 meantone, there are again 8 good major thirds (this time not perfect but off by 7 cents). And again, the remaining major thirds are bad, but better than in the 1/4 comma meantone. In our simplified[sic] account, the next step is the Valotti temperament: it is also mathematically oriented (it was re-invented later by the celebrated physicist XXX Young). It is accomplished by setting 6 consecutive fifths to be pure, and the other 6 tempered, all 6 by the same amount (HW: what is the amount?). This results in a remarkably symmetrical temperament, with some 3M a little better than ET, some a little worse. The division between the better and worse can be placed at will Valotti placed the bad region to coincide with the bad regions of the meantones. This is important to what follows. In Valotti s time, there were numerous attempts at defining a suitable temperament. The all ended up looking similar to Valotti - everything reasonable but unequal. The possibility of the ET was already known but not widely accepted. There were claims that the inequality is very desirable, since it gives each key a different character or color. But another major reason was the difficulty of tuning ET: since not a single interval there is pure, tuning has to proceed by laborious counting of beat rates (recall that as you go up on the scale, equal widths in cents do not produce equal beat rates!). And in 1722, in the middle of all this, comes Johann Sebastian Bach. Without saying anything or writing or arguing about it, he composes his monumental Well Tempered Clavier : 24 incredible Preludes and Fugues, one pair in every Major/minor key.

108 CHAPTER 5. MUSIC, CONSONANT AND DISSONANT 5.3.7 An organ fingerprint Many of the issues of tuning are summarized on Figure 5.16 this is a tool that I developed in order to see most of the aspects of the tuning of the thousands of pipes in a given pipe organ on one page. For each of the twelve Major chords separately, the tuning of each triplet of pipes of that chord (all octave positions and all stops) are represented as a single point on a scatterplot. The horizontal axis shows the deviation of the chord s 5th from just, and vertical show the corresponding deviation for the Major 3rd. The result is reminiscent of target shooting: the center of each cluster shows the intention of the tuner, and the spread of the cluster shows the degree to which the intention has been realized. 5.3.8 Tuning and Temperament conclusion In our times, classical music is not exactly what fills arenas and stadiums with young people. Perhaps one should not exacerbate the problem by exaggerated criticisms and opinionated judgments, and apply some common sense. For a piece composed with a specific tuning scheme in mind, it is correct and usually preferable to perform that piece in that tuning. This may be the case of much of truly ancient music. It is also perfectly reasonable and often artistically valid to experiment with new sound effects involving microtuning. However, for the bulk of Western music the Equal Temperament is the temperament of choice. Starting with Bach, composers have discovered powerful ways of using harmony with results so dramatic that the small effects of tuning are negligible (and often detrimental) in comparison with the artistic freedom ET provides. Today it is a bad advice to tune a new pipe organ (intended for the whole range of organ literature) in anything but ET. And it is irresponsible to publish books with titles such as How Equal Temperament Ruined Harmony.

5.3. TUNING AND TEMPERAMENT 109 Figure 5.1: Two octaves of an embryonic keyboard based on the 7 Pythagorean notes discussed in the Text. The nominal values of the frequencies (in cents) are shown in the Equal Temperament, but they could be in Just values instead (again, see Text)..

110 CHAPTER 5. MUSIC, CONSONANT AND DISSONANT Figure 5.2: Basic music concepts.

5.3. TUNING AND TEMPERAMENT 111 Figure 5.3: The famous Circle of Fifths. The large letters in the outer ring (e.g. C) label the perfect 5ths (e.g. C-G), the corresponding Major chord (e.g. C-E-G) and the scale (e.g. CDEFGABC). The next label in the clockwise direction (in this case G) labels the 5th G-D, major chord G-B-D. The corresponding G-major scale starts at G and its penultimate step is raised by a semitone (in this case F becomes F # ). Going in the opposite (counterclockwise) direction (in our case from C to F) involves lowering the 4th step of the new scale (in this case in the resulting F-major scale the 4th step (B) is flattened to B b etc). This process can then be repeated until all 12 possible Major scales are created; it requires a minute of thought to verify that the additions of the sharps are indeed consistent with reduction of the flats as illustrated at the bottom of the circle. The construction of minor scales is slightly more complicated. For our purposes we only note that on the standard circle of 5ths the minor chord associated to C-major is A-minor as both of these contain the same Major third (C-E). In this sense, the minor chords are interesting hybrids: the C-minor chord has a fifth from the C-major, major third from the E b major chord, and the minor third from A b Major.

112 CHAPTER 5. MUSIC, CONSONANT AND DISSONANT Figure 5.4: Result of the classical experiment on the dissonance of pure tones by Plomp and Levelt.

5.3. TUNING AND TEMPERAMENT 113 Figure 5.5: The consonance curve (first used by the great Helmholtz) in its modern form. One tone with six harmonics is fixed at 250 Hz; the other is varied in frequency from 250 to 500 Hz. A simple algorithm evaluates contribution of each pair of partials. As expected, there are peaks of consonance when the two frequencies are in the ratio of small integers...

114 CHAPTER 5. MUSIC, CONSONANT AND DISSONANT Figure 5.6: Comprehensive summary of the physics aspects of musical intervals. Top: the harmonic spectrum, illustrating the numerous pure intervals that exist between various harmonics of a single tone. Table: The 12-tone scale, with frequency ratios and cents for Equal temperament, as well as for just tuning. Bottom: Summary of musical cents.

5.3. TUNING AND TEMPERAMENT 115 Figure 5.7: A little composition illustrating a consequence of the fact that the addition of the fact that the addition of a lesser diesis and greater diesis is very close to 100 cent. Within just a few bars the pitch climbs by a semitone, unbeknownst to anyone but possessors of the absolute pitch. I was quite fond of this exercise when I first wrote it some 30 years ago. But I soon found out that similar little pieces are known to have been written since at least the 17th century :-(

116 CHAPTER 5. MUSIC, CONSONANT AND DISSONANT Figure 5.8: Construction of the decaus tuning. The original book of Solomon decaus (1615) is really dense to read and interpret, so we gratefully follow the clear prescription by Donald Hall: build two major thirds up from F to get three notes F, A and C #. Then make a string of three fifths up from each of these three notes. First verify that this procedure indeed produces all the 12 notes of an octave, and determine the position of these notes in Equal Temperament, where a Major third is = 400 cents and each fifth is = 700 cents (first two columns). Then repeat the procedure in the Just Tuning, where the Major third = 5/4 and a fifth = 3/2. We see immediately that we get, by construction, 8 perfect major thirds and 6 perfect fifths. Note that continuing from the end of one column to the beginning of the next one, we do get a complete, contiguous circle of twelve 5ths.

5.3. TUNING AND TEMPERAMENT 117 Figure 5.9: A simple EXCEL evaluation of the quality of all the 12 Major and 12 minor chords in the decaus tuning. The meaning of the columns is: B: root of the chord of the name in column A C: deficiency of the major thirds of the major chord D: deficiency of the 5th (equal to the major and minor chord with the same root) E: deficiency of the major third of the minor chord.

118 CHAPTER 5. MUSIC, CONSONANT AND DISSONANT Figure 5.10: The Gotheborg Console while being completed, with extra keys such as were used in 18th century. Figure 5.11: A 19-tet design of Woolhouse (1835). Many such keyboards were contemplated or even built as early as 16th century. There are even two examples still in existence of a 53-tet monster (for centipedes, I guess).

Figure 5.12: Consonances and dissonances in a C Major chord as function of frequency. The harmonics C 0 1 C0 38 of the implied root C 0 1 of the C-major chord are equidistant and contain, as subsets, all harmonics of C1, E1 and G1. For simplicity the note C1 is tuned to 400 Hz. If the notes E1 and G1 are tuned in Equal Temperament relative to C1, their harmonics will shift as indicated (the marks pointing upwards correspond to the just tuning, the pointing downwards are for ET) Figure 5.13: The same display as above, but as function of log of frequency. The octaves are now equidistant but the harmonics are logarithmic. 5.3. TUNING AND TEMPERAMENT 119

Figure 5.14: Example of a Major chord with dissonances enhanced by suitable positions of individual notes 120 CHAPTER 5. MUSIC, CONSONANT AND DISSONANT

5.3. TUNING AND TEMPERAMENT 121 Figure 5.15: Circle of Fifths Revisited: History of Tuning and Temperament on one Figure. The deviations of the width of Major third in each of the 12 chords from the ratio 5/4 (in cents) for three tunings and three temperaments: 600 BC Pythagoras 14xx 1/4 comma meantone 15xx 1/6 comma meantone 17xx Valotti 18xx Equal 2004 Lehman

122 CHAPTER 5. MUSIC, CONSONANT AND DISSONANT Figure 5.16: An organ fingerprint of the small UW pipe organ (about 1,000 pipes) showing the tuning intent as well as the precision of its implementation, for all the Major chords of a pipe organ, on one slide. This was done just before a fresh tuning as is obvious from the scatter of the points. See the test for the description of the plots.