Lecture 5: Tuning Systems

Lecture 5: Tuning Systems In Lecture 3, we learned about perfect intervals like the octave (frequency times 2), perfect fifth (times 3/2), perfect fourth (times 4/3) and perfect third (times 4/5). When tuning the strings of a musical instrument like a guitar or a piano, it s preferable to use these perfect intervals, because two strings with fundamental frequencies a perfect interval apart will share many harmonics. Thus, we can tune them to a perfect interval by using beats to make sure that the harmonics line up. So, perfect intervals are a natural choice for dividing the octave into a musical scale. How do we do this? That is, if you set the frequency of middle C, how should the frequencies of A, B, D, E, F, and G be chosen? The following construction of an ancient Greek version of the 7-note major scale, called Pythagorean tuning, is based on a discussion in Measured Tones. The Greeks began their scale on what is now the note D in the standard C-major scale. Suppose that this starting note has frequency 1; it won t really, but the frequency of every other note will be obtained by multiplication from this one. We want to construct a scale of notes whose frequencies are between 1 and 2 (an octave above). Let s take as our basic note-construction procedure going up or down a perfect fifth; this means multiplying the frequency by either 3/2 or 2/3. If a note gets too high or too low, we ll use octave equivalence (i.e., multiplying or dividing by powers of 2) to put the frequency back in the interval [1, 2]. First, we add a note A a perfect fifth above D, with frequency 3/2, and a note G a perfect fifth below D, with frequency 2/3: note G D A frequency 2/3 1 3/2

Next, we add notes E a perfect fifth above A and C a perfect fifth below G. The notes C and G are below the frequency of our starting D, so we multiply their frequencies by 2 enough times to get them into the desired octave. We do the same to the frequency of E, bringing it down an octave. Now we have a scale with 6 notes from D to D: note D E G A C D frequency 1 9/8 4/3 3/2 16/9 2 This is known as a pentatonic scale because it has 5 distinct pitch classes. This scale occurs with surprising frequency in both Western and Asian folk music. For example, the use of the pentatonic scale is so universal in Chinese folk music that they just use numerals 1 though 5 to stand for notes, and dispense with staves. To get the full Pythagorean scale, we just need to go one perfect fifth above the E (to get B) and one perfect fifth below C (to get F). Then we have the following 7-note scale: note D E F G A B C D frequency 1 9/8 32/27 4/3 3/2 27/16 16/9 2 This scale has some nice properties: First, there are only two sizes of steps from one note to the next. The steps D-E, F-G, G-A, A-B and C-D all come by multiplying by 9/8. The steps E-F and B-C come by multiplying by 256/243. Moreover, since (256/243)^2 1.11 while 9/8 =1.125, two small steps are approximately the same as one large step. So, it makes approximate sense to call them half-steps and whole-steps. Note that the scale we have constructed is called the Dorian mode; other modes of ancient music come by taking the Dorian mode and shifting the starting point to a different note. In particular, the Aeolian mode is the analogue of the modern C-major scale. We can construct a Pythagorean major scale, starting on a note C with frequency 1, by taking appropriate whole- and half-steps up from C:

note C D E F G A B C frequency 1 9/8 81/64 4/3 3/2 27/16 243/128 2 If we wanted to tune a piano according to this Pythagorean major scale, this tells us how to tune all the white keys, but what about the black keys? In other words, if we ve divided the octave into 5 whole steps and 2 half-steps, then we want to find the notes that fit inside the whole steps. To continue with our basic procedure, we go up a perfect fifth down from F, to get a note B with frequency 16/9, between A and B. Similarly, we add a note F# between F and G, by going a perfect fifth up from B. Since 243/128 =3^5/2^7, this F# has frequency 3^6/2^9. Next, go a perfect fifth down from B to get a note E, between D and E, with frequency 32/27=2^5/3^3 Then, go a perfect fifth up from F#, to get a note C# between C and D, with frequency 3^7/2^11. Next, go a perfect fifth down from E to get a note A, between G and A, with frequency 2^7/3^4=1.5802 Then go a perfect fifth up from C# to get a note G#, also between G and A, with frequency 3^8/2^12=1.6018. So, the Pythagorean procedure of continually adding new notes eventually generates inconsistent ways of filling in the gaps. What happened is that going up 6 perfect fifths from D and going down 6 perfect fifths from D generates two notes that are almost, but not quite, octave equivalent. Put another way, going up a perfect fifth 12 times gets you to a note which is not quite 7 octaves above where you started: (3/2)^12 2^7, but (3/2)^12 is about 1.3% more than 2^7. [see Benson, figure 5.3] This ratio (3/2)^12 / 2^7 = 1.0136 is known as the Pythagorean comma; it s the ratio of G# to A in the scale we just constructed. All right, suppose you ignore these difficulties, because you just want to play pieces in one key, without using exotic notes like G# or A.

Even so, there are still unsatisfactory features of the Pythagorean scale. Because we tried to have as many perfect fifths as possible, we don t have perfect thirds or sixths. Recall that a perfect third is the ratio 5/4 and a perfect sixth is 6/3. The interval from C to E in the Pythagorean scale is 81/64 = 1.265 whereas 5/4=1.25. That s about a 1% error, and the sixths are off by the same amount. This actually makes an audible difference when you play a major triad. [play Mathematica example: Pythagorean versus perfect CEG] Again, it s more convenient to have perfect 3rds so that you can easily tune a musical instrument to the scale. So, in order to have some perfect thirds and some perfect fifths, we might choose a different compromise. Suppose we begin at C, pretending it has frequency 1, and then require that notes E and G above it be a perfect third and perfect fifth away: note C E G C frequency 1 5/4 3/2 2 Next, starting on G we construct another triad G-B-D that s made up of a perfect third and a perfect fifth. (Because D is beyond an octave above C, we divide by 2 to put it in range.) Now we have note C D E G B C frequency 1 9/8 5/4 3/2 15/8 2 Of course, we get the same D as the Pythagorean scale, but the B is different. Then, we construct another perfect triad that s a perfect fifth below C. note C D E F G A B C frequency 1 9/8 5/4 4/3 3/2 5/3 15/8 2 Now we have a 7-note scale like the Pythagorean major scale. This is called the Ptolemaic major scale (see Johnston) and is the starting point for a family of tuning systems known collectively as just intonation. Notice the following interesting feature of this scale: there are 3 different possible values for the ratio of one note to the next one. The steps C-to-D, F-to-G and A-to-B are 9/8 ratios (same as the Pythagorean whole tone) but the steps D-to-E and G-to-A are

10/9, a slightly smaller ratio. The ratio between the larger and smaller steps is (9/8)/(10/9) = 81/80, and this is known as the syntonic comma. The smallest steps E-to-F and B-to-C are 16/15, a bit larger than the Pythagorean semitone of 256/243. [play scales in both systems] There are many ways to fill out a 12-note chromatic scale starting from here, some of which we will discuss later on. But I want to pass on to the system in use today. Historically, the Ptolemaic scale (in various forms) was commonly used in Western music up until the 18th century. Notice that this scale is designed so that the three most commonly used major chords---on C, G and F---sound perfect. But what if you want to modulate, or play a song in some other key, using the same instrument? What usually happens is that suddenly those major chords which sounded good in the key of C sound increasingly awful in other keys. For example, the interval D-to-A is (5/3)/(9/8)=40/27=1.481, about 1.2% flatter than a perfect fifth, so any attempt to play in the key of D will fall flat (so to speak). For example, compare the triad on A (a pure minor chord) with the triad on D: [play in Mathematica] The solution to this problem, which allows us to move freely among keys, is to give up on having perfect intervals. How do we do this? We saw when constructing the Pythagorean chromatic scale that going up by 12 perfect fifths is not quite equal to 7 octaves. (The difference is the Pythagorean comma.) So, as our scale-construction ratio, let s replace the 3/2 by a number whose 12th power will be exactly 7 octaves, namely 2^(7/12), or the 7th power of 2^(1/12) For convenience, let the letter r stand for 2^(1/12). In this system, known as equal temperament, we go up a fifth by multiply by r^7. Starting on C with frequency 1, G has frequency r^7, the D above has frequency r^14 (which we divide by 2=r^12 to get r^2 for a lower D), the next note A has frequency r^9, and so on. We get the following

chromatic scale: note C C# D D# E F F# G G# A A# B C freq 1 r r^2 r^3 r^4 r^5 r^6 r^7 r^8 r^9 r^10 r^11 r^12=2 In the equal temperament system, every semitone is equal to the ratio r, and every whole tone is equal to r^2. Comparing the semitones and whole tones with the other systems, we have Pythagorean Ptolemaic Equal Temp. semitone 256/243=1.0535 16/15=1.06666 r=1.05946 whole tone 9/8=1.125 9/8 or 10/9=1.11 r^2=1.1225... so that the scale steps for equal temperament are between those for the previous two systems. The real advantage of this compromise is that all the keys have exactly the same interval ratios, given by powers of r. In every key, a major third is r^4 (4 semitones) and a major sixth is r^9. So, the scale does not change shape when you change keys. The disadvantage is that basic intervals like the third and the fifth are slightly discordant. The difference is 3/2 /(r^7) = 1.00113..., about a 10th of 1 percent. Can you hear the difference? [play examples] ============================== Irrational Numbers As you know, rational numbers are fractions of integers; these play a key role in constructing the Pythagorean and just intonation tuning systems. Irrational numbers are everything else. How do we know such numbers exist? Because we can describe them, and prove that they can t be fractions of integers. Here is the grandfather of all examples of irrational numbers: the square root of 2. We prove that this can t be a rational number by assuming it is, and then reaching a contradiction.

Assume that 2 = p/q, where p and q are relatively prime integers (i.e., gcd(p,q)=1). Squaring both sides and clearing denominators gives 2 q^2 = p^2. Now, because 2 divides the LHS it must divide the RHS, i.e. 2 p^2. This means that p^2 is an even number, so p must be an even number. Hence, 4 divides p^2 and so 4 divides 2q^2. Hence 2 divides q^2, and so 2 divides q. But we assumed that p and q have no common factors! Now, any rational should be expressible as p/q with no common factors; but somehow the specific equation 2=p/q contradicts this. So, we must be incorrect in assuming that 2=p/q is possible for integers p and q. Aside: One important fact that s behind the above argument is that if 2 divides p^2 then 2 must divide p. This is a special case of the following fact about prime numbers: Proposition. If k is a prime number and k divides the product of integers p and r, then k divides p or k divides r. So, now we know that 2 is irrational. This in turn implies that the ratio 2^(1/12) that s at the heart of the equal temperament system is also irrational. Again, we can prove this by contradiction: Suppose that 2^(1/12) equals a rational number p/q. Then raising it to the sixth power gives 2 = p^6/q^6. But this is impossible since we already know that 2 can t be equal to a fraction of integers! ============== Cents As you have seen, musical intervals are really ratios of frequencies, and taking successive intervallic steps really means multiplying the corresponding ratios. In order to turn these operations into addition, we introduce a way of measuring intervals using logarithms, and these are measured in units called cents. This is specifically adapted to the equal temperament system, so that 100 cents is the same as a semitone (i.e., the ratio 2^(1/12)); this is where the name cent comes

from. If x is a ratio of frequencies, then we convert that into cents using the formula c = 1200 log_2(x), where log_2 indicates log base 2. In terms of the usual base 10 logarithm, log_2(x) = log(x)/log(2). For example, a Pythagorean semitone 256/243 corresponds to 1200 log(256/243)/log(2) = 90.2 cents, which shows that the Pythagorean semitone is about 10% flatter than the 100-cent semitone. Assignment: Read Harkelroad, Chapter 3 1. Suppose you want to tune a musical instrument to a major scale starting on the note A with frequency 220Hz. To what frequency would you tune the note F# above A, in the Pythagorean, Ptolemaic and equal temperament systems? 2. Calculate the values of the Pythagorean comma and the syntonic comma in cents. 3. Suppose we fill out the Ptolemaic major scale to get a 12-note chromatic scale in the following way: say that F# is a perfect third above D, C# is a perfect third above A, E is a perfect third below G, A is a perfect third below C, and B is a perfect third below F. (a) Express the difference between E in this system and in Pythagorean system in cents, and in syntonic commas. (b) Do the same for C#. 4. Write your own description (about 300 words long) describing the differences between the Pythagorean, Ptolemaic and equal temperament tuning systems. 5. Prove that 5 is irrational, and explain why this implies that the golden ratio phi must also be irrational.