Scales_4 (2).pdf

Scales and Tuning, Part 4 Physics of Musical Sound, 33-114, Spring Semester, 2014 1. Crucial Intervals and Chords In Chapter 19 of the textbook, Hall talks about the important note progressions and chords in Western music. This includes the idea of a “tonic” note, which is the key for a major scale, and the “dominant” note, which is a Perfect Fifth above the tonic. The two most important 3-note chords are the Major and Minor triads. The Major triad is defined as a Major Third interval (4 semitones), followed by a Minor Third interval (3 semitones), for example C −E −G. The Minor triad is just the reverse, a Minor Third, followed by a Major Third, for example, C − E ♭ − G. In both cases, the interval between the lowest and highest notes is a Perfect Fifth (7 semitones). From our discussion of Just intervals, we know that the intervals in the Major and Minor triads use small integers in their frequency ratios, and are therefore the most consonant sounding. So we would like these very important intervals to have their Just ratios. In Part 3 of this note, we found that we cannot change keys in a Just diatonic scale and keep the frequencies of all the notes the same. But perhaps we can at least keep the Major and Minor triads in every key with their correct Just ratios of 5:4 for the M3, 6:5 for the m3 and 3:2 for the P5. 2. The Impossibility of Perfect Tuning The question we are asking is: can we produce a tuning scheme such that, in all 12 diatonic major scales, the Major and Minor triads have Just ratios? This question basically revolves around the tuning of the organ and the harpsichord. For 1000 years the organ was the dominant instrument for religious music, and for 500 years the harpsichord was the dominant instrument in secular music. So clearly their tuning scheme was very important. Neither instrument can be easily re-tuned, especially the organ where, once a length of pipe is cut, the pitch for that pipe is set forever. To accompany singers and other musical instruments, it was very desirable for the organ and harpsichord to be able to change keys. Yet we still want the pure tones of the Just intervals in the Major and Minor triads. This is our quest. In Section 18.4, Hall does a good job explaining the impossibility of our task. Please read that section. We will summarize the main arguments here. A) 3 Just Major Thirds 6= Octave Consider starting at any C note and going up by Just Major Thirds with a ratio of 5:4, corresponding to 386 cents. You would go through the notes C − E − G♯ /A♭ − C ′ . So three Major Thirds makes an Octave, which makes sense because we know a M3 is 4 semitones, so 3 Major Thirds would be 12 semitones, which is an Octave. However, the frequency ratio would be ( 54 )3 = 125 6= 2. In cents, this would be (3)(386 cents) = 1159 cents 6= 1200 cents. So 3 Just 64 Major Thirds are 41 cents short of a true Octave.

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B) 4 Just Minor Thirds 6= Octave We go through a very similar argument with Just Minor Third intervals. Since they are 3 semitones each, four of them should give an Octave: C −E ♭ −G♭ /F ♯ −A−C ′ . But (4)(316 cents) = 1263 cents, which is 63 cents larger than an Octave. C) 12 Just Perfect Fifths 6= 7 Octaves Imagine starting at some note, let’s use F3 as an example, and going up by 12 Perfect Fifths. You would get the following sequence: F3 − C4 − G4 − D5 − A5 − E6 − B6 − F7♯ − C8♯ − G♯8 − D9♯ − A♯9 − F10 . So a sequence of 12 Perfect Fifths gives 7 Octaves, which makes sense because a Perfect Fifth is 7 semitones, an Octave is 12 semitones, and (12)(7) = (7)(12). However, 12 Just Perfect Fifths corresponds to 12(702 cents) = 8424 cents 6= 7(1200 cents) = 8400 cents. We are 24 cents too high in this case. D) 4 Just Perfect Fifths 6= 1 Just Major Third Plus 2 Octaves The 3 problems which we found above all involved at least some notes with sharps or flats. So you might think that the problem is somehow related with the chromatic scale and not the diatonic scale. However, we can find another basic problem using only the notes in the diatonic scale in the key of C. Consider starting at the note C4 and going up to the note E6 using two different paths. In the first path, go up through four Just Perfect Fifths: C4 −G4 −D5 −A5 −E6 , which is a jump of 4(702 cents) = 2808 cents. For the other path, go up by a Just Major Third and then up 2 Octaves, giving the sequence: C4 − E4 − E5 − E6 . This is equal to (386 cents) + 2(1200 cents) = 2786 cents. The two paths should give the same frequency for the E6 note, but the two methods differ by 22 cents. Thus, we have found another discrepancy, this time using only the white keys on the piano. Note that A), B), and C) all involve “spelling” problems. In A), we need G# to finish a M3 starting on E, but Ab to start the one ending on C. If we didn’t make use of the enharmonic equivalent, we would get: C − E − G♯ − B ♯ . There is a similar problem in B), which would lead to using a Dbb . And in C), the last note should really be E # , not F , since a P5 above some type of A must be some type of E. Case D) actually spells out correctly. I don’t know of any deep significance to this, though it is the smallest of the four mis-matches in our list. Altogether, we have three examples of issues with proper interval spelling, and four examples related to very practical tuning problems. 3. The Tuning Mosaic From the 4 examples above, we see that our goal of making the Major and Minor triads Just in all 12 diatonic keys is impossible. Simply stated, the Just Perfect Fifth and Just Major Third interval ratios are incompatible with the 2:1 Octave ratio that we require. Or equivalently, no power of 3 is ever exactly some power of 2, etc. This means we are going to have to make compromises. The various discrepancies which we found above have to be resolved somewhere in any tuning scheme. It’s just a question of where we put them. (In 12-tone systems, it’s a question of where to put them. Or we could use more than 12 notes per octave, which has been tried over the years but is not a very easy keyboard to play. Electronic music offers new 2

opportunities for using such extended tunings, but we will not say more about them here.) Over the years, musicians have devised dozens of different tuning schemes to try to make the best compromise. We would like to have a way of graphically displaying these different tuning schemes, so that we can easily compare them. Hall uses one such pictorial representation in the textbook, but we will use a simpler version. We will show only P5 and M3 tunings: dropping the explicit m3 in Hall allows for a simpler rectangular grid. And we will show actual cents values, rather then deviations from Just, a minor (sic) change. The “tuning mosaic” we will use is shown as Figure 1. It is made up of 4 rows of notes, with 5 notes in each row. To reproduce the mosaic, you start at the left-hand end of the bottom row, with the F note. Each successive note in that row is then a Perfect Fifth higher (F − C − G − D − A). Once you have 5 such notes, the last note in the row repeats as the first note on the left end in the next higher row. You then continue making Perfect Fifth intervals, always repeating the last note on the right in any row with the first note on the left in the next higher row. The top row of the tuning mosaic is identical to the bottom row. You will not be expected to recreate the tuning mosaic, but it is important that you understand how it is constructed. Each note in the mosaic (for example, the E note) is a Perfect Fifth above the note to the left of it (A) and a Perfect Fifth below the note to the right of it (B). It is also a Major Third above the note directly below it in the same column (C), and a Major Third below the note directly above it in the same column (G♯ /A♭ ). Thus, if we take any note in the mosaic, along with the note to its right and the note above it, the 3 notes in the resulting right triangle are the notes in the Major triad for that key. For example, in the key of C, we have the diagram in Figure 2.

F

C

G

D

A

C#/Db

G#/Ab

D#/Eb

A#/Bb

F

A

E

B

F#/Gb

C#/Db

F

C

G

D

A

Figure 1: Blank tuning chart showing all perfect fifths and major thirds. The dashed line separates out the main portion and the repeated portion to make the repetitions clearer. In the little boxes between each pair of notes both horizontally and vertically, we will put the cents value for that interval. It is crucial that you memorize the cents values for 3

E m3 M3 C

P5

G

Figure 2: Tuning triad. the Just Perfect Fifth (702 cents) and the Just Major Third (386 cents). The boxes between notes in the same row will have a cents value close to the Just Perfect Fifth value of 702, and the boxes between notes in the same column will have a cents value close to the Just Major Third value of 386. Note that there are 24 boxes, not counting repeated ones, but there are only 11 choices to make in a 12-tone scale (given that an octave equals 1200), so there must be lots of constraints relating the values! That’s our next topic. 4. Filling in the Tuning Mosaic For any particular tuning scheme, there is always some general principle which assigns the cents values of many of the intervals in the tuning mosaic. The most obvious example of this is the 12-tone equal-temperament tuning scheme of the piano. As we know, in this system all Perfect Fifths are 700 cents and all Major Thirds are 400 cents. So its tuning mosaic looks like the pattern in Figure 3 repeated over and over. We won’t show the full 12-tet mosaic; it’s a bit boring.

E 400 C

700

G

Figure 3: 12-tet triad. To find the correct cents values for the tuning mosaic in other tuning schemes, we will use 5 different “rules” given below. Study them carefully, since you will be expected to know them and how to use them. 1. The cents values of the 12 Perfect Fifths in the mosaic must add up to 7 Octaves for a total of 8400 cents. If you start at the F note in the lower left-hand corner of the mosaic and go up 12 Perfect Fifths to the F note at the right end of the second row, you have gone up 7 octaves. Therefore, the cents values in the 12 boxes between those notes must add up to 8400 cents. Thus, if you know 11 of the values, you can then solve for the value 4

of the 12th , using this constraint. Mathematically, we could write this rule as: Σ12 i=1 P 5i = 8400.

(1 constraint)

Notice that the 12-tet scale satisfies this rule, since 12(700) = 8400. 2. The cents value for the Major Third interval between any 2 notes must equal the sum of the cents values of the 4 Perfect Fifths between those notes, minus 2 Octaves (2400 cents). To understand this rule, consider the F note. If we go vertically upward in the mosaic, we get to the A, which is a Major Third higher. But if we go horizontally by 4 Perfect Fifths we also get to the A, but this time 2 Octaves higher. So if we add up the cents values of the 4 Perfect Fifths and subtract 2400, we should get the same value as the Major Third. Thus, if you know 4 out of 5 of these values, you can solve for the fifth. Mathematically, we could write this as: M 3 = Σ4i=1 P 5i − 2400.

(12 constraints)

All the 12-tet Major Thirds satisfy this rule, since 400 = 4(700) − 2400. 3. The values of the 3 Major Third intervals in any column of the mosaic must add up to 1 Octave, or 1200 cents. Start with any note in the bottom row of the mosaic. Go up by 3 Major Thirds and you will reach the same note an Octave higher in the top row. So the sum of the 3 values must equal 1200. If you know any 2 of them, you can solve for the third. Mathematically, this is: Σ3i=1 M 3i = 1200.

(3 constraints)

The 12-tet Major Third intervals of 400 cents each obviously satisfy this rule since 3(400) = 1200. 4. In going from any note to the note diagonally above and to the right of it, you can go 2 different ways: either up and then over, or over and then up. The sum of the Perfect Fifth and Major Third values must be the same for both ways. Again, consider the F note. If we go up a Major Third to A and then up a Perfect Fifth, we get to E. But we can also first go up a Perfect Fifth to C and then up a Major Third to E. The two paths must give the same total, so the sum of the Perfect Fifth and Major Third values must be the same. Mathematically, this is: P 51 + M 31 = P 52 + M 32

(12 constraints)

5. The last rule is so obvious that sometimes you forget to use it. Any Perfect Fifth value you find in the bottom row of the mosaic also applies to the top row, since the 2 rows are identical. Similarly, any Major Third value you find for the left-hand column can also be put in one of the boxes in the right-hand column. I’ve ignored these trivial repeats when counting the number of constraints for the previous rules. As an example of how to use these rules, let’s find the tuning mosaic for the ancient tuning scheme called Pythagorean Tuning. Its general philosophy is to make as many of the Perfect Fifth intervals Just (i.e. equal to 702 cents) as possible . We cannot make all 12 Just, since we would violate the rule that the 12 P5’s must add up to 8400 cents. But we can make 11 of them 5

Just. Using rule 1), the 12th P5 then has a value of 8400 − 11(702) = 678 cents. This P5 can go anywhere in the mosaic, but it is usually put between G♯ (A♭ ) and E ♭ (D♯ ). This one distinct Perfect Fifth is called a “wolf”, because it is so flat (24 cents below Just) that it sounds as bad as a howling wolf. We can now use rule 2) to find the values of the Major Thirds. There are two situations. When you have 4 Just Perfect Fifths in a row, each equal to 702 cents, then the corresponding Major Third equals: 4(702) − 2400 = 408 cents. There are 8 such M3’s. However, when the “wolf” is one of the 4 P5’s, then the Major Third equals: 3(702) + 678 − 2400 = 384 cents. There are 4 such M3’s. So the Pythagorean tuning mosaic looks like the following:

F

702

384

C

702

384

G 408

C#/Db 702 G#/Ab 678 D#/Eb 408 A

408 702

408 F

E

702

C

B

702 A#/Bb

G

702

702 F#/Gb

702

D

F 384

702 C#/Db

408 702

A 408

384

408 702

D 408

384

408 702

702

408 702

A

Figure 4: Pythagorean tuning chart showing all perfect fifths and major thirds. It is sometimes useful (say, for doing sums in your head) to subtract off the average values and look only at differences, resulting in smaller numbers to work with. For example, the rule that 12 P5 sum to 8400 cents can be thought of as: Σ12 i=1 (P 5i − 700) = 0. This does not mean that the 700 cent value of 12-tet is “best”, but merely that it set to the average value required for any temperament. Thus in Pythagorean tuning, the 11 P5’s with an extra 2 cents imply that the final one must be 22 cents lower than the average of 700, or 24 cents lower than the Just value of 702 cents. Be very careful to not confuse the deviation from the average (−22 ¢) with the deviation from Just (−24 ¢) ! This distinction is even more significant for the M3 interval.

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5. Rating the Major Triads Once we have calculated the tuning mosaic for a certain tuning scheme, we can use it to determine how well all the 12 Major triads have been tuned. The criteria is based on how close the Perfect Fifth and Major Third in each Major triad come to their Just values of 702 and 386 cents. How close the interval must be in order to be considered good, acceptable or bad is somewhat subjective. However, because the Perfect Fifth has more higher harmonic frequencies which are close to overlapping than the Major Third, it’s clear that we demand a smaller deviation from Just for the Perfect Fifth to sound “good” than for the Major Third. We will use the criteria given by Hall in Problem 17) from Chapter 18 in the textbook. For a particular Major triad, first define m and n as the difference between each interval and its Just value: m = | P 5 − 702 | and n = | M 3 − 386 | . We then rate the Perfect Fifth and Major Third intervals using the following criteria: Perfect Fifth Major Third Rating Good m < 6 cents n < 11 cents Acceptable 6 ≤ m < 12 cents 11 ≤ n < 30 cents Bad m ≥ 12 cents n ≥ 30 cents Finally, the overall rating for the Major triad is the poorer of the two ratings for the Perfect Fifth and Major Third intervals. This means for the triad to be considered “good”, both intervals must be good, and to be considered “acceptable”, both intervals must either be acceptable, or one must be good and the other acceptable. If either is “bad,” then so is the triad. Instead of the designation “acceptable”, Hall uses “dubious,” but I think this is too harsh since all the 12-tet Major triads, with n = 400 − 386 = 14 cents, are certainly considered “acceptable.” Using this procedure for the Pythagorean tuning scheme from above, we find that there are 3 good triads with m = 0, n = 2: C ♯ /D♭ , B and F ♯ /G♭ . There are 8 acceptable triads with m = 0 and n = 22: F, C, G, D, A, E, D♯ /E ♭ and A♯ /B ♭ . And there is 1 very bad triad with m = 24 (the wolf) and n = 2: G♯ /A♭ . I will give you the table above, but be sure you understand the procedure for rating the 12 Major triads in any tuning scheme. 6. Various Tuning Schemes Hall does a good job describing the different classes of tuning schemes in section 18.5 of the textbook. Please read this section carefully. He shows his tuning mosaic for many of them, and also in Table 18.3 gives the actual cents values of each note in the scale and the quality of their triads. Note that Hall determines the quality of not only the 12 Major triads but also the 12 Minor triads. But since every interval in a Major triad is also in some other Minor triad, the total number of good, acceptable and bad triads is just 2 times the number for the Major triads. There are 4 main classes of tuning schemes, each having a different underlying philosophy or method. They include (I interchanged the second and third compared to Hall):

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A) Pythagorean tuning, which we have already discussed. It tries to make as many Just Perfect Fifths as possible. The Good/Acceptable/Bad results for the Major triads are 3/8/1, with the one bad triad being very bad. This was a standard tuning in early Medieval music. B) The second class, called Just tuning, uses a mixture of Just Perfect Fifths and Just Major Thirds. Hall describes two different schemes, named after their supposed inventors, Ramos and de Caus. There are many other possible variations; for example, our list of Just intervals in these notes is equal to the version of de Caus transposed down by a M3. Other varieties have more essential differences. Ramos is unusual in some fascinating ways, but we don’t have time to delve into that in detail here. The Major triad rating in the de Caus scheme, for example, is 6/0/6. C) The third class is called meantone tuning. There are several variations on this scheme. Perhaps the most common, 1/4-comma meantone tuning, sets 11 of the 12 Perfect Fifths to the same value (which turns out to be 697 cents). It does so to make as many Just Major Thirds as possible, which turns out to be 8. The resulting triad quality is 8/0/4. As long as you stay away from the 4 horrible keys, this is a good tuning which was often used in Elizabethan music of the early Renaissance. In general, the various meantone schemes are labeled according to how much lower than Just the Perfect Fifths are made as a fraction of the 22 cents discrepancy we found earlier between a sequence of four Just Perfect Fifths and a Just Major Third plus two octaves. This discrepancy is given the obscure name of “syntonic comma.” So, for example, in the 1/4-comma meantone tuning described above, the 11 Perfect Fifths are about 5 cents below Just (697 cents), which is 41 of the 22 cent comma. The other part of the name (meantone) arises since the D note is midway between C and E in cents (unlike the Just diatonic scale). So the two tones C − D and D − E are set to the mean value allowed by the just C − E interval. 1 −comma Meantone In this language, the 12-tet tuning scheme of the piano is approximately 11 1 tuning, since all of the Perfect Fifths are adjusted to 2 cents below Just, which is about 11 of 1 22 cents. Strictly speaking, they are flat by exactly 12 of the 24-cent “Pythagorean comma”. D) The last tuning class is given the general name of “temperaments,” a term we have seen before with the 12-tet system. The name means to mis-tune or “temper” the notes away from their Just ratios. Meantone schemes other than simple 1/4-comma don’t have many if any just ratios, and are perhaps better thought of as temperaments. But they have a more regularity than most temperaments, which are often more “irregular”. The Holy Grail of tuning was to find a scheme in which all 12 Major triads would be either good or, at least, acceptable, and also have as many Just intervals as possible. If this were accomplished, then all the major and minor keys could be used to compose a piece of music. Such a tuning system is called a “circulating temperament.” The names can be a bit fluid. 12-tet is a temperament, but in many ways like meantone. Pythagorean is sort of a special type of Just, obsessed with Fifths and not concerned at all about Major Thirds. Let’s discuss one example of a circulating temperament in detail. In 1/4-comma meantone, we set 11 P5 equal and this gave many Just M3. The P5 were flat by 1/4 of the 22-cent syntonic comma. The result is that the final P5 is a very bad wolf of 738 cents. This is not acceptable in a circulating temperament. Our 4P 5 = M 3 + 2 · 8ve rule also implies that this wolf ruins 8

four M3 as well, leading to the 8/0/4 rating. Since we have to absorb the 24 cent Pythagorean comma (the difference of 12 P5 and 7 Octaves) somewhere in the circle of fifths, circulating temperaments are generally easier to discuss in terms of this slightly larger comma (which can be confusing, but it is only slightly different). So to avoid the wolf, one must temper the P5 by less than 1/4 comma (such as 12-tet), or temper fewer of the P5 by some amount, as in the following example. A circulating temperament was found by Andreas Werckmeister in 1691. It might be called a Modified 41 −Comma Meantone tuning, since 8 of the Perfect Fifths are Just and the other 4 are tuned 24/4 cents flat. The tuning mosaic for it is shown below. The Major triad rating for this tuning is 4/8/0, so it is a circulating temperament. Many people believe that this is the tuning used by Johann Sebastian Bach in composing his famous Well-Tempered Clavier for keyboard (1722). This piece consists of 24 preludes and fugues, each pair of which is written in a different one of the 24 major and minor keys, showing that a tuning scheme had been found which allowed such a composition. Please be aware that Werckmeister created many temperaments; Hall’s version is different than the one given here, for example. Even when they are numbered, the nomenclature for numbering them varies. And many other proposals for Bach’s actual tuning have been made (by Kellner, for example) and speculation continues to this day. We will probably discuss some recent work of Bradley Lehman in class.

F

702

408

C

696

408

G 402

C#/Db 702 G#/Ab 702 D#/Eb 402 A

402 702

390 F

E

702

C

B

702 A#/Bb

G

696 F#/Gb

696

D

A 390

702

F 408

702 C#/Db

396

Figure 5: Werckmeister tuning chart.

9

696

408

396 696

D 396

402

390 702

696

402 696

A