Language Log

Pavarotti and the crack to chaos

The death of Luciano Pavarotti has brought out some of the hallucinatory anatomy, physiology and acoustics that singers and their teachers use to describe what they do. But it also brought a note from a reader with an interesting question about the emotional content of musical intervals and scales.

Daniel J. Wakin's "High C: The Note That makes Us Weep", NYT, 9/9/2007, contains a few choice examples of vocal metaphors and allusions:

“The reason it’s so exciting to people is, it’s based on the human cry,” said Maitland Peters, chairman of the voice department at the Manhattan School of Music. “It’s instinctual. It’s like a baby. You’re pulled into it.” When a tenor sings a ringing high C, it seems, “there’s nothing in his way,” Mr. Peters said. [...]

Mr. Peters ... said the chest voice, the strongest source of sound, and the head voice, where the sound vibrates in the head’s cavities, must be perfectly balanced. The base of the tongue, the jaw, the larynx must all lie in just the right position, unrestricted by tension.

Mr. Pavarotti once described the feeling this way: “Excited and happy, but with a strong undercurrent of fear. The moment I actually hit the note, I almost lose consciousness. A physical, animal sensation seizes me. Then I regain control.”

The tenor Juan Diego Flórez, also acclaimed for his top, will sing “Fille” at the Met in April. He said in an interview on Thursday that he imagines a keyboard in his head, and reaches for the note there.

“You think very high,” he said. “You give a lot of space in your throat.”

But what led a reader to write me about this article was not the vocal-technique metaphors, but the echo of Baroque Affektenlehre in this remark about pitch and key:

The pitch, in itself, has a satisfying quality. The key of C major, after all, is a stable, cheerful, happy key, the one with no sharps or flats.

Lane Greene wrote:

As a musician I'm a bit struck by the bit about C major being the happiest of all keys, with no sharps or flats. No individual note, sharp or flat, should be any happier than any other, right? Any major key has the same interval between its notes - A-flat major should be every bit as happy as C major... or is there something I don't know about flats and sharps?

There's the classic line in Spinal Tap where Nigel describes D-minor as "the saddest of all keys". It's ridiculous because anyone knows D-minor shouldn't be any sadder than G-minor. Or not?

Well, not, at least historically. As the article on "Key" in Grove's explains:

Keys are often said to possess characteristics associated with various extra-musical emotional states. While there has never been a consensus on these associations, the material basis for these attributions was at one time quite real: because of inequalities in actual temperament, each mode acquired a unique intonation and thus its own distinctive ‘tone’, and the sense that each mode had its own musical characteristics was strong enough to persist even in circumstances in which equal temperament was abstractly assumed.

To see where this comes from, consider the circle of fifths, which leads us conceptually through the 12 chromatic keys of recent western music:

A perfect fifth corresponds to a pitch ratio of 3 to 2, and successive musical intervals correspond to multiplication of ratios, so that 12 perfect fifths makes a ratio of 3/2 raised to the 12th power, or about 129.7463.

This is unfortunate, because the assumption built into the circle is that 12 fifths equals 7 octaves, bringing us back to the same pitch class we started from, 7 octaves up. But the interval of an octave is a ratio of 2 to 1, and 2 to the 7th power is 128.

This divergence between 12 fifths and 7 octaves -- ((3/2)^12)/(2^7) ≅ 1.013643 -- has been known for more than two millennia as the Pythagorean comma. It's one of three crucial flaws in the mathematical fabric of reality that apparently formed part of the esoteric lore of the Pythagoreans. According to Thomas McEvilley, "The Shape of Ancient Thought: Comparative Studies in Greek and Indian Philosophies",

It requires a leap of horizon to understand the intensity with which such things mattered to ancient thinkers. ... The issue which made [the Pythagorean comma] so pressingly important was nothing less than the question .. whether reality is mathematical or not.

When Pythagoras discovered (or learned) the so-called Pythagorean Theorem, ... it is said that he hastened to sacrifice oxen. he felt that he had touched on a power center in the mathematical fabric of the universe. ...

The Pythagorean Theorem is the threshold to the discovery of irrational numbers and incommensurable lengths -- a discovery which Hellenists attribute to Hippasus of Tarentum, a renegade Pythagorean whom, according to one account, Pythagoras pushed off a boat for revealing to outsides the tragic secret of the Pythagorean Theorem, which was irrationality or incommensurability. ... The discovery that the side and diagonal of a square will always be incommensurable produced an ideological convulsion in the Pythagorean order comparable to the shock conveyed by the discovery of the Precession or the Pythagorean comma. ... Like the Precessional drift and the Pythagorean comma, this apparent crack or gap in the mathematical fabric of the universe seemed ominous, as if such cracks lead through the membrane of order to chaos. They deny that the universe is orderly and hence that it is cognizable, and thereby remove credibility from all human thought. The Precession threatens the calendar and all the depends on it, and through the Pythagorean comma, as through a crack to chaos, the plethora of untuned sounds that could disrupt the harmony of the universe flows in.

The notion that small-integer musical intervals also play an important role in the melody of speech recurs stubbornly. Sometimes the idea is that different emotional or attitudinal states are associated with different musical intervals -- I posted last year about some Dutch research that claimed to find that sad people speak in minor keys ("Poem in the key of what", 10/29/2006). Another recurrent idea is that different languages have different characteristic intervals or scales -- R.A. Hall once argued that Sir Edward Elgar never because popular outside of the U.K. because he favored intervals that are peculiarly common in British speech ("Elgar and the intonation of British English", Gramophone 31(6), 1953). (Actually, in fairness to Hall, he claimed only that Elgar's frequent use of melodic leaps echoed the typically wide pitch range of British speech.)

In evaluating these ideas, I think we can safely say that if Pythagoras had based his cult on the role of number theory in human speech, the problems that led to Hippasus of Tarentum's unfortunate maritime accident would never have arisen. If there is any secret knowledge here, it would have to be a clever method for finding small integers in the intervals of speech, in the face of the straightforward observation that linguistic tone and intonation (appear to?) involve glissandi among freely gradient pitch values.

Returning to that embarrassing Pythagorean comma, to see what it means for the scales in different keys, consider what a properly-tuned (i.e. "just") diatonic scale is like. The scale degrees correspond to small-integer ratios of pitches, as indicated in the table below.

	Interval name	Just interval	Just interval relative to 1
C	unison	1:1	1
D	(major) second	9:8	1.125
E	(major) third	5:4	1.250
F	fourth	4:3	1.333...
G	fifth	3:2	1.500
A	sixth	5:3	1.666...
B	major seventh	15:8	1.875
C	octave	2:1	2

Within the diatonic scale, some of the internal intervals work out exactly -- thus F to C is (2/1)/(4/3) = 3/2, just as it should be, and and G to D is also (18/8)/(3/2) = 3/2.

But there are worrisome symptoms already here. If we go up by a perfect fifth from D to A, for example, we would get (3/2)*(9/8) = 27/16 relative to C, which is not the 5:3 ration of a perfect sixth. Turning it around the other way, the interval between justly-tuned D and justly-tuned A (in the key of C) is (5/3)/(9/8) = (5/3)*(8/9) = 40/27, which is by no means 3/2. We'll see in a minute what this means for other intervals, if we move to the key of D major without re-tuning.

If we build on these intervals to fill in the rest of the justly-tuned chromatic scale we get something like this:

	Interval name	Just interval
C	unison	1/1
C#	minor second	16/15
D	major second	9/8
D#	minor third	6/5
E	major third	5/4
F	fourth	4/3
F#	diminished fifth	7/5
G	fifth	3/2
G#	minor sixth	8/5
A	major sixth	5/3
A#	minor seventh	16/9
B	major seventh	15/8
C	octave	2/1

But now consider the derived ratios in (for example) the key of D major, compared to C major:

	Interval name	Just interval	Just interval relative to 1		Derived interval	Derived interval relative to 1
C	unison	1:1	1	D	(9/8)/(9/8) = 1:1	1
D	(major) second	9:8	1.125	E	(5/4)/(9/8) = 10:9	1.111...
E	(major) third	5:4	1.250	F#	(7/5)/(9/8) = 56:45	1.244...
F	fourth	4:3	1.333...	G	(3/2)/(9/8) = 4:3	1.333...
G	fifth	3:2	1.500	A	(5/3)/(9/8) = 40:27	1.481..
A	sixth	5:3	1.666...	B	(15/8)/(9/8) = 5:3	1.666...
B	major seventh	15:8	1.875	C#	(2*16/15)/(9/8) = 256/135	1.896...
C	octave	2:1	2	D	(2*9/8)/(9/8) = 2:1	2

Four of the seven scale intervals are different!

On this approach, each of the twelve chromatic keys will contain different internal intervals.

Violinists and singers can re-tune the intervals when they modulate, but keyboard players and players of fretted string instruments (like viols) are stuck, and wind players are limited in what they can do to change the pitches they play.

Because some of the keys that result from this problem are not just different, but unpleasantly sour-sounding in some of their crucial internal intervals, various schemes have been developed over the centuries to "temper" the tuning of the different chromatic scale degrees, so as to spread the problem across different key signatures to some extent. The most consistent method for doing this is "equal temperment", in which all twelve semitones are set exactly as the twelveth root of 2, a ratio of approximately 1.0595:1. Thus a tempered fifth is not 1.5:1, but rather 2^(7/12) ≅ 1.498; a tempered major third is not 1.25:1, but 2^(4/12) ≅ 1.260;1

In equal-tempered tuning, the internal intervals of all keys are exactly the same. But equal temperment didn't become the norm until the 19th century -- before that, other systems of temperment were used, in which the internal intervals in different keys in fact were different.

It's probably because of this that the Baroque era's "Theory of the affects" (Affektenlehre) included a component based on choice of key. According to Grove's definition, this was

In its German form, a term first employed extensively by German musicologists, beginning with Kretzschmar, Goldschmidt and Schering, to describe in Baroque music an aesthetic concept originally derived from Greek and Latin doctrines of rhetoric and oratory. Just as, according to ancient writers such as Aristotle, Cicero and Quintilian, orators employed the rhetorical means to control and direct the emotions of their audiences, so, in the language of classical rhetoric manuals and also Baroque music treatises, must the speaker (i.e. the composer) move the ‘affects’ (i.e. emotions) of the listener. It was from this rhetorical terminology that music theorists, beginning in the late 16th century, but especially during the 17th and 18th centuries, borrowed the terminology along with many other analogies between rhetoric and music. The affects, then, were rationalized emotional states or passions. After 1600 composers generally sought to express in their vocal music such affects as were related to the texts, for example sadness, anger, hate, joy, love and jealousy. During the 17th and early 18th centuries this meant that most compositions (or, in the case of longer works, individual sections or movements) expressed only a single affect. Composers in general sought a rational unity that was imposed on all the elements of a work by its affect. No single ‘theory’ of the affects was, however, established by the theorists of the Baroque period. But beginning with Mersenne and Kircher in the mid-17th century, many theorists, among them Werckmeister, Printz, Mattheson, Marpurg, Scheibe and Quantz, gave over large parts of their treatises to categorizing and describing types of affect as well as the affective connotations of scales, dance movements, rhythms, instruments, forms and styles.

Even after the adoption of equal temperment, there remain several possible reasons for associating moods and attitudes with keys. From the point of view of performers, different keys can lie very differently on their instrument, and therefore feel different to play. And from the point of view of some listeners, those with perfect pitch and synesthesia, different keys may also evoke very different associations. Even if all the intervals are tempered and therefore mathematically identical in every key, the pitches themselves are different.

[Jonathan Knibb writes:

This isn't an especially original or deep observation, but as a coda to your excellent post on key and temperament it would be worth mentioning that, whether or not there is any auditory reality to the perceived affective differences between keys, a widespread belief in such differences can become a self-fulfilling prophecy, as composers choose keys they feel appropriate to the nature of their music. Of course, this could happen also without any such belief, as long as a sufficient proportion of influential works appear to support certain associations. To take the quoted example of D minor, many of Mozart's works in that key (the opening of Don Giovanni, the piano concerto in that key, the Requiem, etc.) share a certain affective quality, difficult to define in words but distinct from his use of say G minor, and it would be surprising if that fact did not influence its later use, explicitly or subconsciously, whatever Mozart's own feelings may have been.

]

[Ray Girvan writes:

Another major mover in forming these "pitch = particular emotion" conventions was John Curwen, who popularised the Sol-fa system as a mnemonic to teach music to singers not up to reading a score.

Curwen stated (I don't know on what grounds) mental effects for each note of the scale: "Doh, the strong or firm tone; ray, rousing and hopeful; me, steady and calm; fah, desolate or awe-inspiring; soh, grand or bright tone; lah, sad or weeping tone; te, piercing or sensitive tone".

The Sol-fa system was immensely influential, and so many singers must been have told this as fact that I could well believe, as Jonathan Knibb says, that it could have turned into self-fulfilling prophecy.

]

[Randy Alexander writes:

Even in equal temperament, keys can have different characteristics simply because one key is higher or lower than another key. The dominant to tonic relationship in G is far away from the same dominant to tonic relationship in C. A melody sung in one key has notes that are easy, hard, resonant, not so resonant, etc. The head voice vs. chest voice mix will be different for every note, giving each key a markedly unique quality. Even on a piano, keys that are relatively higher are perceived as hollow, and relatively lower keys are muddy. For composers, choice of key has a strong relationship with what the music is actually doing. Simply because of range, some things sound great in one key but ridiculous in another.

True, but people who feel that different keys have different affects seem to feel that way about (say) B major vs. C major, where the range difference is small.]

Posted by Mark Liberman at September 9, 2007 07:31 AM