The claims for a pitch

The teaching

As the acoustics of pitch and tuning from the monochord to equal temperament teaches it.

What a pitch is

A pitch is a vibration repeated so many times a second, counted in hertz. The ear hears from about twenty to twenty thousand vibrations a second, and it hears pitch by ratio, not by difference: doubling the rate gives the octave, the same note higher, whether from 110 to 220 or from 440 to 880. Every musical system begins from this, and every interval is a ratio between two rates.

The ratios of the monochord

Greek tradition tells that Pythagoras, passing a smithy, heard hammers ringing in harmony and found the reason in their weights. The tale is a legend, but the ratios it teaches are true, and the monochord, a single string over a movable bridge, shows them. Half the string sounds the octave, two to one. Two thirds of the string sounds the fifth, three to two. Three quarters sounds the fourth, four to three. The simpler the ratio, the smoother the interval, because more of the two notes' overtones coincide.

The comma

Build a scale from pure fifths alone, as Pythagorean tuning does, and twelve fifths almost but not quite come back to the starting note seven octaves higher: they overshoot by a small interval, the Pythagorean comma, about a quarter of a semitone. Tune the thirds pure instead, at five to four, and other intervals go sour. No tuning can make every interval pure in every key, and the history of tuning is the history of where to put the error.

Meantone, well and equal temperament

Renaissance keyboards used meantone, which narrows the fifths a little so that the major thirds come out pure, at the cost of a few keys that cannot be used and one fifth so far out that it howls and is called the wolf. Baroque musicians used well temperaments, which spread the error unevenly, so that every key could be played and each kept a colour of its own. Equal temperament, the modern system, divides the octave into twelve equal semitones, each a ratio of the twelfth root of two. Its fifths are a hair narrow, about two hundredths of a semitone, and its major thirds noticeably wide, about a seventh of a semitone; in exchange every key is alike.

Cents

Musicians measure small intervals in cents, each a hundredth of an equal semitone, twelve hundred to the octave. A trained ear can tell two notes a few cents apart when they are played one after the other. The step from A at 440 to A at 432 is about thirty-two cents, a little under a third of a semitone, and either one, heard on its own, sounds simply like an A.

Concert pitch as an agreement

Concert pitch is the A that instruments tune to, sounded by the oboe at the start of an orchestral concert. It is an agreement, not a fact of nature. Many orchestras today tune a little above 440, to 442 or 443, for brilliance, and players of early music choose their pitch by the music they play: commonly 415 for the Baroque, about a semitone lower, 430 for the music of Haydn and Mozart, and 392 for French music of the court of Louis XIV. Voices and instruments are built and trained to the pitch in use, and the ear soon settles into any of them.

What the lower tuning claims

The case for A at 432 says that it is in tune with nature: with the vibration of the earth, with a C of 256, which is a power of two, and with sacred measures. The arithmetic behind it is real but small. A C of 256, the philosophical pitch of the early eighteenth century, gives an A of 432 only in Pythagorean tuning, by the ratio twenty-seven to sixteen; in equal temperament the same C gives an A of about 430.5.

More basic still, a hertz counts vibrations per second, and the second is a human unit, first a fraction of the day and now fixed by the vibration of caesium atoms. A number of hertz can only be natural if the second is, and the second was chosen by people. The earth's lowest electrical resonance, often cited in these claims, wanders around 7.8 hertz and is not a note of any scale.

What the solfeggio tones claim

The solfeggio tones are a set of numbers, 396, 417, 528, 639, 741 and 852 hertz, later joined by 174, 285 and 963, each sold as working a change in body or mind, and 528 above all as a tone of love and repair. They take their name from the sol-fa syllables of the hymn Ut queant laxis. But in medieval music those syllables named steps in a pattern that could begin on more than one note, never a fixed number of vibrations, and no one could count vibrations at all until Marin Mersenne measured the pitch of strings in the seventeenth century. The numbers were drawn from games with digits, not from any chant.

Beats and binaural beats

When two tones close in pitch sound together in the air, they reinforce and cancel each other in turn, and the ear hears a regular throb at the difference between them: 440 and 442 beat twice a second. Tuners use this, slowing the beat until it vanishes as a string comes into tune. A binaural beat is different. Each ear is given its own steady tone through headphones, and the brain, combining them, hears a slow pulsing that is in neither ear. It is heard best with low tones, under about a thousand hertz, and with differences from a few hertz up to about thirty.

The chant traditions and pitch

The chant traditions themselves never fixed a number. The Gregorian books mark only the shape of the melody against a movable clef, and the cantor sets the first note where the voices of that choir sit easily. Vedic recitation keeps its raised, low and falling accents relative to the reciter's own voice, and a Byzantine choir sets its drone to its singers. What these traditions ask of pitch is that it suit the voices and serve the words.