Jan 092011
 

Equal Temperament

Pure Pitch vs. Equal Temperament

Equal temperament (ET) means the same distance between each of the 12 notes of the chromatic scale, as a ratio. Pure pitch is tuning slightly altered from ET. The reason it sounds more “pure” is that there is no degrading of the harmonic overtone series, as there is in ET; Although ET’s 4th, 5th, intervals are nearly identical to pure pitch; these intervals in ET do not degrade the harmonic series.

A Do-It-Yourself-er Finds a Problem

When I was young and foolish (now I’m old and foolish) I once tried to tune my piano by ear. I could tune any chord, say, an F, A, C, F to perfection and it would sound great…until I tried to play something. Most other chords sounded horrible. That same F chord on a properly tuned piano doesn’t sound quite as pure, yet we don’t notice.

An a cappella choir, an orchestra, or any configuration of instruments can sound pure chords in a piece of music that goes through many chords, changes key, and so on, as long as there is not a piano or fretted instrument in the ensemble. What about piano concertos? And other piano/instrument configurations?

The differences between pure pitch and ET are subtle. The other instruments can adjust pitch, and the “off-pitchness” of the piano is so slight that it could be looked at as different keys having different “colors”. Most people can’t notice a difference–this includes many musicians. The fretless and variable pitched instrumentalists use their ears to make subtle shifts (from a piano-pitch). For example, a Cb is flat from a piano B, assuming we are not in the key of E, F#, or B (perfect 4th/5ths, although for those keys, unless modulating, the note would be written as “B”, not “Cb”). Similarly, an A# is sharper than Bb. Some types of wind instruments, and single-reeded instruments can only approximate the actual pitch; the same situation as a piano.

Enter J.S. Bach

.Until the 18th century, the problem for music and composition was that until ET was discovered, a piece of music had a very limited harmonic pallet. The research, discovery, and gradual implementation,  must have facinated J.S. Bach. One of his major groundbreaking works is the “24 Preludes and Fugues for the Well-Tempered Clavier“. A series of pieces first in the key of C major, then C#major, then their corresponding minor keys. Twelve Major-key preludes and fugues, and twelve minor key preludes and fugues.

A Brief Introduction to the Math Behind It

There has been a lot of research into the history of ET. The formula needed to derive it, most likely discovered by trial and error, or applying the theories of the day to practice, was a major mathematical insight. The formula involves the 12th root of 2. Twelve, because there are 12 tones that span an octave, and two, becauuse the highest note is double the frequency of the lowest note. Also, we are concerned about the pitch-difference (measured in cycles per second) between 2 adjacent notes. The result is a decimal number, which is a multiplier constant applied to cycles/second.

The 12th root of 2 is algebraically equivalent to its inverse: [12√2 ➾ 2^(1/12) = 1.0594630943593]. The decimal number result is the multiplier. It can be thought of as a ratio,. Logarithmically, the distances between pitches are equal. We multiply, say, 440 * 1.0594630943593 and the result is 466.16 (rounded to 2 decimal places). This is the pitch for the next chromatic note up, in this case, Bb, or A#.

The table below shows the octave from A440–just above middle C–to  one octave higher, “A” at 880 cps. Incidentally, by tradition, A440 is the pitch the concertmaster plays for the orchestra to tune up to before they begin to play. The numbers in the 2nd column are the result of the adjacent note’s pitch times the ET multiplier, 1.0594630943593. For purpose of illustration, the result is rounded to 2 decimal places.

The practical implication of equal temperament is that each note is “compromised”, but just barely, for the best fit–and the fit works out for any piece of music no matter what key it may modulate to.¹

Equal Temperament

Note NameCycles per Second (Hz)
A
440.00
A#466.16
B493.88
C523.25
C#554.37
D587.33
D#622.25
E659.26
F698.46
F#739.99
G783.99
G#830.61
A880.00

The Big Bangs of Music

Howard Goodall, a musicologist and composer, has an excellent DVD collection called “Howard Goodall’s Big Bangs”. His idea is that there were a number of historical events without which we wouldn’t have music as we know it today. These “Big Bangs” were originally aired on BBC TV.

The five “Big Bangs” were

  1. Notation and Sheet Music, so music could be distributed and assimilated across generations and geography;
  2. Equal Temperament tuning, so compositions of music could venture into many keys;
  3. Opera, which started to spread music to the masses, and large music halls were built;
  4. The invention of the Piano, which could handle all the elements of music, i.e., melody, harmony, and rhythm, and also began to bring music into the living rooms of the (initially, “upper”) middle class;
  5. The invention of recording–first wax, then tapes which could be played on the phonograph or tape deck. Recording technology made it possible to make records that could be mass produced from a master recording. About the same time, the microphone came to be in use, and that allowed for more control over balance to produce better ensemble recordings. An up-close-and-personal style of singing began to be developed, as opposed to the operatic type of singing, which necessarily requires volume and projection from the singer in order to be heard. Of course, before recordings, you could only hear music performed by live musicians.
You can watch a brief intro of Howard Goodall’s Big Bangs here:


Howard Goodall’s Big Bangs

Equal Temperament explained on WikiPedia

¹. In practice, when tuning a piano, the theoretical ET is not followed precisely. Called “scaling”, the notes are gradually stretched (the intervals widened) across the keyboard. A concert grand, because of it’s longer strings can be tuned in ET so that for example, a 3-octave unison pitch does not “beat”; whereas in smaller pianos, as the tuner moves higher up the keyboard the stretching increases. This is partly to compensate for the perception that higher notes sound flat. In the keyboard’s upper reaches, the ear can tolerate the “beating”, i.e., the off-pitchness found in the higher tones. The overtone series of the highest C, for example, soon get out of range for human hearing.

It’s just my opinion, but I believe an experienced tuner who tunes by ear will give the best results. This is due to the previous explanation, as well as the fact that every piano is different;  A machine can only measure empirical data, whereas the tuner can apply human perception.