Tuning and temperament

The harmonic series gives pure intervals: the octave at the ratio 2:1, the fifth at 3:2, the fourth at 4:3. It seems natural to tune an instrument by them and be done. But when you try it with the twelve keys of the octave, the sums do not close. The whole history of Western tuning is the history of where to hide the leftover.

Tuning by fifths and fourths

The octave is the easiest interval to tune, but it brings no new note: notes an octave apart share a name. The ratios 3:2 and 4:3 are almost as easy, and each gives a useful interval, the perfect fifth and the perfect fourth. That is why most string instruments are tuned in a series of fifths or of fourths (Blatter 2007, part 6, appendix III).

The violin is tuned in fifths:

Hear the passage

The guitar in fourths, with a major third between the third and second strings. Its part is written an octave above the sound:

Hear the passage

Stacking fifths from one note, the first five already make a pentatonic scale, the scale many cultures have used over the centuries (Blatter 2007, part 6, appendix III).

The sum that does not close

The problem shows when you keep going. Starting from the lowest A on the piano and going up twelve pure fifths, you pass through all twelve notes and arrive, in theory, at the highest A, seven octaves up. Blatter works it out in frequencies. The low A vibrates at 27.5 Hz; each pure fifth multiplies the frequency by 1.5; after twelve fifths the result is 3,568.02 Hz. Taking the same A up seven octaves, each multiplying by 2, the result is 3,520 Hz (Blatter 2007, part 6, appendix III).

The two sums should give the same A, and they miss by 48 Hz. The difference is audible, about a quarter of a semitone, and is called the Pythagorean comma, since it was identified by Pythagoras (Blatter 2007, part 6, appendix III).

The same comma turns up in the note names. Going up in fifths from C you reach G sharp; going down in fifths, A flat. With pure fifths the two notes do not coincide:

Hear the passage

On the piano they are the same key. Tuned by pure fifths, they are two different sounds.

The natural system and the tempered one

Med separates two systems. The natural one, based on acoustic calculation, fixes each note's vibrations by exact ratios, and belongs to theorists such as Pythagoras and Zarlino. In it the semitones are not all equal. Med measures the difference in commas, each a ninth of a tone: from C to C sharp there are five commas, from C to D flat four, so C sharp sits a little higher than D flat (Med 1996, ch. V).

The tempered system makes every semitone equal, four and a half commas each, and divides the octave into twelve equal parts. Med describes it as a renunciation: the perfection of natural tuning is given up in favour of using every chromatic note, to make harmony easier (Med 1996, ch. V).

Blatter reaches the same conclusion through the keyboard. A keyboard giving every pitch the music asks for would need dozens of keys per octave; experimental keyboards like that have been built and proved unwieldy. The way out was to keep twelve keys and temper the tuning, that is, to make compromises (Blatter 2007, part 6, appendix III).

Cents

To compare systems, acousticians divide the octave into 1,200 cents. In equal temperament each semitone is exactly 100 cents (Blatter 2007, part 6, appendix III). With that ruler the differences become visible:

Interval	From the harmonic series	By pure fifths	Equal temperament
perfect fifth	702	702	700
perfect fourth	498	498	500
major third	386	408	400
minor third	316	294	300

The tempered fifth loses only 2 cents against the pure one (Blatter 2007, part 6, appendix III). The thirds lose much more: the tempered major third is 14 cents wider than the harmonic series one, and the minor third 16 cents narrower.

What the piano gives up

In equal temperament every fifth and fourth on the keyboard is slightly flawed, and the thirds are the wrong size. Each fifth is narrowed so that the Pythagorean comma disappears; near the middle of the keyboard it beats three to four times every five seconds. The major thirds are too wide, the minor thirds too narrow. Blatter says no professional vocal or instrumental ensemble would tolerate such intonation, but the keyboard player has no choice (Blatter 2007, part 6, appendix III).

In return, the piano gains what no pure system gives: every key sounds equally good, and G sharp can be A flat without trouble. That is what makes a closed circle of fifths, and modulation to any key, possible.

Before equal temperament

Equal temperament was not the first solution. Between the Pythagorean system and it came others, which spread the comma unevenly.

• Meantone, widely used in the sixteenth and seventeenth centuries, narrowed the fifths a little more to get pure major thirds. Keys with few accidentals sounded very good; distant ones were unusable, and one fifth, the one left over, sounded so out of tune it earned the name wolf fifth.
• The well temperaments of the late seventeenth century, such as Andreas Werckmeister's, spread the comma so that every key could be used, each with a slightly different colour.

Med records Werckmeister's treatise, published in 1691, and says Bach enshrined the tempered system in The Well-Tempered Clavier, the 48 preludes and fugues in every key (Med 1996, ch. V).

Who does not have to temper

Med divides instruments into two groups. Tempered instruments have fixed pitch, like the piano and the organ, and produce the notes of the tempered scale. Untempered ones, like the violin, the trombone and the voice, have no fixed pitch and can produce the notes of the natural system. So, Med says, they combine the two: they tune naturally when the harmony allows, and in a tempered way when playing with a tempered instrument (Med 1996, ch. V).

Blatter describes this as active listening. Singers and string and wind players adjust every note as they play. In controlled tests, string players tended to use Pythagorean tuning, with major thirds wider and minor thirds narrower than those of the harmonic series (Blatter 2007, part 6, appendix III).

And some adjustments depend on where the music is going. A diminished interval tends to close, and sounds better with its two notes a little closer; an augmented one tends to open, and sounds better a little wider. Blatter works out that the diatonic semitone, C to D flat, is about 10 percent smaller than the tempered one, and the chromatic semitone, C to C sharp, almost 15 percent larger (Blatter 2007, part 6, appendix III). It is the same conclusion as Med's commas, with a different ruler.

That gives spelling a weight the piano hides. Writing C sharp or D flat is not just a matter of reading: for anyone tuning freely, they are two different pitches.

What comes next

With the harmonic series and tuning in hand, you can return to harmony with fresh eyes: why chords built in thirds sound the way they do, and why some chords seem related to others.