The design of this leaded piece is my invention, inspired by a TV programme about equal temperament. The musical history below is mainly derived from Barbour (1972), and I hope I have got the details right.
The design:
This comprises 84 squares, arranged as 7 rows of 12. There are seven colours repeated in the same order row by row, so that each colour is repeated 12 times. The resulting pattern makes each colour a ‘knight’s move’ away from itself. The table below replaces the seven colours by the numbers 1 to 7.
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 1 | 2 | 3 | 4 | 5 |
| 6 | 7 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 1 | 2 | 3 |
| 4 | 5 | 6 | 7 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 1 |
| 2 | 3 | 4 | 5 | 6 | 7 | 1 | 2 | 3 | 4 | 5 | 6 |
| 7 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 1 | 2 | 3 | 4 |
| 5 | 6 | 7 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 1 | 2 |
| 3 | 4 | 5 | 6 | 7 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
Each colour occurs once and only once in each column. For example, consider the number 7. If the squares of the table are numbered 1 to 84 in sequence, then 7 occurs in squares 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84. The column in which each of these squares occurs is given by the remainder when dividing the number of the square by 12, i.e. 7, 2, 9, 4, 11, 6, 1, 8, 3, 10, 5, 0 (or 12). Thus each colour appears once and only once in each column – a property resulting from 7 and 12 being ‘coprime’ (ie having no factors in common except 1 and -1 ). It also means that each block of seven columns forms a highly-structured Latin square (in which each colour appears once and only once in each row and column).
Pythagorean tuning:
Pythagoras observed that the note made when a piece of metal was struck, or a string was plucked, rose by an octave when the length of the piece was halved (i.e. the frequency of vibration doubled). When a third was cut off the piece (i.e. the frequency of vibration rose by a proportion 1 divided by 2/3 = 1.5), the note was different but harmonized. This is a ‘perfect 5th’. He found that when a third was repeatedly cut off the piece, twelve different notes occurred until the original note was almost exactly returned to (see below for a precise definition of ‘almost’). Since Pythagoras believed that whole numbers underlay music (and everything else) he brushed over this gap and assumed there were 12 notes (semitones) in an octave, and that cutting a third off the length of the metal or string raised the note by 7 semitones (4 tones).
Pythagorean tuning therefore starts at some baseline note and tunes up and down perfect octaves and fifths. The table above can be used as a map for this tuning – moving to the next occurrence of a colour is tuning up a fifth, while dropping a row is tuning down an octave. Thus to tune up a single tone (two spaces), we would need to tune up two fifths and down one octave (which may be written T = 2 F – O ). To tune up a semitone (one space), we would need to tune up three octaves and down five fifths ( S = 3 O – 5 F). Of course, these eight steps can be carried out in any order. The single occurrence of each number in each column corresponds to his mapping out all 12 semitones.
Ratios:
In terms of ratios of frequencies, tuning up a fifth changes the frequency by the ratio 3/2, while tuning down an octave changes the frequency by ½. Thus T = 2 F – O is equivalent to a ratio of 3/2 x 3/2 x ½ = 9/8, while S = 3 O – 5 F is equivalent to 23 x (3/2)-5 = 256/243 . Table 2 shows the resulting intervals and ratios when tuning from a baseline C.
Note that the fifth is perfect (3:2) as designed, and the fourth (C to F) is also a perfect harmonic (4:3). However two tones (a major third C to E) is 81/64 = 1.266 rather than the ‘perfect’ 5/4 = 1.25 – thus this tuning produces very sharp thirds. This degree of sharpness is known as the syntonic comma, with ratio 81/64 divided by 5/4 = 81/80 = 1.0125.
| Notes | C | D | E | F | G | A | B | C | |||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Intervals | 9/8 | 9/8 | 256/243 | 9/8 | 9/8 | 9/8 | 256/243 | ||||||||
| Ratios from C | 1 | 9/8 | 81/64 | 4/3 | 3/2 | 27/16 | 243/128 | 2 | |||||||
| Decimal ratios | 1 | 1.125 | 1.266 | 1.333 | 1.5 | 1.688 | 1.898 | 2 |
According to Pythagorean tuning, raising 12 fifths followed by dropping 7 octaves should return to the original note (ratio 1), but in fact 12 F – 7 O has ratio (3/2)12 x 2-7 = 531441/524288 = 1.0136 which is the Pythagorean comma (approximately 74/73).
Meantone temperament:
Tempering is the art of adjusting the pure harmonics of the Pythagorean scale so that the instrument could play in a number of keys. Instruments such as violins can be tuned in fifths but can be adjusted when played, but tempering needs to be built into keyboard and fretted instruments. Aron’s meantone temperament of 1523 flattened fifths but kept pure thirds as 5/4, with a tone set as the mean of a third, i.e. with ratio square root of 5/4 = 5½/2= 1.118. Barbour suggests the fifths were flattened by a quarter of a syntonic comma, or (81/80) ¼. Thus a fifth is reduced from the Pythagorean 3/2 to 3/2 x (81/80) ¼ = 5 ¼ = 1.4953 . Table 3 shows the resulting intervals and ratios when tuning from a baseline C.
| Notes | C | D | E | F | G | A | B | C | |||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Intervals | 5½/2 | 5½/2 | (8/5)5/4 | 5½/2 | 5½/2 | 5½/2 | (8/5)5/4 | ||||||||
| Ratios from C | 1 | 9/8 | 81/64 | 4/3 | 3/2 | 27/16 | 243/128 | 2 | |||||||
| Decimal ratios | 1 | 1.125 | 1.266 | 1.333 | 1.5 | 1.688 | 1.898 | 2 |
Meantone temperament was only used for keyboards, and Bach’s ‘Well-tempered Clavier’ was probably tuned in a similar form of temperament that enabled it to be played in all keys (although the chosen temperament may have suited the commoner keys better).
Equal temperament:
The spacing of the frets in instruments such as lutes and viols is common to all strings, and it was realized by 1577 that there was a need for equal temperament, in which all semitones are of exactly equal ratio. Since there are 12 semitones in an octave, the ratio must be 2 to the power 1/12 = 1.0595. Ideally, each fret should therefore be placed a fraction 2-1/12 = .9439 of the length of the string, which Galilei in 1581 approximated by 17/18 = .9444. Thus each fret in a lute would be placed one 18th of the current length of the string – this leads to the 12th fret being a fraction (17/18)12 = .50364 of the length of the string rather than the correct .5, and so it is likely that a small additional spacing was introduced. Numerical calculations of 21/12 were obtained in China in 1595 and published by Stevin in 1596, while Mersenne (1636) gave detailed numerical accounts of numerous temperaments. In fact, examination of paintings such as Holbein’s The Ambassadors (1533) suggests that lutes were being tuned in equal temperament even before published calculations of the ratios.
| Notes | C | D | E | F | G | A | B | C | |||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Intervals | 21/6 | 21/6 | 21/12 | 21/6 | 21/6 | 21/6 | 21/12 | ||||||||
| Ratios from C | 1 | 21/6 | 21/3 | 25/12 | 27/12 | 23/4 | 211/12 | 2 | |||||||
| Decimal ratios | 1 | 1.122 | 1.256 | 1.335 | 1.498 | 1.682 | 1.888 | 2 |
Table 4 shows the tuning for equal temperament, which Barbour takes as the ‘gold standard’ for other tuning systems. It is remarkable how well these integer powers of 21/12 approximate the pure harmonic ratios, as shown in Table 5.
| Intervals | Pure harmonic ratio | Semitones r | Equal temperament ratio 2r/12 |
|---|---|---|---|
| Octave | 2 | 12 | 2.000 |
| 5th | 3:2 = 1.5 | 7 | 1.498 |
| 4th | 4:3 = 1.333 | 5 | 1.335 |
| 3rd | 5:4 = 1.25 | 4 | 1.256 |
| Major tone | 8:9 = 1.125 | 2 | 1.122 |
A consequence of these different tunings is that it was difficult to tune fretted and keyboards in performance. However, this problem disappeared with the increased use of unfretted string instruments, which can more easily adapt to alternative tunings.
References:
Barbour J M (1972) Tuning and Temperament – A Historical Survey. Da Capo Press. New York.