This entry continues on from 'Sol-fa (1): the key to the riddle of staff notation' and briefly discusses the subject of temperament in music, using Sol-fa for reference.

Proportion

The six notes of the Guidonian hexachord¹ fall into the kinship of pure mathematical relations. The medieval theorists didn't know about frequency, which is how we define pitch nowadays, but they knew the underlying maths. Indeed it had been known since Pythagoras, c500 BC, who worked out the mathematical relations for pipe lengths, string lengths, and bell weights. These all hold the same relationships as the inverse of the frequencies.

Frequency is measured in cycles (vibrations) per second, units named after the scientist Hertz. A standard tuning fork defines the note A above middle C as 440 c/s (cycles per second), which can also be written as 440 Hz. Non-standard forks can also be bought.

If Ut has a frequency of 8 times x then Re is 9x, and Mi is 10x.

The same relations bind Fa with Sol and La: if Fa is 8y, Sol is 9y and La is 10y.

Furthermore, with Fa at 8y, Ut will be 6y; a relationship² of 4 to 3.

To put them all in a tidy line, we need to use bigger numbers. Taking x for granted from now on, we get these harmonious proportions:

Now comparing those numbers we see that:

Ut-Re and Fa-Sol are both 8:9

Re-Mi and Sol-La are both 9:10

Ut-Mi and Fa-La are both 4:5

Ut-Fa, Re-Sol and Mi-La are all 3:4

Ut-Sol is 2:3 and

Ut-La is 3:5.

The relation of any note to its octave above (e.g. C-c, D-d) is always 1:2

Harmony everywhere!
Well, almost: the only pairs whose relationships don't reduce to smaller numbers are Re-Fa and Re-La.

The names given to the intervals are:

By tradition the thirds, fourths, fifths and octaves are concordant⁴, and the tones and semitones are discords, when two or more notes are sounded together. The same consonance and dissonance apply to the inversion of each interval, that is, the intervals occurring when one of a pair of notes is flipped to its octave, making thirds become sixths, seconds become sevenths and so on.

Prime numbers greater than five are not dealt with in classical European harmony at all, nor are their multiples⁵.

The syntonic and Pythagorean commas

Temperament issues arise from the fact that some notes require retuning if they are to join with others in perfect harmony. This happens not only between different hexachords, but even within the same hexachord: as we saw above, Re-Fa and Re-La are not harmonic intervals within a hexachord.
Here are some notes for comparison between the original medieval 'hard', 'natural' and 'soft' hexachords:

The notes B Fa and B Mi⁶ are a semitone apart, as shown in the first entry in this series. This is not a question of temperament. There are however other differences, small but problematic, between notes that are treated as equal.

Two As, two Ds

'A La Mi' is not as high as A Re; and D La is not as high as D Sol.

If F Fa is 64 and G Re is 72 then A Mi is 80; but with G Ut still at 72, A Re is 81.

Similarly if C Sol is 72 then D La is 80; but with C Fa at 72, D Sol is 81.

This difference, the difference in size between the greater and lesser tone, was known to the ancient Greeks as the syntonic comma. It is about a fifth of a semitone, which is clearly audible to untrained ears.

Flats versus sharps: never the twain shall meet

Additional hexachords can be generated in both directions: denoting F as Sol will invoke a hexachord starting on Bb, introducing the note Eb, and so on towards the flat side; denoting G as Fa will produce one on D and introduce F#, and so on to the sharp side.

The sharps will never meet the flats; on the one side we reach Ab and on the other G#, but there is no reason why these two should be at the same pitch; and indeed they are not. The difference between Ab and G#, going round the cycle of fifths, is slightly larger⁸ than a syntonic comma, and was called the Pythagorean comma by the ancient Greeks.

Practical solutions

Some instruments such as concertinas have been traditionally made with separate buttons for Ab and G#.
If one pitch is to serve for both then either

Some temperaments

The tuning system most recommended in the Medieval period was Pythagorean which disregards the consonance Ut-Mi altogether and simply makes all the fourths pure (3:4)⁹. Taking the Fa as a new Ut and tuning another pure fourth, and so on, we end up with a 'wolf' fourth between our first note and the twelfth one we tune: too big by a Pythagorean comma. The major thirds (Ut-Mi) that come out as a result are too big by a syntonic comma, since we have only greater (Ut-Re) tones, and no lesser (Re-Mi) tones. However, a strange coincidence happens towards the end of our tuning: a third that straddles the wolf fourth will become wonderfully pure, as the two wrongs in this case make a right and the Pythagorean comma almost exactly cancels the syntonic comma. Since we can choose which will be our first and last notes to tune, we can place the wolf fourth wherever we like. Putting it between E and B will result in beautifully tuned chords of G major, D major, and A major; this is what we get if we tune only the highest two strings¹⁰ of a guitar purposely wide.

Starting from pure intonation as in the first list above, and simply dividing Ut-Mi (4:5) into equal steps for Ut-Re and Re-Mi (it involves a square root) gives a partial solution, widely used in the Renaissance: Mean-tone tuning. It sounds very pleasant in keys with few sharps or flats, but it still leaves you with unequal semitones. The resulting anomalies gave rise to many different versions of mean-tone: quarter-comma, fifth-comma, sixth-comma and so on.

Tuning a guitar to four perfect fourths and a pure major third will leave the top E flatter than the bass E by exactly a syntonic comma. Widening all the intervals equally will approximate to fifth-comma meantone tuning. This is complicated by the placing of the frets, which are normally equal-tempered, though many guitar makers (particularly in Spain) flatten¹¹ the second fret; experience shows that an equal-tempered second fret always sounds too sharp. String physics rather than mathematical considerations may account for this. A solution that many guitarists use is to tune all the bottom strings fairly pure and tune the top two wider; this will sweeten the chord of A, the guitar's home key, and also D and G; but the E chord will have a bit of a rasp with its flat fifth.

Temperament makes a great difference to an organ, since a triumphant long-drawn-out final organ chord is heard without any decay of the sound, and without any chance of adjustment. For this reason organ tuners in England held on to mean-tone tuning, and only gave in to equal temperament around 1850.

Some viol players split their first fret, to give a sharp Bb on an A string but a flat C# on a C string. Others adjust the tuning as they go, as a large alteration can be made by pushing or pulling a string with the fretting finger.

Equal temperament was proposed by Aristoxenus around 350 BC, rejected for almost two thousand years, then championed by Rameau in the eighteenth century and only more-or-less-universally applied since around 1850; by which time the early-music movement had begun, which has reinstated unequal temperaments for performing period music. Equal temperament is arrived at by dividing the octave into twelve equal semitones, using the twelfth root of two¹².

In equal temperament every interval except the ocatve is compromised: the harmonious mathematical relationships are hinted at rather than sounded. The biggest difference equal temperament makes to the sound of a chord is that the major thirds are too wide, by about one seventh of a semitone. Musicians other than keyboard players can and do bend the tuning of these notes to add a glow to significant chords. Of course they also bend notes for expressive reasons, as well as harmonious ones.

Plucked instruments work well in equal temperament, perhaps because their sound dies out and seems to adjust itself miraculously in the process. Pianos are in the worst case; because of the high tension on their frames, they suffer from a condition called 'inharmonicity' which is offset by tuning their octaves wide. String players accordingly need to adjust their intonation when playing with piano accompaniment.

Many varieties of mean-tone temperament were used in the renaissance, and the seventeenth and eighteenth centuries saw even more inventive systems come into vogue. Theorists searched for a system that would give as many good chords as possible in as many keys as possible. J S Bach wrote two sets of preludes and fugues in all twelve major and twelve minor keys, known as the 'Well-Tempered Clavier' (keyboard); it is now generally thought that equal temperament was not Bach�s ideal, but one of the 'well-tempered' systems such as those of Werckmeister or Niedhardt. These systems sought to keep a pleasing variety between keys; others purposely made distant keys more discordant than familiar keys, so that a composer could increase the drama of his compositions by straying into those outlandish keys before returning to a comfortable conclusion.

Vallotti's 1754 system, now favoured by many viol players, makes several of the commonly used thirds harmonious by dividing the Pythagorean comma between all the 'natural' notes (six fifths: F-C-G-D-A-E-B), while the six fifths incorporating 'black¹³ notes' (B-F#-C#-G#-Eb-Bb-F) remain pure.

Further information

A practical demonstration of 34 historical temperaments, with a program to create and hear temperaments of your own, is to be found here.