The circle of 5ths is the progression of twelve pure 5ths, each having a value of 701.955 cents: C -G-D-A-E-B-F#-C# -G# -D# -A# -F-C. The problem discovered by Pythagoras, is that twelve pure ascending 5ths equal 7 octaves and 23.5 cents, (12 X701.955 = 8423.46 cents). 7 pure octaves equal 8400 cents. In other words, the 5ths are slightly too large to fit the octaves.How does this affect the tuning of a musical instrument, such as an organ, piano, or harpsichord? If you begin on middle C ( 261.63hz), this "fundamental" has an interval sounding a pure 5^th above: (392.2hz) G +2 cents. By tuning the fundamental of the string or pipe a 5^th above exactly in tune with this harmonic, you have corrected the G to a pure 5^th. Now correct in tune the D a 4^th below (D +4) in tune with the G pipe or string. Continue tuning up a 5th and down a 4^th, until you arrive at the C an octave above middle C, having completed the tuning of a chromatic octave. You have now moved clockwise round the circle of 5ths, ascending 701.955 cents on each 5^th. On listening to the octave, it is immediately apparent that the C above middle C is very sharp, in fact by 23.5 cents. In most instruments, it is very important that the octaves are not “stretched”, so clearly some TEMPERING is necessary to restore the true octave. One or more 5ths must be tempered or flattened by a total of 23.5 cents, or 1 PYTHAGOREAN COMMA.

The Syntonic Comma

What is the syntonic comma? simply the difference between four pure 5ths (4 X 701.955=2807.82 cents) and two octaves and a pure major 3^rd (1200+1200+386.3= 2786.3 cents): the difference being 21.52 cents. The purity of the 3rds are judged by how sharp they are in relation to this comma. Three thirds must add up to 1200 cents in order to fit an octave (e.g. C-E, E-G#, G# -C), however, three pure thirds (386.3 X 3 = 1158.93 cents) are 41.07 cents too narrow. This is called the LESSER DEISIS. In order to fit the octaves, one or more thirds must be sharpened by a total of 41.07 cents. Like the 5ths, major 3rds do not fit the octaves, although in this instance, the 3rds are too narrow. An advantage of using the Syntonic Comma is that by dividing it by four, and then tempering four consecutive 5ths by a quarter of a comma each produces a pure major 3rd. For example, temper the 5ths: C-G; G-D; D-A; A-E by 1/4 of a syntonic comma will give a pure C-E 3rd.

Arnout Van Zwolle and Pythagorean tuning

The simple solution to eliminate the Pythagorean comma is to tune eleven 5ths pure and flatten the remaining 5^th by 1 comma. In the early 15^th century, Arnout Van Zwolle devised a temperament with the comma on B-F#; thus, this 5^th is 23.5 cents flat. However, although this temperament gives good intonation in some keys, it is a RESTRICTED TEMPERAMENT: some of the 24 keys ( 12 major and 12 minor) being unusable because the intonation is poor. Popular keys, such as C ,Eb, F and Bb major are quite poorly intonated keys. This tuning suited the modal music of the 14th and 15th centuries.

Pietro Aaron's Mean Tone (1/4 Comma Mean Tone)

In 1523, Pietro Aaron devised a tuning system which produced 8 pure major 3rds in each octave, leaving four wide 3rds, each 41 cents sharp.  This temperament is called: ¼ Syntonic Comma Mean Tone. The mean tone interval is the mean of the whole tone ratios 9/8 ( 204 cents) and 10/9 (182 cents): (204+182)/2=193 cents; this is convenient, as 2X 193=386 cents, or two meantone whole tones equal a true major 3^rd. Aaron discovered that flattening 4 ascending 5ths by 1/4 of a syntonic comma each (21.5/4=5.375 cents) produced a pure major 3rd. Beginning on middle C+0 cents, we have: C+0; G-3.5; D-7; A-10.5; E-14. We have ascended four 5ths, or in tuning a keyboard, we have ascended a 5th (C-G); descended a 4th (G-D); ascended a 5th (D-A); and descended a 4th (A-E). C+0 to E-14 equals 386 cents. If we carry on with this process of flattened 5ths, moving on from E, we have: E-B; B-F#; F#-C#; C#-G#. We have now ascended four more 5ths from E-14 to G#-28; in other words, by again squeezing the 5ths to fit the 3rds, we have produced another pure 3rd. Looking at G-3.5 and ascending four 5ths to B-17.5, we have another pure 3rd. By continuing to temper or squeeze all but one of the 5ths to fit the 3rds, Aaron succeeded in producing a temperament with eight pure major 3rds, and their inversions: the minor 6ths. The tuning also gives nine minor 3rds only 6 cents flat of just minor 3rds. In flattening eleven of the twelve 5ths by 1/4 of a syntonic comma each, the Pythagorean comma has not only been eliminated, but exceeded by about 1 1/2 times which is about 36 cents. To counteract this, one 5^th: G# -Eb is tuned 1 1/2 commas wide, or 36 cents sharp. This interval is known as a wolf 5^th, because it produces a distinctive howl. Temperaments containing wide 5ths are known as non-circulating temperaments, as the circle is broken. Additionally, as I have mentioned earlier, three major 3rds must fit an octave; with eight pure 3rds, the remaining four 3rds must take up the shortfall, and are each 41 cents sharp, or 427 cents instead of 386. The position of the "wolf" determines which 3rds are the "discords".

Today, this temperament initially sounds very odd to musician’s accustomed to equal temperament, as it has many much flatter intervals, as well as the very wide 5^th. In most keys, the leading notes ( 7ths) are flatter than just values, and very much flatter than in equal tuning. In C#, E, F#, G# and B major however, the leading note to octave is only ¾ of a semitone (76 cents) which sounds quite startling! Yet despite these oddities, this temperament was popular for more than three centuries, and is considered to be an idealistic tuning today . In total, eight keys are unusable, whilst the best keys do not differ in key “flavour,” or character, and all contain the same number of errors: 32 cents per diatonic octave when compared with just values. This may seem a lot of errors, but in equal temperament, there are 50 cents of errors per diatonic octave, regardless of key. Although popular, it was to some extent superceded by 1/5 comma mean tone; this being slightly less radical, and whilst there is still a wolf 5^th, this is only one comma (24 cents) sharp.

Interestingly, there are an increasing number of musicians and theorists, that consider that ¼ comma mean tone as the ideal scale. Some even suggest that it is far better to tune an instrument in ¼ comma, in order to enjoy the excellent intonation that it produces, and to accept the discordant 3rds and wolf 5^th.

Well Temperament

Towards the end of the 17th century, composers, including Bach and Pepusch were pushing the boundaries of the mean tone temperaments, and were looking at ways of being able to modulate freely between the keys, and to be able to compose music which could be played in the (then) badly out of tune keys of C#,F#,Ab, and B.In order to be able to play in all keys, without the intonation (or intervals) sounding too discordant, various theorists began to evolve modified mean tone temperaments with the "wolf" spread over two or more 5ths, and with some pure 5ths which reduced the size of the wolf. Circulating temperaments were also evolved with no "wolf". By closing the circle, and accounting for the Pythagorean comma only, it was found that all 24 keys could be played, without incurring harsh discords. The positioning of the tempered 5ths around the circle and the number of 5ths that the comma was spread over determined which were the good keys, and which were the "spicy" keys. The more 5ths that the comma was distributed upon, the milder the "key character" or "key flavour". Tradition dictated that the rarely used major keys, with the greatest number of accidentals (sharps and flats) would be the poorer intonated keys, with the sharpest 3rds, 6ths and 7ths; the well intonated keys were the popular, most commonly used ones, such as C,D,F,G, and Bb. The neutral "grey" keys which are similar though not identical to equal temperament are Eb, E, and A. Well tempered tunings Include: Kirnberger III, Young II, Vallotti, Neidhardt, Bendeler, Stanhope and Werckmeister III. Unequal or well temperaments are so called because the 5ths are tempered by irregular amounts, or that some 5ths are tempered by an equal amount, but some 5ths remain pure. Temperaments which contain these attributes are also known as circulating temperaments, because the circle is closed and there are no "wolves". On equal temperament, all twelve 5ths are flattened by two cents (1/12 of a Pythagorean comma). Equal temperament is thus a circulating temperament, but not a "well tempered" system because the keys do not differ in character. Werckmeister III has four 5ths: C-G, G-D, D-A and B-F#, each flattened by a ¼ comma (6 cents). The result is very good intonation in most keys, many of the intervals being close or in tune with true just values. Even the worst intonated keys are perfectly usable, but have a distinctive key colour or "key character". One of the obvious ways of creating a circulating temperament, was to begin as with 1/4 comma mean tone: flatten the four ascending 5ths from C to E. In doing so, the syntonic comma has been accounted for, and there remains only the schisma: the two cent difference between the Syntonic and Pythagorean commas. This entails flattening one of the remaining 5ths by 1/12 (2 cents) of a Pythagorean comma. Strangely enough, no such tuning was published until the 1780's; first by J.P. Kirnberger (Kirnberger III, with the schisma on the 5th F#-C#; and around 1810 by Prinz, with the schisma on B-F#.

Equal Temperament

Equal tempering was understood by the ancient Greeks, but not until the late 15^th century was it employed for some fretted instruments: guitars and lutes. A number of 17th century theoroticians calculated the ratio for the equal tempered semitone. It is said that Mersenne, Salinas, Zarlino, Frescobaldi and Petronius amongst others proposed equal temperament, but what I would like to know is what term they used? There was no specific term for equal temperament in the 17th century, and it seems very unlikely that the out of tune major 3rds,6ths, and minor 3rds produced by equal temperament would have appealed to them. They all devised temperaments which are radically different from equal temperament. What I suspect is more likely, was that they were proposing circulating "well-temperaments". Do any writings of these theorists still exist? The rigid, mathematical system used by most piano tuners today was not practised until the beginning of the 20^th century. The advantages of equal temperament are that firstly, it is very neat mathematically, with the pythagorean comma spread equally over all twelve 5ths (hence “equal” temperament). All whole tones are 200 cents, and therefore semitones are 100 cents. It is possible to play in all 24 keys without fear of intervals sounding odd or badly out of tune; on an instrument tuned to this temperament, it is possible to play any piece of music, from any era, and the music will sound tolerably acceptable. In effect, it is a safe, utilitarian tuning, with a bland, neutral tonal colour. In fact, apart from pitch changes, key changes do not result in a change of key colour as the intonation in all 24 keys is exactly the same. The disadvantages are, the complete lack of key colour, and that apart from the octaves, no other note is in tune. Indeed, several intervals are badly out of tune: minor 3rds are 16 cents flat, major 3rds are 14 cents (2/3 of a SYNTONIC COMMA) sharp, 6ths are 16 cents sharp and 7ths are 12 cents sharp, the 7^th being the leading note. Sharp leading notes destroy the brightness of octaves. As so many keyboard notes (fundamentals) are out of tune with the pure, just harmonics within the strings or pipes, this results in pulses or beats, which can be heard whenever two notes are sounded which vary slightly in frequency. On organs tuned to equal temperament, slow passages sound beat ridden and fast passages produce a not very pleasent howl. Because all the 5ths are out of tune, the mixtures (which are tuned as pure intervals) sound very harsh and tend to screech.

Although considered aurally unsatisfactory on keyboard instruments for many centuries, it gradually became accepted in Europe in the late 19^th century, and became the standard tuning in the early part of the 20^th century at the same period in which attempts were made to standardise pitch. The reasons for the wholesale adoption of equal temperament are a little obscure, although as many English organs were tuned to 1/5 comma mean tone (a restricted temperament), equal tuning appeared to some as an improvement. The new fashion of scientific principles superseding "outdated" traditional thinking was undoubtedly a significant reason for the rise of equal temperament, which was perceived as a scientific, mathematical system and therefore had to be superior! The adoption of an international pitch standard however did (and still does) make good sense. In the 19th century, opinions differed as to which well-temperament was the best; equal temperament being wholly neutral was seen as a fair compromise at the time. In retrospect, it is a great pity that a well-temperament such as Vallotti, or Young I or II was not adopted as the standard tuning instead.

Up to the turn of the 20^th century, equal temperament, as we know it today, was not a common practice. There were various forms of pseudo equal tunings, and well-tempered systems still in vogue. Equal temperament is not an easy system to tune accurately, and in fact, it is rare to find a piano that has been tuned to equal temperament where the middle octaves are tuned to an accuracy of within 2 to 3 cents. This is mainly owing to the fact that there are no pure intervals apart from the octaves with which to check the tuning accuracy. The practice of tuning to specific beat rates did not become standard until the early 20^th century. Tuners used to define the interpretation of a given tuning system with their ears. Even those that claimed to tune to equal temperament, would pay careful attention to biasing the tuning in favour of the popular keys, generally along the lines propounded by Thomas Young. Relative beat rates, rather than the current rigid practice of tuning specific beat rates as accurately as possible, were the the accepted practice.

Table Showing The Intervals Of The Major 3rds: C-E,E-G#,G#-C, In Some Well Known Tunings