The figure above may, at a superficial glance, be recognised as a 'Great Circle of Fifths' (from music theory).

A little close examination will show that this particular Circle, is not by the rules of tonal "grammar", a complete circle of fifths. The interval, in this particular example, between G# and E-flat is of course not a fifth, but a diminished minor sixth, according to tonal grammar. The same would be true had there been an alternative interval from C# to A-flat.

A perfect fifth in Western tonal "grammar" must be notated with two alphabet letters that are 5 letters apart in the "musical alphabet" (which only goes from A to G and then recurs). (For the rules of musical interval notation see here). Therefore, circulating right round the Circle in perfect fifths, we would never end up back at the same starting note name. For example, if we start at C and circulate clockwise right around the Circle, when we arrive back at the position of the C, we would have to be writing Bx (B double sharp) rather than a C, if all the intervening perfect fifths were correctly notated as perfect fifths.

You might say, from familiarity with keyboard instruments and other "fixed pitch" instruments, that Bx is in any case the "same note" as C, acoustically, so the difference is merely one of note names, and not one of actual musical difference, or one of any musical consequence. Certainly, on the piano, for example, you can name a given note whatever you like, and it will still be the same note, acoustically. 

So, it does not matter then, on the piano, whether you call the A-flat an "A-flat" or a "G-sharp", because it will still be the same note, acoustically. However, consider the interval between this note and the E-flat. In musical grammar the interval A-flat to E-flat is not the same interval as G# to E-flat. One is a perfect fifth whilst the other is a diminished minor sixth. The question then naturally arises:

When do intervals with different names in musical grammar represent acoustically different intervals, and when do they not?

The answer, of course, is that note names in tonal grammar can be swapped with an "enharmonic equivalent" without changing the acoustical interval that the note names represent. At least, this is true on the piano keyboard. The particular chosen enharmonic name in any instance has much more to do with the implied "tonal center" than with the actual pitches of the notes.  

This is in direct contrast to the use of an interval Circle in the past. Historically, Circles of intervals had a great deal to do with the precise pitches of the notes, and originally, nothing to do with "tonal center" or tonality. The modern Circle of Fifths is a device that is consistent with modern tonal grammar, complete with its principle of enharmonic equivalents. It is a device used primarily to serve the aims of tonal theory.

The Circle is often, by convention, called the Pythagorean Circle, but really, this is more than a little misleading. Pythagoras was born in the 6th century BC, whilst tonality was not formally developed until the 17th / 18th centuries AD. Not surprisingly, there is some considerable difference between the "Pythagorean" Circle and the modern, so-called "Circle of fifths". In fact, the common point between them, is simply that if one produces a series of 12 rising perfect fifths, one ends up, approximately, back at the same "note", but 7 octaves higher. (Try confirming this on a piano keyboard with a full sized compass). 

For the purposes of modern tonal theory's use of the Circle device, the approximation is ignored, or circumvented through the principle of enharmonic equivalents. This is completely opposed to the significance of the "Pythagorean Circle" itself. In other words, the whole point of the Pythagorean Circle lies in the fact that the 12 fifths it represents, do not really circulate. Pythagoras (according to the available evidence) was certainly not prepared to simply ignore the approximation, or to "gloss over" it by using suitable "rules" for the exchanging of "enharmonic equivalents". Rather, he was concerned with the precise relationship between 12 perfect fifths and 7 octaves, and in particular with the 

precise "difference" between them. 

Pythagorean perfect fifths do not circulate. Without getting technical, we can say that a Pythagorean perfect fifth is in practice the same perfect fifth that string players would recognise and tune and a "perfectly tuned", beautiful, consonant, perfect fifth. The "Pythagorean Circle", as it is called, represents 11 of these perfect fifths, but it cannot represent 12 of them, and Pythagoras, according to the attestations, was perfectly aware of this. One arrives after a complete circulation, at a note that is not only approximately 7 octaves higher, but is different to the starting note, and cannot simply have its name changed to an "enharmonic equivalent" to get round the problem.

There is no primary evidence that Pythagoras ever used the visual Circle device itself, to represent the 11  perfect fifths, although it seems likely from the evidence that Circles, generally, were held to have special significance.      

The "difference" between the final note and a note 7octaves above the starting note, when one circulates round the Circle completely, is known in mathematical terms as the Pythagorean comma. This is also the "difference" between a Pythagorean perfect fifth, and the interval that remains when one circulates round 11 perfect fifths, leaving this last remaining interval to form it own "size".

For example, in the Circle illustrated, circulation of perfect fifths starting at C, goes clockwise as far as G#, and anticlockwise as far as E-flat. The interval G# to E-flat, differs in size from a Pythagorean perfect fifth, by the micro-interval called the Pythagorean comma. This difference is by no means musically negligible. It is approximately a quarter of a semitone, and if a perfect fifth is "de-tuned" by this amount, it sounds grossly mistuned, in Western music. The tone qualities of the perfect fifth's consonance are destroyed, and the interval sounds dissonant. Historically, such an interval was (and in fact still is) known as a "wolf" interval, the precise origin of the term "wolf" being obscure.

The music theory book "Great Circle of Fifths" is not really about how intervals are tuned or what their precise "sizes" must be. However, if it is to apply to practical music making in which we do not have to re-tune fixed pitch instruments in order to play in certain keys, then it actually implies a particular kind of tuning, a tuning in which all or a number of the perfect fifths are not Pythagorean perfect fifths. In other words, the Great Circle of Fifths as it appears and is used in modern music theory, is not really the Pythagorean Circle at all. In fact, a Circle of fifths can only exist where some or all of the fifths round the Circle are not Pythagorean.            

Temperament

Tuning considerations cannot simply be ignored on Western keyboard instruments, or on many other instruments. 

The "complete" Circle from which the music theory Great Circle of Fifths is derived, is the Temperament Circle. When no temperament is present, the Circle will be a Pythagorean Circle, one example of which is :

The difference between this and the first Circle illustrated, is that this one has numbers between the note letter names. The numbers indicate the amount of tempering in each interval round the Circle. A tempered interval is one that has had its size deliberately increased or decreased from its Just Intonation size. Zero means no tempering, in other words, tuning by Just Intonation. The zeros here mean that the Pythagorean Circle is not really a Temperament Circle because no tempering is present. The '1' between G sharp and E flat is not a deliberately tempered amount - it is merely the consequence of tuning the other 11 intervals with zero tempering or Just Intonation. A Circle in which all 12 tempering values are zero, is impossible. The total amount of tempering round the Circle must always add up to a minimum of 1. Why ?

The answer lies in harmonic ratios. Harmonic ratios are simple whole-number ratios that can be associated with musical intervals. They can be applied to musical string or pipe lengths, or to frequencies, or to wavelengths. They are a good guide to the size of musical intervals produced from strings and pipes, or from any other source whose acoustical tone structure follows very similar frequency rules to those of strings and pipes. 

Author's special note:

Harmonic ratios are not universally applicable, and they are not inviolable laws! Many people feel that the naturally occurring association of a simple ratio with a musical interval, for example, 3/2 with the perfect fifth, is somehow magical, mysterious, arcane or Divine. Such a view is in effect a continuation of the Pythagorean tradition. The ratios are a perfectly natural consequence of natural physical laws. Their appearance is an elegant and beautiful manifestation, but equally beautiful and elegant manifestations occur in many other places in natural phenomena, science and mathematics. If you would like to know a little more about how harmonic ratios arise, click here.

It is surely somewhat narrow minded to be stubbornly Western-centric in our thinking. Other cultures recognise musical intervals that are explicitly not based on the harmonic ratios associated with ours. Furthermore, in many cases, the recognition of a musical interval, of consonance, and of good tuning, cannot be reduced simply to single ratio, in the context of modern acoustics. These things depend on tone structure, which is complex, involving many ratios, even just for a single musical interval. In sound, simple ratios can only be applied to simple, single tones called pure tones, and for these, the reliability of the association of a given harmonic ratio with a given musical interval, is limited. The relationship only works only within certain frequency ranges and perception can vary from person to person. It can even vary between the ears of one person. Harmonic ratios, whether applied to frequencies, string lengths or pipe lengths, are quite accurate, but they are still only an approximation. Harmonic ratios are useful, but they are far from being the last word on musical structure.

Harmonic theory shows that musical intervals are characterized by these harmonic ratios. Being a system of ratios, when musical intervals are "added" by being put together to create a new, bigger interval, their ratios are multiplied. 

The harmonic ratio for a perfect fifth is 3:2. Twelve consecutives perfect fifths (if they circulated right round the Circle) would produce a compound interval whose ratio is :   

The final note would be in the same position on the Circle as the starting note. However, 

it would not actually be precisely the same note, 'transposed' up a number of octaves. The interval formed by the 12 perfect fifths is not quite equivalent to 7 octaves. We could have divided the Circle into seven segments each representing an octave. The harmonic ratio of the octave is 2:1. The total compound octave produced round the Circle would then have the size :

These two large intervals do not mutually agree in size. There is a small interval between them, whose size is :

This musical interval difference between 12 consecutive perfect fifths and seven consecutive octaves, is called the Pythagorean comma, a comma being a microtonal interval difference between sequences of consecutive harmonic intervals.

The '1' in the Pythagorean Circle above, thus refers to 1 Pythagorean comma. There will always be in total at least 1 Pythagorean comma implicit in a Circle of 12 fifths. In the Pythagorean Circle the comma appears in its totality in just one interval, in this case between G sharp and E flat. By deliberately "mistuning" some fifths so that they are less than "perfect", or narrower than just intonation, the comma can be "distributed" in smaller quantities, so that its total appears divided amongst more intervals. This is the principle of a temperament.

Any fifth that, as a result of tempering in other intervals, becomes increased or decreased in size by a whole comma, ceases to be consonant - it becomes "wildly mistuned" and is not really recognisable as a beautiful musical fifth at all. Such an altered interval is called a wolf interval. It is impossible to construct a twelve-note-to-the-octave chromatic scale entirely from justly intoned perfect fifths and/or perfect fourths (inverted fifths). One of the fifths will always become a wolf, if the other 11 are perfect. There is no wolf in the modern piano scale, because in fact none of the fifths are perfect fifths, except "grammatically" speaking, using tonal grammar. Acoustically, they are tempered fifths - they are deliberately "mistuned", each by a very carefully controlled small amount - one twelfth of a Pythagorean comma.

The convention of calling the black (raised) notes on the piano by the default names of F sharp, A flat, B flat, C sharp and E flat, in the absence of any other tonal demand, is historical. In earlier systems of tuning keyboard instruments, such distinctions would be meaningful, and would refer to actual differences in note pitches. One would choose, for example whether the A flat would indeed be an A flat, or whether it would be a G sharp, and a change of choice would involve retuning the note.

Much of the modern piano repertoire is consequently notated in a way that is, as far as the actual acoustical notes are concerned, anachronistic and adhering to complications that are only necessary in the sense that they preserve historical practice. They preserve features that would only become objectively, and acoustically, significant should the music be played using a tuning system other than that employed on the modern piano. In other words, should a note be "incorrectly" written in the music as an A sharp rather than a B flat, then apart from the incorrect "grammar", the only possible difference this makes in the modern tuning practice is psychological - there is no objective, acoustical difference as far as the note produced by the piano is concerned. That is not to say a psychological difference is unimportant, but that would be a separate issue. 

All complications and enharmonic distinctions adherent in tonal "grammar" are entirely lost in the physical mapping from written music to piano keyboard, in the modern tuning system. The specific enharmonics used in the written music, may be psychologically significant, and may affect the way in which the music is interpreted and performed by the pianist, but it cannot affect the pitches of the notes or the intonation of the intervals. In terms of pitch and intonation, rather than contrived grammar, written enharmonic differences actually amount to nothing, on the piano.   

The Pythagorean Circle and comma is concerned with the 'perfect' intervals - fifths, octaves, and the inversion of the fifth within the octave, the perfect fourth. The 'imperfect' intervals thirds and sixths are in temperament theory dealt with primarily through another comma, similar in size, called the Syntonic comma.

The straight lines in the Circles above connect notes that define compound major thirds, unless the line subtends the small arc containing the interval G sharp to E flat. In the scale that the Pythagorean Circle defines, 8 of the major thirds are 'Pythagorean thirds' which are much larger intervals than the major thirds found on a modern piano, whilst the remaining 4 major thirds (defined by an arc containing the 'wolf') are practically harmonic, and called Justly Intoned, or pure.