Wikipedia
"In twelve-tone equal temperament all semitones are equal in size (100 cents). In other tuning systems, "semitone" refers to a family of intervals that may vary both in size and name. In Pythagorean tuning, seven semitones out of twelve are diatonic, with ratio 256:243 or 90.2 cents (Pythagorean limma), and the other five are chromatic, with ratio 2187:2048 or 113.7 cents (Pythagorean apotome); they differ by the Pythagorean comma of ratio 531441:524288 or 23.5 cents." Wikipedia

Ramsay
"Again, had the scales been properly derived from each other, the sharp of the one would always have been higher than the flat of the other;² and instead of the flat being two commas, less the apotome minor, higher than the sharp, the sharp would have been the apotome minor, the ratio of 8192:8201 1/4, higher than the flat; and there would never have been more than the apotome minor between the sharp of the one and the flat of the other, except where the scales were nine and the notes themselves were twelve fifths apart, and then the difference would have been a comma and the apotome minor." [Scientific Basis and Build of Music, page 14]

Getting Fifths as we ascend toward the number twelve they are in themselves the same, but with regard to their relationships they are quite different. Before and up to the twelfth fifth no scale has all the notes at the same distance above the first scale of the series. But after twelve, the thirteenth scale for example, B#, supposing the scale to be marked by sharps only, is a comma and a very small ratio above C; Cx is the same distance above D of the first scale; Dx the same above E; E# is the same distance above F; Fx the same distance above G; Gx the same distance above A; and Ax the same distance above B. So the scale of B# is just the scale of C over again at the distance of twelve-fifths, only it is a comma and the apotome minor higher; and each series of twelve-fifths is this distance higher than the preceding one. [Scientific Basis and Build of Music, page 30]

The simple natural scale is the fifth; the compound natural scale is the octave; the harmony scale, or chord-scale, is the three fifths; the great genetic scale is six octaves; for, like the six creation days, it takes the six octaves to give birth to the elements of which the wondrous structure of our music is built up; the birthplace of B, the seventh of the octave scale, is the sixth octave of the great genetic scale. The area of the twelve major and twelve minor scales is twelve fifths or seven octaves, the twelfth fifth being a comma and the apotome minor in advance of the seventh octave. This is a quantity so small that it can be ignored in real music; and the two notes, say E# and F, joined to close the circle of this horizon of our music world. E# is the top of the twelfth fifth, and F is the top of the seventh octave; and they are practically, though not exactly mathematically, the same note. Illustrations of this will be found among the plates of this work. [Scientific Basis and Build of Music, page 79]

G# as it occurs in the scales of A, E, and B major, and A? as it occurs in the scales of F and B? minor, are only distant the apotome minor, and are well represented by one key of the piano. It is only G# as it occurs in the scale of F six sharps major, and A? as it occurs in the scale of E six flats minor, that is not represented on the piano. These two extreme notes F# and E? minor are at the distance of fifteenth fifths and a minor third from each other. This supplies notes for 13 major and 13 minor mathematical scales; but as this is not required for our musical world of twelve scales, so these far-distant G# and A? are not required. The piano is only responsible for the amount of tempering which twelve fifths require, and that is never more than one comma and the apotome minor. [Scientific Basis and Build of Music, page 80]

The difference between B# and C? is the apotome minor - a very small difference - and this can only occur in the mathematical scales. In tempered scales, such as are played on the piano, one key serves equally well for both. Although seven sharps may be employed, seven black keys are necessary. As F# and G? have the same relation to each other as B# and C?, and as B# does not require a black key but is found on a white one, so all the semitonic necessities for twelve tempered scales are fully supplied by 5 black keys, since the white keys are as much semitonic as the black ones. [Scientific Basis and Build of Music, page 80]

the apotome minor; but one of these is the original comma which is genetically between the two D's; and it occurs here again at the 13th scale, the first of a new circle; it really corresponds to the two D's at the beginning of this first series. Whenever there is more than one comma and the apotome minor between G# and A?, it is because there has been a mistake in counting this one over again; or some other mistake. [Scientific Basis and Build of Music, page 86]

The reason why there are 13 mathematical scales is that G? and F# are written separately as two scales, although the one is only a comma and the apotome minor higher than the other, while in the regular succession of scales the one is always 5 notes higher than the other; so this G? is an anomaly among scales, unless viewed as the first of a second cycle of keys, which it really is; and all the notes of all the scales of this second cycle are equally a comma and the apotome higher than the notes of the first cycle; and when followed out we find that a third cycle is raised just as much higher than the second as the second is higher than the first; and what is true of these majors may be simply repeated as to the D# and E? of the minors, and the new cycle so begun, and all successive minor cycles. Twelve and not thirteen is the natural number for the mathematical scales, which go on in a spiral line, as truly as for the tempered scales, which close as a circle at this point. [Scientific Basis and Build of Music, page 89]

The middle portion with the zigzag and perpendicular lines are the chromatic chords, as it were arpeggio'd. They are shown 5-fold, and have their major form from the right side, and their minor form from the left. In the column on the right they are seen in resolution, in their primary and fullest manner, with the 12 minors. The reason why there are 13 scales, though called the 12, is that F# is one scale and G? another on the major side; and D# and E? separated the same way on the minor side. Twelve, however, is the natural number for the mathematical scales as well as the tempered ones. But as the mathematical scales roll on in cycles, F# is mathematically the first of a new cycle, and all the notes of the scale of F# are a comma and the apotome minor higher than G?. And so also it is on the minor side, D# is a comma and the apotome higher than E?. These two thirteenth keys are therefore simply a repetition of the two first; a fourteenth would be a repetition of the second; and so on all through till a second cycle of twelve would be completed; and the thirteenth to it would be just the first of a third cycle a comma and the apotome minor higher than the second, and so on ad infinitum. In the tempered scales F# and G? on the major side are made one; and D# and E? on the minor side the same; and the circle of the twelve is closed. This is the explanation of the thirteen in any of the plates being called twelve. The perpendicular lines join identical notes with diverse names. The zigzag lines thread the rising Fifths which constitute the chromatic chords under diverse names, and these chords are then seen in stave-notation, or the major and minor sides opposites. The system of the Secondary and Tertiary manner of resolution might be shown in the same way, thus exhibiting 72 resolutions into Tonic chords. But the Chromatic chord can also be used to resolve to the Subdominant and Dominant chords of each of these 24 keys, which will exhibit 48 more chromatic resolutions; and resolving into the 48 chords in the primary, secondary, and tertiary manners, will make 144 resolutions, which with 72 above make 216 resolutions. These have been worked out by our author in the Common Notation, in a variety of positions and inversions, and may be published, perhaps, in a second edition of this work, or in a practical work by themselves. [Scientific Basis and Build of Music, page 115]

This diagram shows pictorially the open in the spiral of the mathematical scales, in which, if written in sharps only, B# is seen a little, that is, a comma and the apotome minor, in advance of C, and as the first scale of the new cycle; for it is a violation of Nature's beautiful steps to call it a thirteenth scale of this order, since every scale in the order is 31 commas in advance of the preceding, whereas B# is only one comma and a small fraction in advance of C. If the scales be written in ?s and #s for convenience of signature, then G# is seen a comma and apotome in advance of A?; while the whole circle of keys advancing by fifths are each 31 commas in advance of the preceding. We may therefore cast utterly from us the idea of there being more than twelve mathematical scales, and view the so-called thirteenth as simply the first of a new round of the endless spiral of scales. There is, however, in this note a banner with the strange device, "Excelsior," for it leads us onward into ever-advancing regions of vibrations, and would at last bring us to the ultimate and invisible dynamic structure of the visible world. The tempered system of 12 keys, as in Fig. 1, is by causing the G# and A? to coalesce and be one, as the two D's are already literally one by Nature's own doing. [Scientific Basis and Build of Music, page 118]

The inner stave contains the chromatic scale of twelve notes as played on keyed instruments. The flat and sharp phase of the intermediate notes are both given to indicate their relation to each other; the sharpened note being always the higher one, although seemingly on the stave the lower one. The two notes are the apotome minor apart overlapping each other by so much; ?D is the apotome lower than C#; ?E the apotome lower than D#; F# the apotome higher than ?G; G# the apotome higher than ?A; and A# the apotome higher than ?B. The figures for the chromatic scale are only given for the notes and their sharps; but in the mathematical series of notes the numbers are all given. [Scientific Basis and Build of Music, page 120]

See Also

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apotome
apotome higher
apotome lower
apotome minor
apotome scales
apotomes and skismas
Comma
Diesis
Interval
Note
pythagorean komma
Semitone