mathematical scales

Ramsay
"The ratio of 2:3 twelve times, in Fifths, is so near the ratio of 1:2 seven times, in Octaves, as to allow this cycle of the mathematical scales to be closed without losing any of its vitality. The reason why there are thirteen instead " [Scientific Basis and Build of Music, page 29]

of twelve mathematical scales is that F# and G?, which in the tempered system are one, being counted the same, are made two scales in the mathematical; but it is a needless nicety. Twelve is the natural number and period for both mathematical and tempered scales. And as the system of twelve Fifths contains the key system of music four times, only three of these twelve Fifths being required for any one key, it follows that the tempered key is affected by only one-fourth part of the small amount to be tempered into the whole twelve. [scientific Basis and Build of Music, page 30]

The mathematical scales, if followed out regardless of other laws which rule in music, would read like a chapter in Astronomy. They would lead us on like the cycles of the moon, for example. In 19 years we have 235 moons; but the moon by that time is an hour and a-half fast. In 16 such cycles, or about 300 years, the moon is about a day fast; this, of course, is speaking roughly. This is the way seemingly through all the astronomical realm of creation. And had we only the mathematical ratios used in generating the notes of the scale as the sole law of music, we should be led off in the same way. And were we to follow up into the inaudible region of vibrations, we should possibly find ourselves where light, and heat, and chemical elective motions and electric currents are playing their unheard harmonies; or into the seemingly still region of solid substances, where an almost infinite tremor of vibrations is balancing the ultimate elements of the world. Music in this case would seem like some passing meteor coming in from among the silent oscillations of the planetary bodies of the solar system, and flashing past with its charming sound effects, and leaving us again to pass into the higher silence of those subtle vibrations to which we have referred, having no infolding upon itself, no systematic limit, no horizon. But music is not such a passing thing. Between the high silence of these intense vibrations, and the low silence of oscillating pendulums and revolving planets, God has constituted an audible sphere of vibrations, in which is placed a definite limit of systematic sounds; seven octaves are carried like a measuring line round twelve fifths; and motion and rest unite in placing a horizon for the musical world, and music comes [Scientific Basis and Build of Music, page 39]

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 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 Plate shows the Twelve Major and Minor Scales, with the three chords of their harmony - subdominant, tonic, and dominant; the tonic chord being always the center one. The straight lines of the three squares inside the stave embrace the chords of the major scales, which are read toward the right; e.g., F, C, G - these are the roots of the three chords F A C, C E G, G B D. The tonic chord of the scale of C becomes the subdominant chord of the scale of G, etc., all round. The curved lines of the ellipse embrace the three chords of the successive scales; e.g., D, A, E - these are the roots of the three chords D F A, A C E, E G B. The tonic chord of the scale of A becomes the subdominant of the scale of E, etc., all round. The sixth scale of the Majors may be written B with 5 sharps, and then is followed by F with 6 sharps, and this by C with 7 sharps, and so on all in sharps; and in this case the twelfth key would be E with 11 sharps; but, to simplify the signature, at B we can change the writing into C, this would be followed by G with 6 flats, and then the signature dropping one flat at every new key becomes a simpler expression; and at the twelfth key, instead of E with 11 sharps we have F with only one flat. Similarly, the Minors make a change from sharps to flats; and at the twelfth key, instead of C with 11 sharps we have D with one flat. The young student, for whose help these pictorial illustrations are chiefly prepared, must observe, however, that this is only a matter of musical orthography, and does not practically affect the music itself. When he comes to the study of the mathematical scales, he will be brought in sight of the exact very small difference between this B and C?, or this F# and G?; but meanwhile there is no difference for him. [Scientific Basis and Build of Music, page 108]

SYSTEM OF THE THREE PRIMITIVE CHROMATIC CHORDS.

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]

THE MATHEMATICAL AND TEMPERED SCALES.

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]