**Ramsay**

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]

character of its own. And as Nature has constituted them, these various forces all converge to the Center of the Tonic Chord, and, with the exception of the interval of the octave itself, the notes of the **tempered scale** being a little nearer each other than the mathematically perfect notes, these converging forces and this tempering mutually assist each other, and give a greater decision to the resolution of chords. [Scientific Basis and Build of Music, page 36]

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 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]

See Also