# Mathematical Questions and Solutions in Continuation of the Mathematical Columns of the Educational Times Volume 28

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This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1878 edition. Excerpt: ... necessarily obtain differential tones even iroin tlie lowest partials, which not only will not fit into places as partials of any fundamental, but will sometimes be so close to one another or to somo other partial as to produce beats. Thus, if m' (p, q, r) be altered to m' _p, q (l + t), J-(l + s), the differentials of tho former, m (q-p, r--p, r--q), which fit into the fundamental, become m q--p + qt, r--p + sr, r--q + tr--tq, which cannot possibly fit in and may often beat. Thus, if m' (4, fi, 6) become m' 4, 5 + 5t, 6 +6s, the differentials are m l + 5, 2 +6s, l + 6s--5i, and the first and last of these beat m (6s--lOt) times in a second. This is quite distinct in both the equal and mean tone temperament, although the beats are between faint notes. But if ml (3, 5, 8) is altered to m' 3 (1 + s), 5 (1 + f), 8, the differentials are m2 + 5t--3s, 5-3s, 3--5t, and the two latter beat with the lowest partials m (5 + &t) and m (3 + 3s) of the principal notes themselves, the beats being 5t + 3s in each case, so that they reinforce each other and become very conspicuous, making the tempered combination very bad, while the just combination is very good. Similarlyfor the alteration of m' (2, 3, 5) into ml 2, 3 (1 +s), 8(1+!!). 9. Calculations for the "Estimation of Oonsidence and Hissidence.--It may so happen that we do not know of what simple interval any given interval is an alteration. We may then proceed thus, taking as examples dissidences of the form m' p, q where p and q are large numbers. The following is an explanation of the calculation and results. Ex. 1. 1' 264, 394-772, the 1' shows that the absolute pitches of the dissident notes are given within the brackets. The dissident beats arise from the partials which are...show more

## Product details

• Paperback
• 189 x 246 x 2mm | 68g
• United States
• English
• black & white illustrations
• 1236914066
• 9781236914064