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The classical Hebrew lyre was a rectangle, which had 10 strings according to Josephus Flavius, just like the Ten Commandments were written on the Tablets of the Law in 10 vertical columns, known as the 10 Sephirot or spheres. The Hebrew name for a lyre is Kinnor, which was early personified in Greek mythology as Cinyras.

Reprinted from Archives for Mosaical Metrology and Mosaistics (AMMM) vol. 2, no. 3 (1995) p. 29 by courtesy of BAALSCHEM PRESS Verlag,
Bahnhofstr. 19, D-35745 Herborn 1,
GERMANY.

For writing purposes, there were 15 horizontal lines on every tablet, resulting in 150 squares of one tenth of a cubit each. Their musical relevance was first suggested to me by the musical note b, which is a semitone below c and generally defined as 8/15, while the c above it is half the string or 7.5/15. If 15 is the common denominator of the frequency ratios, we immediately get the upper tetratchord of the Octave, namely a = 9/15 = 3/5 and g = 10/15 = 2/3.

Reprinted from Archives for Mosaical Metrology and Mosaistics (AMMM) vol. 2, no. 3 (1995) p. 24 by courtesy of BAALSCHEM PRESS Verlag,
Bahnhofstr. 19, D-35745 Herborn 1,
GERMANY.

Since the division of the Octave (c : c' = 1/2) into any number (= n) of equal intervals is the n-th root of 1/2, and hence an irrational number, the main problem in designing our musical scale by Moses in 1441 B. C. E. was mathematical. The solution was found by mathematical interpolation from a good approximation for the middle of the Octave, and by subdividing each half-octave into three tones or six semitones, thus supplying the precise frequency ratios of our diatonic scale.

Of course, all this was calculable 3 1/2 thousand years ago, for (7/10)2 = 49/100 and 50/100 = 1/2, so that the middle of the Octave is close to 7/10, and each half is easily divided into 3/10 and 3/15. This explains the relevance of Mosaical matrixes, namely of tenths and fifteenths for music. While the horizontal division into fifteenths is used for the upper tetrachord, as explained above, the lower tetrachord makes use of the division of each of the Tablets of the Law into 60 Manah weight-stones by 6 vertical and 10 horizontal lines (see the first three adjoining diagrams).

Reprinted from Archives for Mosaical Metrology and Mosaistics (AMMM) vol. 2, no. 4 (1997) pp. 24 and 25
by courtesy of BAALSCHEM PRESS Verlag,
Bahnhofstr. 19, D-35745 Herborn 1,
GERMANY.

The precision of this method is beyond what the human ear can hear, and what the human voice or musical instruments can produce. The frequency ratios obtained by interpolation bring about an aberration of plus/minus 1 percent or less from the even-tempered standard, as elaborated on the first adjoining chart. Of the frequency ratios, all are less than 1/8-tone off the even-tempered standard, 5 less than 1/16-tone, and 2 less than 1/32-tone, if compared with the values on the second adjoining chart. Already the quartertone between two semitones is hardly appreciated.

Reprinted from Archives for Mosaical Metrology and Mosaistics (AMMM) vol. 2, no. 3 (1995) p. 30 by courtesy of BAALSCHEM PRESS Verlag,
Bahnhofstr. 19, D-35745 Herborn 1,
GERMANY.

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