Eschatum & hyperoche

Paul Erlich found the terms "hyperoche" and "eschatum" in my program,
and I'll explain them here. This comes from Rasch's edition of
Sauveur's work.
They are two intervals named by Conradus Henfling (1648-1716), in a
letter to Sauveur published by the Royal Academie of Sciences of
Prussia in Berlin, in 1710. Henfling and Sauveur were constructing
equal temperaments by assigning numbers of steps to basic intervals.
Henfling started with the diatonic and chromatic semitones (in casu
the Pythagorean limma and apotome) which he called "diatonum" and
"chroma". He called the difference between the two "harmonie" (which
is then the Pythagorean comma). The difference between chroma and
harmonie is "hyperoche" and the difference between harmonie and
hyperoche "eschatum". If you set them equal in size, and both one
step, you get an octave of 50 steps. If one assumes the eschatum to be
zero it gives an octave of 31 steps. Setting the hyperoche to zero
leads to 19 and finally a zero harmonie to 12.
Sauveur used a different method, he set the ratio of the comma to the
chromatic semitone to an integer number. This leads to the systems 12,
19, 31, 43, 55 and 67 (0:1, 1:1, 1:2, 1:3, etc.). Later he made a
table of equal temperaments which also included fractional ratios
between the two, for instance 2:3 gives 50, 3:10 gives 141 and so on.

Manuel Op de Coul coul@ezh.nl

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