What do these unusual signs in the key signature mean?

Question

In Thesaurus of Oriental Hebrew Melodies by Abraham Zevi Idelsohn in 1923, I found this unusual key signature:

I know that the more typical notation for the little X in the key signature is used for double sharps, but in the intro of this volume the author of this work seems to say that he is using that symbol to denote a quarter tone:

For the sake of simplicity I have transposed all the examples into one common scale

which is similar to the scale of the Arabic Maqames Iraq, Siga and Aug and the Greek Dorian in which however steps 1, 4, 5 and 7 have three-quarters instead of a half

which was probably also in the Pentateuchal mode and prevails still among the Oriental Jews, while the European Jews have been influenced by the interval system of European music. The second step is augmented in the Orient by a quarter, in Europe by a half

However, nowhere does he specify what the angled lines mean. Anyone have any clues?

Answer 1

The angled lines are variant accidentals, indicating "one quarter tone sharp."

The notation in question is a variant key signature, not a time signature. (Disregarding the sometimes conflicting information in your second photograph), the key signature is "C double-sharp, B one quarter tone sharp, G one quarter tone sharp."

The several possibilities for quarter tones are given in Gardner Read's "Music Notation: A Manual of Modern Practice", pg. 145 (shown below). Your author's choice is documented on row 5, column 2.

Answer 2

In the Western music tradition we would call this method 'detuning' or 'microtonal shift'

In the western music tradition there are scales which follow the pattern

Whole Whole  Half, Whole Whole Whole Half

This means that from one note to its octave higher we can play a "scale" that sounds like beautiful travel up the notes to the next tonic (like from D up to D, or C to C)

If you look at the staff with the word "common line" to the left and you start on the higher D, you can see the scale wrap around in the fashion

1 1 .5, 1 1 1 .5 (whole whole half...)

Now look at the staff below this one and start on D again and wrap around

1 .75 .75, 1 1 .75 .75

This means that the Ds will sound the same as in the western piano tuning, as well as G and A.

The remaining notes of this scale ([d] E F [g] [a] B C) ... E, F, B, and C are all slightly different than in the western tuning. How are they different?

The X means to detune the "western traditional note" by 25 cents or 1/4 tone.

Detuning C, E, and F by 25 cents should make everything work because the total number of steps in the octave will still be the same as a "western" scale.