Reply to PaulE

Paul: I was referring to the Minkowski "Taxicab" "City Block," or
"Manhattan" metric where the distance function is ds|dx| +|dy| +|dz|...
rather than the square root of the sum of the squares of the Euclidean
or relativistic metrics. Thus 6/5, 5/3, 16/15 and 15/8 would have
the same distance from 1/1 as you point out.

As for "punning," one may map the relevant region of the tonal
lattice onto (or do I mean into?) some ET by using Fokker's
application of the cross and/or box products after choosing the
intervals (kommata) one wishes to set to zero. I think his papers
on "Periodicity Blocks" are in the Bibliography.

Whether one should use a pseudo-Cartesian (right angle) lattice
or a polygonal/polyhedral one is for me mostly a matter of taste.
At Paul E's earlier suggestion, I started plotting 7 and higher
limit scales on oblique, polygonal lattices (the simpler ones are
easily interpretable as polyhedrons in 3 space). I must admit that
I got stunning graphics, though beyond about the 11-limit, I find
the rectilinear mode easier to interpret.

However, I got fascinated by the patterns, so, I wrote a series of
Q&D Basic programs to plot scales (after factoring their ratios) on a
series of polygonal lattices of the type Erv Wilson has used. I stopped
at the centered-tridekagon and the 43-limit. I recommend this
approach to those of you with faster computers, more memory, and higher
resolution graphics than I have.

I might mention the Carter Scholz's latest version of JiCalc plots
scales on a 31-limit lattice (pseudo-Cartesian). It's downloadable
from Mills. Also, it plays each lattice point when the note
is clicked.

-John



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To Graham Breed (2nd attempt, sorry if it appears twice)

Helmholtz devotes a number of pages to Arabic and Persian scales in
"On the sensations of tone".
The scales you gave were catalogued by Safi al-Din (Bagdad, 13th century).
They are among the twelve main maqams that he described.
To answer your second question, yes they are from a reliable source.
A good one is
Liberty Manik: Das arabische Tonsystem im Mittelalter. PhD diss. Freie
Univ. Berlin, 1969. E.J. Brill, Leiden, 1969, 140 pages.
They are also in my list of modes which also contains a few modern Arabic
scales: ftp://ella.mills.edu/ccm/tuning/papers/modename.txt .
A good introductory source for answers to your other questions is
Habib Hassan Touma: Musik der Araber. English translation by Laurie
Schwartz: The music of the Arabs, Amadeus Press, Portland, 1996, 238 pages.
Modern tunings, at least in Turkey, often use a subset of 53-tET.
They also use an interval unit of 1060 parts to the octave (1/20 of a
53-tone comma) there, but I don't know what its name is. Subsets of
24-tET are also used, in more popular music.
As an interesting aside: the word maqam is a cognate of the Jewish nickname
for Amsterdam: Mokum. Its meaning is "the place", or so I'm told.

Manuel Op de Coul coul@ezh.nl

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To Graham Breed:

Helmholtz devotes a number of pages to Arabic and Persian scales in
"On the sensations of tone".
The scales you gave were catalogued by Safi al-Din (Bagdad, 13th century).
They are among the twelve main maqams that he described.
To answer your second question, yes they are from a reliable source.
A good one is
Liberty Manik: Das arabische Tonsystem im Mittelalter. PhD diss. Freie
Univ. Berlin, 1969. E.J. Brill, Leiden, 1969, 140 pages.
They are also in my list of modes which also contains a few modern Arabic
scales: ftp://ella.mills.edu/ccm/tuning/papers/modename.txt .
A good introductory source for answers to your other questions is
Habib Hassan Touma: Musik der Araber. English translation by Laurie
Schwartz: The music of the Arabs, Amadeus Press, Portland, 1996, 238 pages.
Modern tunings, at least in Turkey, often use a subset of 53-tET.
They also use an interval unit of 1060 parts to the octave (1/20 of a
53-tone comma) there, but I don't know what its name is. Subsets of
24-tET are also used, in more popular music.
As an interesting aside: the word maqam is a cognate of the Jewish nickname
for Amsterdam: Mokum. Its meaning is "the place", or so I'm told.

Manuel Op de Coul coul@ezh.nl

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