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
Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Fri, 17 Jan 1997 02:13 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA12385; Thu, 16 Jan 1997 11:01:41 +0100 Received: from eartha.mills.edu by ns (smtpxd); id XA12285 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id CAA22254; Thu, 16 Jan 1997 02:01:38 -0800 Date: Thu, 16 Jan 1997 02:01:38 -0800 Message-Id: <199701160500_MC2-F50-FF96@compuserve.com> Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu
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
Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Tue, 21 Jan 1997 17:18 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA10671; Tue, 21 Jan 1997 17:21:49 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA10634 Received: from by ella.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id IAA07459; Tue, 21 Jan 1997 08:21:46 -0800 Date: Tue, 21 Jan 1997 08:21:46 -0800 Message-Id: <009AEA8BF326C8C0.043E@vbv40.ezh.nl> Errors-To: madole@mills.edu Reply-To: tuning@ella.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@ella.mills.edu
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
Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Wed, 22 Jan 1997 00:33 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA11128; Wed, 22 Jan 1997 00:36:44 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA01950 Received: from by ella.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id PAA15460; Tue, 21 Jan 1997 15:36:39 -0800 Date: Tue, 21 Jan 1997 15:36:39 -0800 Message-Id: <009AEB7312CC8240.0638@vbv40.ezh.nl> Errors-To: madole@mills.edu Reply-To: tuning@ella.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@ella.mills.edu