Triadic or tetradic infill?

On Dec. 11, Daniel Wolf (sorry for being slow again, Daniel) said that he
was curious why a triadic infill should be preferred to a tetradic one.
I believe this is simply due to reasons of practicality.

Let me show this at two examples. To leave the well-trodden tracks a
little bit behind us, lets design an 11-limit scale (thank you, Manuel Op de
Coul, for introducing me into these parameters) within a twelfth, based
on the tetrad 5/7/9/11. A short way into the attempt we encounter the
interval between two important members of the new scale, 7/5 and
15/11. They are only 45.56 Cent apart. If we do not want to destroy the
harmonic context, we have to use this interval as one of the building
blocks of the scale. Thus we end up with a 39JT version within the
twelfth, containing such nice intervals like 11/9, 9/7, 15/11, 7/5, 11/7, 5/3,
9/5, 15/7, and 11/5, but also a lot of 'ballast intervals" and a variety of 4
different steps between 42.11 and 66.74 Cent. The corresponding ET
with 39 steps of 48.77 Cent is not too bad, but the whole process is
cumbersome, and the result not very inviting to a practical application.

However, if we restrict ourselves to using a triad instead, then we can
be bold and even attempt a 13-limit scale within the twelfth, based on
5/9/13. Our basic building block in this case appears between 9/5 and
15/9, has 133.24 Cent and leads us with little effort to a 15JT scale
within the twelfth, having only two different steps, 114.5 Cent and
133.24 Cent, respectively. The "defect" of the scale's ET (with 126.80
Cent per step) is not larger than the one of the 39-step version above,
despite the much larger step distance, and this is really an inviting scale,
containing in its just version 15/13,13/9, 5/3, 9/5, 13/5 and, if we are not
too puristic, also 4/3 (325/243) and 9/4 (729/325).

Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl
with SMTP-OpenVMS via TCP/IP; Fri, 20 Dec 1996 03:09 +0100
Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA09396; Fri, 20 Dec 1996 03:12:02 +0100
Received: from eartha.mills.edu by ns (smtpxd); id XA10828
Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI)
for id SAA09292; Thu, 19 Dec 1996 18:11:59 -0800
Date: Thu, 19 Dec 1996 18:11:59 -0800
Message-Id: <199612200208.VAA24966@copland.udel.edu>
Errors-To: madole@ella.mills.edu
Reply-To: tuning@eartha.mills.edu
Originator: tuning@eartha.mills.edu
Sender: tuning@eartha.mills.edu