Pentatonic source scale

John Chalmers wrote:

> Other interesting pentatonics are slendro-like septimal scales such as
> 1/1 7/6 4/3 3/2 7/4 2/1 and neutral ones such as 1/1 11/9 4/3 3/2
> 11/6 2/1, etc. and their modes.

If you take the 13-limit pentad 1/1--26/21--14/11--14/9--8/5 and duplicate
it in four parallel 3/2's, i.e.

28/27--14/9---7/6-----7/4

104/63--26/21--13/7--39/28

4/3-----1/1---3/2----9/8

16/15----8/5----6/5---9/5

56/33---14/11--21/11--63/44

you get a fairly evenly spaced 20-tone scale that contains thirty-two
different pentatonics of the form

1/1--X--4/3--3/2--(3/2)X--2/1

where "X" ranges from a minor second (giving you "descending" or
koto-like pentatonics), through major seconds and then minor and major
thirds (for "ascending" or gamelan-like pentatonics), including the
two scales John mentions.

This comes about because the pentad contains eight different major or
neutral thirds, ranging from 11/9 (347c) to 84/65 (444c). Adding the
3/2's thus creates eight (each) major seconds, minor seconds, and
minor thirds for a total of 32 values of "X". (Of the 32 six are just
a few cents apart, so the number of really distinct pentatonics is 26.)

The scale can be tempered and evened out a bit by using the
corresponding tones from scales like 72-TET or 75-TET, which will give
you just the 26 distinct pentatonics. Since twenty is a nice
manageable source scale size for fixed-pitch instruments like guitar
or metallophone, I thought this might be of interest.

Ken Wauchope




Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl
with SMTP-OpenVMS via TCP/IP; Wed, 30 Oct 1996 20:16 +0100
Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA30288; Wed, 30 Oct 1996 16:02:50 +0100
Received: from eartha.mills.edu by ns (smtpxd); id XA04021
Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI)
for id HAA14664; Wed, 30 Oct 1996 07:02:48 -0800
Date: Wed, 30 Oct 1996 07:02:48 -0800
Message-Id: <961030150106_71670.2576_HHB68-10@CompuServe.COM>
Errors-To: madole@ella.mills.edu
Reply-To: tuning@eartha.mills.edu
Originator: tuning@eartha.mills.edu
Sender: tuning@eartha.mills.edu