Intervals list (beating frequency)2

After my last message in reply to Lorenzo, I thought I'd finish the
task. First I'll set out the method more clearly:

f = frequency of lower pitch
t = size of tempered interval
p/q = ratio of (small number) just interval t approximates
B = |f(q.t - p)| = frequency of beating

|x| is the absolute value of x, i.e. any negative sign is disregarded.

So for a 12-TET M3, the calculation runs as follows:

B = |f(4.2^1/3 - 5)|
= 0.03968f (to five places after the point)

The list of intervals Lorenzo wants, from the slowest to the fastest
beating, runs as follows:

12-TET interval p/q B/f

P5 3/2 0.00339
P4 4/3 0.00452
M2 9/8 0.02030
m7 16/9 0.03618
M3 5/4 0.03968
M6 5/3 0.04538
m3 6/5 0.05396
m6 8/5 0.06299
M7 15/8 0.10199
m2 16/15 0.10805
a4 45/32 0.25483

I should stress the theoretical nature of the above results: the value
of f, and the harmonic spectrum of the tones involved can affect the
result in practice. Moreover, the beating given for the aug. fourth is
something of a fiction, since the harmonics that govern the beating are
very high. The M7 will also beat with 2/1 (0.11225) and the m2 with 1/1
(0.05946). I certainly wouldn't recommend that the beating frequency of
tempered intervals, whether in theory or practice, should be employed as
a criterion of consonance and dissonance.
--
Jonathan Walker
Queen's University Belfast
mailto:kollos@cavehill.dnet.co.uk
http://www.music.qub.ac.uk/~walker/

Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl
with SMTP-OpenVMS via TCP/IP; Thu, 31 Oct 1996 22:35 +0100
Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA05439; Thu, 31 Oct 1996 22:36:42 +0100
Received: from eartha.mills.edu by ns (smtpxd); id XA29548
Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI)
for id NAA24337; Thu, 31 Oct 1996 13:36:39 -0800
Date: Thu, 31 Oct 1996 13:36:39 -0800
Message-Id: <009AAB038BD0A780.6694@vbv40.ezh.nl>
Errors-To: madole@ella.mills.edu
Reply-To: tuning@eartha.mills.edu
Originator: tuning@eartha.mills.edu
Sender: tuning@eartha.mills.edu

What I want to mention is not pure jazz but jazzy xenharmonic music
on a record that I found in the library.
Mahmoud Ahmed: Ere mela mela. Crammed Discs CD 047, 1986.
Mahmoud Ahmed is one of the great singers from Ethiopia and a star
for a long time. This music is different and fascinating. His
intonation is unusual and may be related to the pelog-like scales that
are used in middle-Africa. For fans of other African music this modern
Ethiopian music is not to be missed. The recordings were made in
Addis-Abeba, 1975-1978. Crammed Discs, 43 rue Ge'ne'ral Patton, Brussels.

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; Mon, 4 Nov 1996 21:16 +0100
Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA01124; Mon, 4 Nov 1996 21:17:25 +0100
Received: from eartha.mills.edu by ns (smtpxd); id XA01127
Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI)
for id MAA29733; Mon, 4 Nov 1996 12:17:22 -0800
Date: Mon, 4 Nov 1996 12:17:22 -0800
Message-Id: <94961104201449/0005695065PK3EM@MCIMAIL.COM>
Errors-To: madole@ella.mills.edu
Reply-To: tuning@eartha.mills.edu
Originator: tuning@eartha.mills.edu
Sender: tuning@eartha.mills.edu