Tempering issues

Paul Erlich wrote:

> This "sameness" and "in-tune-ness" are surprisingly flexible, especially
> when it comes to multiple octaves. Playing the same melody on opposite
> ends of a piano simultaneously, you can get away with a semitone
> transposition without much disturbing the "sameness" or "in-tune-ness"
> (Yasser gives a melody to use for this, and I didn't believe it until I
> tried it!)

Let's assume that the suitability of a temperament depends upon
the most poorly tuned interval in the relevant consonant limit.
If you take the Partch odd limit as the definition of consonance,
then all octave transpositions of the main intervals have to be
counted. If the octave is not perfect, some equivalent intervals
will be tuned worse as a result. So, the suitability of the
temperament diminishes. If you've got the same pitch set in all
registers, it will be difficult to avoid the bad intervals. A
small octave stretching may work to favour the most frequently
used intervals but, for this to be acceptable, it must have a
very small harmonic effect.

As an example, if you play two 4:5:6:8 chords an octave apart,
you get the intervals 3/1, 5/2, 8/3 and 16/5 coming out. A
mistuning of the octave will become threefold apparent in two of
these intervals. Harmony isn't really octave invariant, as I've
argued before, but all these intervals will occur between melody
and harmony.

How big is a piano keyboard? One semitone at 8 octaves implies a
formal octave 100/8=12.5 cents sharp. That's better than the
thirds in 12-equal.


Paul's example of the 8:10:12:15 chord is a good one. Taking 15/8
as a dissonance, this chord will still work best the better the
fifths and thirds are tuned, so that their consonance is more
evident. I think 15/8 has some degree of consonance, though, so
if this chord is to be used frequently this interval should be
considered, so that the difference tones are in tune.

While experimenting with this, I found that 8:10:15 sounds better
than 4:5:7, probably because of the fifth. 4:6:7 is preferable to
8:12:15, so 7-limit still rules!

If the presence of consonances can make up for a dissonance in
a chord, it follows that a strong consonance is important even
in the presence of other consonances. So, octaves and fifths
should be considered first in choosing a temperament. I think
12-equal works partly by 3-limit harmony plus poorly tuned thirds
functioning as weak dissonances. Also, because memory corrects
the tuning of familiar chords. You're still going to notice if
one interval is way out, though. The worst tuned interval in a
chord is still the most important.


SMTPOriginator: tuning@eartha.mills.edu
From: mr88cet@texas.net (Gary Morrison)
Subject: CDRs and DVDs
PostedDate: 17-01-98 15:18:37
SendTo: CN=coul1358/OU=AT/O=EZH
ReplyTo: tuning@eartha.mills.edu
$MessageStorage: 0
$UpdatedBy: CN=notesrv2/OU=Server/O=EZH,CN=coul1358/OU=AT/O=EZH,CN=Manuel op de Coul/OU=AT/O=EZH
RouteServers: CN=notesrv2/OU=Server/O=EZH,CN=notesrv1/OU=Server/O=EZH
RouteTimes: 17-01-98 15:18:37-17-01-98 15:18:38,17-01-98 15:17:40-17-01-98 15:17:41
DeliveredDate: 17-01-98 15:17:41
Categories:
$Revisions:

Received: from ns.ezh.nl ([137.174.112.59]) by notesrv2.ezh.nl (Lotus SMTP MTA SMTP v4.6 (462.2
9-3-1997)) with SMTP id C125658F.004E9A42; Sat, 17 Jan 1998 15:18:32 +0100
Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA28007; Sat, 17 Jan 1998 15:18:37 +0100
Date: Sat, 17 Jan 1998 15:18:37 +0100
Received: from ella.mills.edu by ns (smtpxd); id XA28050
Received: (qmail 4376 invoked from network); 17 Jan 1998 06:18:34 -0800
Received: from localhost (HELO ella.mills.edu) (127.0.0.1)
by localhost with SMTP; 17 Jan 1998 06:18:34 -0800
Message-Id:
Errors-To: madole@mills.edu
Reply-To: tuning@eartha.mills.edu
Originator: tuning@eartha.mills.edu
Sender: tuning@eartha.mills.edu