Topic: Two well-temperaments

2 scales

Thread (1 messages)

From: Herman Miller (2003-09-24)
Subject: Two well-temperaments

All the recent talk about well-temperaments, and especially the references
to Neidhardt's tunings and Paul Poletti's "Temperaments for Dummies"
article, got me playing around with well-temperaments again. I've been
using Werckmeister and Kirnberger tunings lately (the ones built in to the
DX7II), and the really sharp Pythagorean thirds in the distant keys still
bother me. There's one place in the WTC that's really unpleasant, with two
octaves between the root and the major third. But I like the effect of the
pure fifths in the distant keys. I never noticed Neidhardt's tunings
before, but they look pretty good. Neidhardt's 1724 "Village" / 1732 "Small
City" temperament has tolerable major and minor thirds, and at least sóme
of the distant keys have pure fifths. So I started playing around with some
tunings, and using a spreadsheet to make diagrams like the ones in
Poletti's article. Here are a couple that look good (and sound good, as far
as I can tell so far).

"Arrow I" (named from the shape of the fifth deviations on the chart)

C-G-D tempered by 3/20 Pythagorean comma
D-A tempered by 1/4 Pyth. comma
A-E-B tempered by 3/20 Pyth. comma
B-F#-C#-G# just
Ab-Eb tempered by 3/20 Pyth. comma
Eb-Bb-F-C just

The nice thing about this one is that it has lots of just fifths in distant
keys, and two of the major thirds are pretty good (only about 5 cents
sharp). The tempered Ab-Eb in the middle of the just fifths keeps the major
thirds in the distant keys from getting more than 18 cents away from just,
which is more tolerable than the 21.5 cents of Pythagorean major thirds.

! arrow1.scl
!
"Arrow I" well-temperament
 12
!
 93.75000
 196.87500
 294.14063
 391.40625
 498.04688
 591.79690
 698.43750
 795.70313
 892.96875
 996.09375
 1089.84380
 2/1

"Arrow II"

C-G-D tempered by 3/20 Pythagorean comma
D-A tempered by 1/5 Pyth. comma
A-E-B tempered by 3/20 Pyth. comma
B-F# tempered by 1/20 Pyth. comma
F#-C#-G# just
Ab-Eb tempered by 1/10 Pyth. comma
Eb-Bb-F just
F-C tempered by 1/20 Pyth. comma

This one has a couple of major thirds that are 19 cents sharp, and the best
thirds aren't as good as Arrow I. But it has a smoother progression of
"good" to "bad" major thirds, and the worst minor thirds aren't as bad as
Arrow I.

! arrow2.scl
!
"Arrow II" well-temperament
 12
!
 93.75000
 196.87500
 295.31250
 392.57813
 499.21875
 591.79688
 698.43750
 795.70313
 894.14063
 997.26563
 1091.01560
 2/1