Well Temperaments, Just Diminished

RE: Well Temperaments

The following table is the adjustment from equal temperament (in cents)
used by Johnny Reinhard for the Werckmeister III temperament used in last
week's performance by the Soho Baroque Opera. He got it from an
unpublished English translation of Werckmeister's treatise. (Beware of
other listings of Werckmeister III -- they contain errors!)

C = +12
C#/Db = +2
D = +4
D#/Eb = +6
E = +2
F = +10
F#/Gb = 0
G = +8
G#/Ab = +4
A = 0
Bb/A# = +8
B = +4
C = +12

As I stated in the review of this performance which I posted to Tuning
Digest, tunings are usually described as starting on C (which would make
C the note with O cents adjustment), but as there was no standard
reference pitch in use in Europe at the time these temperaments were in
use, Reinhard felt it best to tune to A-440 and calculate the adjustments
from there.

>From the research I've done, Werckmeister III seems to have been the
most popular of the well temperaments. (Anyone else know otherwise?)

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RE: Diminished-7th Chords

Diminished-7th chords were used by Bach, so they definitely pre-date the
establishment of equal-temperament, although they became much more
widespread after the acceptance of 12-equal, as they are singularly
fitted to that tuning because of their equal intervals. Note that in
reference to a 12-equal diminished 7th chord, transposition to any of the
chord members, and inversion, are equivalent.

When constructing diminished-7th chords in just-intonation, several
different approaches may be used.

As far as 3-limit goes, I've never experimented with diminished-7ths in
this tuning, nor have I ever heard of anyone else doing so. The minor
3rd has the ratio 32/27 (294 cents). Anyone out there try it?

For 5-limit, the minor 3rds may be either 32/27, or 6/5 (316 cents). The
particular arrangement of these determines the sound of the chord. There
are a few different constructions that can be used; here are the most
consonant :

A succession of three 6/5s is one method. This gives, obviously, the
most consonant minor 3rds, with a diminished 5th with the ratio 36/25
(631 cents), and diminished 7th of 216/125 (947 cents).

Another possibility is a 6/5, a 32/27, and another 6/5. This gives a
diminished 5th of 64/45 (610 cents) and diminished 7th of 128/75 (925
cents).

Of course, allowing for enharmonic equivalence, an augmented 4th could be
used instead of a diminished 5th. Interestingly, using the intervals
6/5, 75/64, and 6/5, gives a chord with exactly the same interval
structure described in the example above, just transposed. In this
chord, the diminished 5th is technically an augmented 4th of 45/32 (590
cents), and the diminished 7th is technically a Pythagorean 6th of 27/16
(906 cents).

All of these are fairly dissonant, and generally they only work as
"resolving" chords, that is, they must move to another, more consonant
chord.

As more and more prime numbers are allowed in the factors of the ratios,
the number of possible notes that may be used is increased exponentially,
making it impractical to list them all in a posting. However, allowing
higher primes also gives more consonant diminished-5th and diminished-7th
intervals.

In the 7-limit, the minor 3rds may be either 32/27, 6/5, or 7/6 (2.67
cents). Various different combinations of these give interesting and (to
varying degrees) pretty good diminished-7th chords.

In mid-1800s Austrian harmonic theory (beginning, I believe, with
Sechter, and continuing all the way to Schoenberg, and to a large extent
even today), the diminished-7th chord was considered to be a dominant-7th
flat-9th with the root omitted. As first noted by Ellis in his
translation of Helmholtz's "On the Sensations of Tone", the most
consonant tuning for the diminished-7th chord is to use the intervals 6/5
- 7/6 - 17/14. This gives a chord with the chord identities 5, 3, 7,
and 17, which are indeed the 10th, 12th, 14th and 17th harmonics of the
missing fundamental, and in *perfect* tuning, they would emphasize that
missing fundamental. Note also that all the difference tones of this
chord are octaves below the chord members, which serves to further
increase the feeling of both consonance and the fundamental.

Joseph L. Monzo
monz@juno.com
4940 Rubicam St., Philadelphia, PA 19144-1809, USA
phone 215 849 6723

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>While an important book in awakening the electronic musicians
>to microtones, Scott Wilkinson's _Tuning In_ has some wild notes listed
>for Werckmeister III. I wonder, Gary, do they match those of Werckmeister
>IV or V?

I've heard of that book, but I'm afraid I haven't seen it.

>>While an important book in awakening the electronic musicians
>>to microtones, Scott Wilkinson's _Tuning In_ has some wild notes listed
>>for Werckmeister III. I wonder, Gary, do they match those of Werckmeister
>>IV or V?
>
> I've heard of that book, but I'm afraid I haven't seen it.

Here are Scott's numbers for Werckmeister III:

Note Cents
C 0.00
C# 93.67
D 192.18
D# 277.69
E 390.23
F 474.58
F# 588.27
G 696.10
G# 790.23
A 888.28
A# 974.79
B 1092.19
C 1200.00

Since he doesn't give any clues about how he derived these numbers, I can't
say why there are discrepancies between his, and, say, Barbour's table for
"Werckmeister's Correct Temperament #3":

C 0
C# 96
D 204
Eb 300
E 396
F 504
F# 600
G 702
G# 792
A 900
Bb 1002
B 1098
C 1200

Bill

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