source file: m1363.txt Date: Mon, 23 Mar 1998 17:26:27 -0500 Subject: Well Temperaments, Just Diminished From: monz@juno.com (Joseph L Monzo) 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 He got it from an unpublished English translation of Werckmeister's treatise.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?) ---------------------------------------------- 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 _____________________________________________________________________ You don't need to buy Internet access to use free Internet e-mail. Get completely free e-mail from Juno at http://www.juno.com Or call Juno at (800) 654-JUNO [654-5866]