Why I hate Vallotti – cont.
I believe the answer is 1/6 comma meantone, a regular meantone system where all the fifths but one (the wolf) are narrowed by 1/6 of a sytonic comma. The resultant temperament has several keys that are equally consonant, several M3rds of good quality, predictable semitones of two different sizes–one for diatonic and
one for chromatic semitones–as well as other musical assets. Threaded throughout the discussion in Bruce Haynes’s superb 1991 Early Music article (pp. 357–81), “Beyond temperament: non–keyboard intonation in the 17th and 18th centuries,” is the frequent re–appearance of 1/6 comma meantone. Already widely known today as Silbermann’s temperament (after the organ builder and friend of J. S. Bach), it corresponds to ideals for temperaments given by Telemann, Tosi, Quantz, and Geminiani, among others, and is described by the French theorist Sauveur (1707) as the temperament of “ordinary musicians”. The historical evidence as presented by Haynes speaks for itself, and is a stunning endorsement. What better temperament to use as a standard for Baroque music than the one employed by everyday musicians at the time? So, from an historical standpoint for music from the earlier 18th century, 1/6 comma beats Vallotti and Young. Its cents chart (with rounded approximations) is given here:

Fig 1. Vallotti and Young “Circulating” Temperaments in Cents
Fig 2. 1/6 Comma Meantone Temperament in Cents
C C# D Eb E F F# G G# / Ab A Bb B C
0 88.6 196.7 304.9 393.5 501.6 590.2 698.4 787 / 806.5 895.1 1003.3 1091.9 1200
Procedure: Temper all 5ths by 1/6 of a syntonic comma, leaving the wolf at Eb—G# or at Ab—C#.
Listen to theabove example with Ab (click below)

Listen to the above example with G# (click below)

Furthermore, it is clear from the many references to major and minor (diatonic and chromatic) semitones and particularly the distinction between sharp and flat versions of a note in woodwind fingering charts well into the 18th century–not to mention Peter Prelleur’s violin fingerboard illustration of 1731–that some form of regular meantone temperament is being assumed as a standard. Vallotti and Young have major and minor semitones in some places, it’s true, but the whole point of a circulating temperament is that the notes can remain fixed and still be usable. Distinctions between G# and Ab are unnecessary because the compromise position of the note in Vallotti and Young is meant to serve for both harmonic contexts. In other words, if Baroque musicians were thinking in terms of circulating temperaments as a standard, then they wouldn’t need to distinguish between enharmonic notes in fingering charts, and they wouldn’t need to build keyboards with split keys, for which there is still some scattered evidence in the 18th century. In addition, the difference between the major and minor semitone in 1/6 comma meantone is very close to a syntonic comma, which lends an extra degree of melodic elegance in a system based on divisions of that interval. The fact that diatonic semitones are always the same size and chromatic semitones are always the same size, however, creates a predictability as to where the notes will be found that Vallotti and Young simply cannot provide. Furthermore, this regularity enables fretted instruments to play fairly easily with the temperament–setting major and minor semitones on the fingerboard–whereas trying to set frets to match an irregular temperament can be hopelessly frustrating.

1/6 comma semitone with Ab

flowerLG
Click image for semitones playlist.

 Or download this interactive version

1/6 comma semitone with G#

flowerLGClick image for 
semitones playlist.

 Or download this interactive version

 

The large number of good M3rds, meanwhile, and the keys that use them, mean that the possibilities for transposition are superior in 1/6 comma over Vallotti and Young. Pitch standards like Chorton (A=ca.460) and Kammerton (A=ca.415) in Germany must have required frequent transposition by whole tone up or down. In 1/6 comma, this can often be done with no change in the relative position of the notes. But in an irregular system, the tempering is always different for a transposed key from its written version. “Hey, don’t forget, we’re playing in G at A=415 but the organ is in Young at A=460 with the player transposing to F, so that G is going to be lower (or the B and D higher) than you expect, heh, heh.” Such conversations were unnecessary with 1/6 comma, but must have taken place, even among muscians familiar with the temperament, when irregular temperaments were in use.

 

   
1/6 comma 5th diagram with Ab 1/6 comma 5th diagram with G#
  Click on a diagram to view a full size image

The large number of good M3rds, meanwhile, and the keys that use them, mean that the possibilities for transposition are superior in 1/6 comma over Vallotti and Young. Pitch standards like Chorton (A=ca.460) and Kammerton (A=ca.415) in Germany must have required frequent transposition by whole tone up or down. In 1/6 comma, this can often be done with no change in the relative position of the notes. But in an irregular system, the tempering is always different for a transposed key from its written version. “Hey, don’t forget, we’re playing in G at A=415 but the organ is in Young at A=460 with the player transposing to F, so that G is going to be lower (or the B and D higher) than you expect, heh, heh.” Such conversations were unnecessary with 1/6 comma, but must have taken place, even among musicans familiar with the temperament, when irregular temperaments were in use.

 

1/6 comma M3 with Ab

flowerLG
Click image for semitones playlist.

 Or download this interactive version.

1/6 comma M3 with G#

flowerLGClick image for
semitones playlist.

 

Or download this interactive version.

 

 

The good M3rds of 1/6 comma are, of course, balanced by four that are so wide as to be unusable (although only 5 cents wider than the worst and supposedly usable M3rds of Vallotti and Young). The classic case is the need for both G# and Ab. Most instruments (and singers) aren’t going to have a problem with that since they can make diatonic semitones, etc. wherever they are needed–hence the distinctions in the fingering charts noted above and references by some theorists to a “55 division” where the octave is divided into 55 commas (basically a fully extended 1/6 comma meantone)–but continuo keyboards with only twelve notes to the octave are stuck with whichever note was chosen when the temperament was set. There are some practical solutions to this problem in performance, some of which may be possible in various circumstances, and all of which can help to extend the utility of 1/6 comma:

1. Choose pieces that need only one note or the other.
2. Retune the note as needed for a piece or group of pieces.
3. Avoid the “wrong” notes and leave them to more flexible instruments.
4. Tune the notes differently in various octaves and avoid the “wrong” versions in the realization.
5. Tune the notes differently on each manual, using one manual for flat–leaning keys and another for sharp–leaning keys.
6. Use a keyboard with split keys.

Finally, there is one last musical reason why I think 1/6 comma is so satisfying. Complex harmonies so typical of Baroque music–including 7th chords and diminished triads–simply sound more satisfying in 1/6 comma. I could never understand why this should be so until I realized that the tritone in 1/6 comma meantone has about as much claim to harmonic purity as could be made. At 590.2 cents (again, a rounded approximation), the 1/6 comma tritone is virtually a combination of the pure (5:4) M3rd at 386.3 cents, and the pure (9:8) major whole tone at 203.9 cents. This is a rationalization, of course, since the interval itself (45:32) does not occur at an audible place in the harmonic series, and yet it may explain why the interval and its reciprocal diminished 5th (64:45) are so completely satisfying, and thus why the tension and release of Baroque harmonic progressions sound especially fine in 1/6 comma.

Vallotti Tritones


Click image for tritone playlist.

 Or download this interactive version.

Young Tritones

Click image for
semitones playlist.

 

Or download this interactive version.

1/6 Tritone with Ab


Click image for tritone playlist.

 Or download this interactive version.

1/6 Tritone with G#

Click image for
semitones playlist.

 

Or download this interactive version.


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Footnotes
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1. Meantone Temperament – One based on narrowing the 5ths by some fraction of the syntonic comma (resulting in a whole tone exactly half—the mean—of the M3rd)
2. Regular Temperament – One in which all 5ths are tempered by the same amount (with the exception of the wolf, if any)
3. Wolf – the one dissonant “5th” in a regular meantone system where it is wide, or in a Pythagorean system where it is narrow
4. Diatonic Semitone – semitone from one scale degree to another: the larger semitone in meantone based temperaments (e.g. C#–D) 
5. Chromatic Semitone – semitone within one scale degree: the smaller semitone in meantone–based temperaments (e.g. C–C#)
6. Syntonic Comma – The interval by which four pure 5ths exceed two 8ves and a pure M3rd.
Ex. (4 x P5ths) – ((2 x 8ves) + PM3rd)
(4x 701.95c) – ((2 x 1200c) + 368.3c)
2807.8c – 2786.3c + 21.5c
7. Major Semitone – larger semitone (diatonic in meantone–based temperaments, e.g. C#–D)
8. Minor Semitone – smaller semitone (chromatic in meantone–based temperaments, e.g. C–C#)
9. Circulating Temperament – One that can be used without adjustment in the complete circle of keys. Sometimes called a “well” temperament
10. Irregular Temperament – One in which the 5ths are of different sizes (e.g. some tempered, some pure)
11. Pure M3rd – interval with the harmonic ratio 5:4, or 386.31 cents, often rounded to 386c
12. Major Tone – interval with the harmonic ratio of 9:8, or 203.9 cents, often rounded to 204c
13. Harmonic series – The overtones belonging to any musical sound.  Ex.: First 10 notes of the C harmonic series.

Harmonic_series