In Extended 1/6 Syntonic Comma Meantone by Ross W. Duffin
I have elsewhere explained why I believe that a flexible 1/6 syntonic comma meantone temperament was (and should be today) the standard tuning system used for ensemble music in the baroque era (see my Why I hate Vallotti (or is it Young?) which appeared inHistorical Performance Online from Early Music America, Vol. 1, No. 1 (2000)). What I mean by “flexible” is that ensemble players (and singers), unlike keyboard players who were usually stuck with 12 notes to the octave, had the ability to sound the proper chromatic version of any note they encountered, distinguishing, for example, between G# and Ab. This is supported by woodwind and string fingering charts which continue to give separate pitches for such notes even though most keyboard players today use irregulartemperaments like Vallotti which compromise the position of notes in the interests of serviceability.
The advantage of a flexible 1/6 comma system such as I propose is that the thirds are always good, the fifths are all acceptable, the chords—even diminished and seventh chords—sound wonderful, and the keys are equally consonant. Even more important, as far as performers are concerned, the melodic intervals are always the same size. To take Vallotti as a counter-example, there are three different sizes of whole tone and six different sizes of semitone. In meantone, there is simply one size of whole tone and two sizes of semitone. Furthermore, the large and small semitones in meantone are predictable as to where they occur: all diatonic semitones—involving movement from one pitch name to another, like C# to D or A to Bb—are large; all chromatic semitones—involving movement within a pitch name, like C to C# or Bb to B—are small. Simple. And better in tune.
There are two main reasons that ensemble players have not been using this advantageous system, however. The first is that keyboard players are generally responsible for choosing performance temperaments and, because of ease of tuning and flexibility of key choice in a 12-note octave, they tend to choose irregular systems like Vallotti. The second reason is that, until now, there has been no way for ensemble performers to hear and practice extended meantone tuning. On my own, I can’t do anything about the first reason, but these exercises are an attempt to address the second.
It may be helpful to know the sizes of 1/6 comma intervals in relation to acoustically pure (Just) intervals, to ET (Equal Temperament) and to Vallotti. Table 1 below gives the values in cents, a logarithmic ratio that divides the octave into 1200 units, each ET semitone being 100 cents.
|1/6 Comma size||55-Division size|
|minor semitone||C – C#||92.2||100||90-110||88.6||87.3|
|major semitone||C – Db||111.7||100||90-110||108.2||109.1|
|minor tone (10:9)||C – D||182.4||200||196-204||196.7||196.4|
|major tone (9:8)||C – D||203.9||200||196-204||196.7||196.4|
|diminished 3rd||C# – Eb||223.5||200||196-204||216.3||218.2|
|augmented 2nd||C – D#||274.6||300||294-306||285.3||283.6|
|minor 3rd||C – Eb||315.6||300||294-306||304.9||305.4|
|major 3rd||C – E||386.3||400||392-404||393.5||392.7|
|diminished 4th||C – Fb||427.4||400||392-404||413.0||414.6|
|augmented 3rd||C – E#||478.5||500||498-502||482.1||480.0|
|“perfect” 4th||C – F||498.0||500||498-502||501.6||501.8|
|augmented 4th||C – F#||590.2||600||588-612||590.2||589.1|
|diminished 5th||C – Gb||609.8||600||588-612||609.8||610.9|
|“perfect” 5th||C – G||701.96||700||698-702||698.4||698.2|
|augmented 5th||C – G#||794.1||800||792-808||787.0||785.4|
|minor 6th||C – Ab||813.7||800||792-808||806.5||807.3|
|major 6th||C – A||884.4||900||894-906||895.1||894.6|
|augmented 6th||C – A#||976.5||1000||996-1004||983.7||981.8|
|minor 7th||C – Bb||1017.6||1000||996-1004||1003.3||1003.6|
|major 7th||C – B||1088.3||1100||1090-1110||1091.9||1090.9|
|diminished 8ve||C – Cb||1107.8||1100||1090-1110||1111.4||1112.7|
|augmented 7th||C – B#||1180.4||1200||1200||1180.4||1178.2|
|8ve||C – C||1200||1200||1200||1200||1200|
The most striking thing about this table, to me, is the daunting range of possibilities for each interval in Vallotti. Think of a diatonic scale in each system: In ET, there are two different sizes of interval (100c and 200c); in 1/6 comma meantone, there are two different sizes of interval (108.1c and 196.7c); in Vallotti, there are potentially nine different sizes of interval (90c, 94c, 98c, 102c, 106c, 110c, 196c, 200c, and 204c)! That’s easy enough for the keyboard player who doesn’t have to think about it, but for the other instrumentalists and singers, it’s frightening. Similarly, a chromatic scale in ET has one size of semitone (100c); 1/6 comma meantone has two in predictable places (88.6c and 108.1c); Vallotti has six in arbitrary places (90c, 94c, 98c, 102c, 106c, 110c)! How can performers find all those different intervals on the fly? With much practice in using the same temperament, of course, they can learn to do so, but they’re not often given the chance since they may have to perform in the preferred temperaments of several different keyboard players, or in a variety of temperaments chosen by a single keyboard player.
The second most striking thing about the table is the lack of differentiation among intervals in ET. ET therefore lacks both the expressive quality of distinct melodic intervals, and the harmonic differentiation of chords with notes of different spellings. In addition, the basic building blocks of all chords—the major and minor 3rds—are farther from Just in ET than they are in meantone, so chords in meantone sound sweeter.
Finally, in 1/6 comma meantone, the augmented 4th (590.2c) and the diminished 5th (609.8c) are acoustically pure, which makes them harmonically much more satisfying than in any other tuning system. Since those two intervals are so important in baroque music, constituting the most crucial features of diminished chords and seventh chords, 1/6 syntonic comma meantone has a natural and insurmountable harmonic advantage in making baroque music sound euphonious. In fact, because the augmented 4th and the diminished 5th in 1/6 comma are both good harmonic intervals, the disadvantage in a 12-note system is somewhat reduced, since G# in a diminished triad will sound just as euphonious as Ab, even though the harmonic tendency of the chord may be subtly different. Thus, where G# might sound awful in a simple F minor triad, it is acceptable in a D diminished or even a Bb dominant seventh chord—although, as Bb-D-F-G#, that chord is really a German sixth chord rather than a dominant seventh chord (Bb-D-F-Ab), and would normally resolve differently. Non-keyboard players don’t have to make that compromise, however.
There are still drawbacks in trying to make 1/6 comma meantone work in a 12-note system (see the online Vallotti article cited above), but instruments and voices with the flexibility to adjust are not bound by those limitations. They simply need to develop a sense of the basic melodic intervals—especially the major and minor semitones and the whole tone—and an understanding of when each type of semitone is required. Everything else flows from that awareness. The first step to making extended meantone work in practice, in fact, is the recognition that sharp notes are always low and flat notes are always high. This is counter-intuitive for modern string players, especially, who are frequently taught to play high leading tones, but it’s consistent with baroque fingering charts (which, for example, show G# at a lower pitch than Ab) and with the general baroque tendency to favor harmony over melody.
But what about the amount of differentiation between sharps and flats in 1/6 comma meantone? How do we know that it represents what musicians of the time would have understood by those distinctions? The differentiation of the accidentals is discussed by, among others, two of the most important instrumental theorists and one of the most important vocal theorists of the mid-18thcentury: violinist Leopold Mozart (1756), flutist Johann Joachim Quantz (1752), and singer Pierfrancesco Tosi (1723). Mozart remarks that, according to their “right” ratios, notes with flat signs are “a comma higher” than those in the same position with a sharp sign. Similarly, in discussing temperament adjustments in performances with a keyboard instrument, Quantz says “notes like D# and Eb, etc. are differentiated by a comma.” These two comments are clear in the light of Tosi’s earlier writing: “A whole tone is divided into nine almost imperceptible intervals which are called commas, five of which constitute the major semitone, and four the minor semitone.… An understanding of this matter has become very necessary, for if a soprano, for example, sings D# at the same pitch as Eb, a sensitive ear will hear that it is out of tune, since the latter pitch should be somewhat higher than the former.” With this information, we can calculate that the octave is divided into 55 commas: a major scale, for example has five whole tones of 9 commas each (45 commas) and two major semitones of 5 commas each (10 commas) for a total of 55 commas. It’s called the 55-Division since it divides the octave into 55 equal parts. Not having the “cents” system, that is the “rough and ready” way they thought about the relationship between various intervals. Its close correspondence to extended 1/6 comma meantone can be seen in the two right columns of the table above, where almost all of their respective intervals are within a cent or two of one another. While I prefer 1/6 comma meantone because of its acoustically pure tritone and diminished 5th, these two temperaments are, for all practical purposes, the same.
Thus, extended 1/6 syntonic comma meantone and the 55-Division are virtually the only tuning systems to satisfy Tosi’s, Mozart’s, and Quantz’s definition of the difference between sharped and flatted notes like D# and Eb, G# and Ab, etc. For those outstanding virtuosi of the mid-18th century, those theorists whose influence is felt to the present day, extended 1/6 comma meantone is the embodiment of proper ensemble tuning.
It remains only for modern performers to learn how to use extended 1/6 comma meantone in real-life situations. This has not really been possible until now because there have been no resources available for the purpose. To remedy that, I have created a group of twenty “exercises” which can be heard with all parts, individual parts, or in music-minus-one format. Scores, including individual parts for the longer pieces, can be downloaded as pdf files. The pieces are chosen entirely from the works of J.S. Bach since he uses so many chromatic inflections and thus provides opportunities for a variety of scalar situations within each piece. The first group consists of seventeen 4v chorales in a range of signatures and finals. These are divided into three categories: those with 12 notes per octave, those with 13 notes per octave (i.e., those with both G# and Ab, for example), and those with 14 notes per octave. The ones with 12 notes per octave simply give a sense of the chromatic tendencies within the different signatures and keys, while the >12 ones obviously require enharmonic distinctions to be made within each exercise. Three highly chromatic contrapuntal works of Bach are given as well, two from Die Kunst der Fuge and one from Das Musikalisches Opfer.
Also linked from that page are major and minor scales in 1/6 comma meantone, prepared by David Dyer for this site.
For those interested in frequencies, cents, and beat-rates for extended 1/6 syntonic comma meantone, a spreadsheet with those values can be downloaded here. The boxed frequency of A=415.304 can be reset to any standard.
If you have questions or comments about this site, please e-mail me at firstname.lastname@example.org.