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Meantone

 

 

Meantone etc.


Glynn Naughton
wrote (November 11, 2003):
For anyone who wants to hear tuning systems such as meantone or whatever (or even create their own), there's a freeware application, Scala: http://www.xs4all.nl/~huygensf/scala/

It's a lot of fun (and very educational) to play around with. The web site also has a huge tuning bibliography, including the texts of some medieval treatises in (yikes!) Latin.

Thomas Braatz wrote (November 11, 2003):
In researching the background for Bach's BWV 243 (D Major version) and Bach's connection with Dresden and its organs with which Bach must have been fairly well acquainted, I came upon this quote by Russell Stinson in the article on Gottfried Silbermann (p. 452) [Oxford Composer Companions: J. S. Bach (Boyd,1999)]: "Bach performed on Silbermann's organs in Dresden, even though he disliked their mean-tone temperament." [See Gress, below] Does Stinson, who took his MA and PhD in musicology at the U. of Chicago and also taught at the U. of Michigan, understand the difference between 'mean-tone' and 'well-tempered' and equal temperament. [Is Stinson translating the word, "mitteltönig" correctly here?] Can we take this statement to mean that, according to his research, Bach had advanced sufficiently beyond 'mean-tone', and having progressed through various experiments with 'well-tempered' keyboard instruments, had arrived at a point which comes closest to what we now understand as equal temperament? [Bach as a pioneer, far advanced for his time?]

In his article on Silbermann in the New Grove (Oxford University Press, 2003), Frank-Harald Gress states the following specifically:
>>The voicing of his [Gottfried Silbermann's] pipes is very powerful by today's standards, with high wind pressure (the Freiberg Cathedral organ originally had 97 mm pressure for the manuals and 109 mm for the pedal). The tuning of most of his organs was originally between a' = 460 and a' = 465 (Chorton), or higher in some of the early organs. The organ of the Sophienkirche in Dresden and several later instruments were tuned to Cammerton (between a' = 410 and a' = 414). Silbermann used a mean-tone temperament that was regarded as controversial by his contemporaries: according to measurements from the Freiberg Cathedral organ, originally nine 5ths were tempered smaller than pure by one-fifth of the Pythagorean comma on average; the C# minor/Eb minor and Eb minor/Bb minor 5ths were pure; the wolf 5th was set at Gb minor/D# minor.<<

According to Brad, as the 5ths get worse, the 3rds get better.

Silbermann's temperament, which Bach disliked, sounds like the 2nd type of mean-tone that Brad was talking about this morning with the 3rds having the 'higher quality', am I right? Why did Bach dislike this type so much? The fact that the 5ths were relatively impure? Was Bach, then, arguing for the 1st type? Didn't Bach want the 5ths purer, as in equal temperament, rather than the 3rds purer, as in mean-tone? Isn't this ideal toward which Bach was moving called 'equal temperament' so that, for Bach, in reality 'well-tempered' = 'equal temperament' although it took a good portion of his life to achieve this equation, an equation that was not yet achieved for many decades after Bach's death, because others, such as Silbermann were still 'lagging behind.' These organs, imperfectly tuned to anything but 'equal temperament' and with pipes that could not be easily modified continued to restrict the movement toward the inevitable 'equal temperament.' Perhaps Bach looked upon this situation with certain dismay as he realized how these new organs would continue to hamper progress because the temperaments used would prohibit the use of certain keys and chords otherwise made possible by the use of a true equal temperament?

The Sophienkirche organ was the one which W. F. Bach played on from his appointment on June 23, 1733 until April 1746. This was the church and organ used by Lutherans who were in the employment of the primarily Catholic court. [It would not appear that this would not be the place where B-Minor Mass (the sections that Bach had copied for performance at the court) nor would the Magnificat have been intended for performance here. Where did the members of the primarily Catholic court worship and hear music performed for their services?

If the Sophienkirche was one of the primary Lutheran court associated churches with a great organ, which organ (and which edifice) served as the primary Catholic court church in the 1730's?

On December 1, 1736 Bach gave a recital on the nearly completed Silbermann organ in the Frauenkirche [after his appointment as "Kirchen" {"Hof"] Compositeur to the royal "Kapelle" in Dresden or "Kapellmeister von Haus aus" to the Dresden court.] Bach had petitioned for this position in 1733 at which time he presented in his application portions of the B Minor mass as evidence. [Possibly he had had in mind, in 1733, to prepare the Magnificat also, but had decided after transposing the score not to submit it after all. There is no evidence whatsoever that parts were ever copied from the transposed score which was written out in Bach's best calligraphic musical score handwriting, an effort which would not be necessary for his usual performances in Leipzig.]

Was there a special chapel organ primarily reserved for the Catholic services of the court where the 'Hofkapelle' (with some of the best musicians of the time) would perform?

Boyd (in the Oxford Composers Companions book referred to above) states:

>>[The questions regarding] What works Bach provided for the Dresden court in his role of church composer remains a matter of conjecture. The 4 "Missae BWV 233-6 may have been among them.<<
To this I would add BWV 243 as a reasonable conjecture.

Again, which venue would Bach have had in mind? The palace Kapelle with its excellent musicians?

I have a note that the palace Kapelle was closed down and secularized in 1737. Had this been the possible 'destination' for the B Minor mass and the Magnificat?
What do we know about this palace Kapelle organ? Was it just a small organ of no significant importance?

I would appreciate any help or suggestions that might lead to resolving these questions.

Bradley Lehman wrote (November 11, 2003):
Thomas Braatz wrote: http://groups.yahoo.com/group/BachRecordings/message/11399
>>(...) I would appreciate any help or suggestions that might lead to resolving these questions.<<
Tom, you're speculating wildly about how much "purer" the fifths sound in equal temperament vs the meantones. You also said that organs are "imperfectly tuned to anything but 'equal temperament'", which is clearly your expectation about relative quality.

And you're projecting Bach's "dismay" at the way things sounded, which (I believe) is really your dismay at even the suggestion of unequal temperaments in practice.

But have you ever actually heard any of these meantone and well temperaments with direct experience at a keyboard, other than equal temperament? If not, don't worry, you're not alone in that: one of the major researchers of temperament earlier in the 20th century was Murray Barbour (from 1930s to 1950s), and he too never heard anything but E.T. His book has very strong biases in that direction: all his measurements are treated as deviation from the E.T. positions of the pitches, and the old temperaments are cast as imperfect approximations of his own ideal, that is, equal temperament.

Suggestion: you could go listen to the fifths and major thirds in 1/4, 1/5, and 1/6 comma temperaments, and compare them against the way their equal-tempered counterparts sound. Those tempered fifths are less nasty than you evidently believe they are; and the major thirds are much, MUCH better (pure, or nearly so) than they are in E.T. The quality of any given triad--its perceptible character--comes almost completely from the major and minor thirds in it, not the fifths. Fifths are neutral!

I'd put it this boldly: wthe music is going along, ordinary listeners DO NOT notice tempered fifths at all, even when they are tempered as tightly as 1/4 comma; but the improvement in the thirds (derived from those fifths stacked up) is remarkable and obvious. And in almost all temperaments, all the fifths are tempered by 1/4 comma or less. The fifths are not a problem as you're making them out to be, with your speculation! Really the only problem with "fifths" is the single diminished-sixth wolf, the place where the enharmonic circle is forced to close (approximately), and that is traditionally dodged by the composers' careful avoidance in the music. In the "well" temperaments there are enough pure fifths (at various places) that the wolf diminished-sixth joining the circle ends up as a pure or nearly-pure fifth!

An even better suggestion: take a course and learn how to do these on harpsichord. That "hands-on" and "ears-on" experience is a lot more productive than speculation is.

If you really want to understand this stuff, and to have your questions answered, you have to hear it in practice on a keyboard: playing through it directly, as Bach did. Theoretical speculation short of that is just a bunch of chatter.

=====

You said, "According to Brad, as the 5ths get worse, the 3rds get better." It's not according to Brad (as if it would suddenly stop being true if I were unreliable). It's according to the laws of physics and mathematics: the superparticular ratios in the overtone series. The closer any interval is to a ratio of small integers, the more nearly in tune it is. Pure major thirds are 5:4. Pure minor thirds are either 6:5 or 7:6. Pure fifths are 3:2. Pure fourths are 4:3. That's not according to some commentator; it's the way it IS.

1/4 comma meantone has a pure major third (5/4) in it, along with a pure augmented fifth (25/16)...two such major thirds stacked together.

1/3 comma meantone has three pure intervals, all from minor thirds stacked up: (6/5), (25/18), (5/3).

1/5 comma meantone has a pure major seventh (15/8) in it, C to B.

1/6 comma meantone has a pure augmented fourth (45/32) in it.

The well temperaments all have various pure fifths and fourths in them. Some of those temperaments also have other intervals that can be expressed as three-digit numeric ratios or less (such as Thomas Young's temperament, which has (256/243), (32/27), (128/81), and (16/9)).

Music sounds more harmonious and resonant in these temperaments because of these pure intervals. When two notes in a pure interval are played together, they reinforce each other (because some of their overtones are vibrating at exactly the same rate). Two notes in an impure interval dissipate one another's energy: the overtones almost but do not quite line up. That difference makes a beat, an audible tremolo (and it's that rate that a tuner counts when tempering such an interval). The two pitches fight it out instead of cooperating.

Equal temperament has no interval in it that can be expressed as a rational number (because it's all based on an irrational number, the twelfth root of 2). Equal temperament is based on a concession that all intervals are imperfect. The major third is (approximately!) 3259034/2586697. Middle C and the E above it beat together at more than 10 times per second...very impure, and more like a steady buzz than a beat. In 1/6 comma temperaments, C-E beats only about 5 to 6 times per second. In 1/5 comma, it's down to slightly over 3. In 1/4 comma it's 0 (pure, no beats).

Because of all this rapid major-third beating in equal temperament, music sounds tense ALL THE TIME. It never really resolves. (It's like the way modern string players tend to use constant vibrato....) Yes, most people are accustomed to that; but it's not the only ideal in town.

Brad Lehman
(...having recently sat through the almost unbearable first fifteen minutes of "Powaqqatsi"--where Philip Glass' music stays on interminable major chords in equal temperament. If they were trying to show the elegant beauty of life in a simpler culture, which appeared to be the point of the cinematography, why didn't they tune the music more simply!?!? Sheesh.)

David Glenn Lebut Jr. wrote (November 11, 2003):
[To Glynn Naughton] What instruments/vocal ranges does it cover?


1/6 comma meantone

Anna Vriend
wrote (June 16, 2004):
Bradley Lehman wrote: < the basic normal temperament of 18th century ensemble musicianship was 1/6 comma meantone (also known as the 55-division) as corroborated by Sauveur (1707), Tosi, Telemann, Quantz, the Mozarts, and others. As Tosi and Quantz both pointed out (for singers and players, respectively), in enharmonic pairs such as G# and Ab, the notes are a [syntonic] comma apart; and that's regular 1/6 comma meantone, right there. >
and further down, about Sauveur: < That's what he called it, "the one used by musicians in general" ("celui dont les Musiciens ordinaires se servent"). >
Is 1/6th comma meantone the same temperament as the one called "temperament ordinaire"?

Thomas Braatz wrote (June 16, 2004):
[To Anna Vriend] According to the article in the New Grove given below, the answer is `no.' The 1/6-comma mean-tone temperament is a temperament with `the corresponding theoretical division of the octave into 55 equal parts,' while the `tempérament ordinaire' had semitones varying in size.

For a quick overview, here are critical excerpts form the article shared in its entirety below:

>>In the 18th century a certain theoretical prestige was enjoyed by 1/6-comma mean-tone temperament and by the corresponding theoretical division of the octave into 55 equal parts, five of which constituted a diatonic semitone and four a chromatic one. References to this division of the whole tone by Sauveur, P.F. Tosi, Nassare, Sorge (who attributed it to Telemann), Romieu, Quantz, Leopold Mozart and others suggest that equal-tempered diatonic semitones were still regarded as smaller than ideal. Neidhardt said so explicitly in 1732.<<

>>In an 18th-century `good' unequal temperament, or `tempérament ordinaire,' the semitones varied in size; those among the diatonic notes (E-F, B-C) were larger than those in the `remote' keys (C-Db, F-Gb or E#-F#, A#-B). Rousseau, for example, said in his "Dissertation sur la musique moderne" (1743) that in `l'accord ordinaire du clavecin' the key of C minor was more tender than D minor partly because the semitone Ab-G was smaller than Bb-A; he also remarked that singers would duplicate such shadings only when so obliged by their accompanying instruments.<<

William Drabkin, Mark Lindley in their article on `Semitone' in the New Grove [Oxford University Press, 2004] state the following (minus examples/illustrations):

>>Semitone [half step](Fr. "demiton, semiton;" Ger. "Halbton;" It."semitono;" Lat. "semitonium, hemitonium").<<

>>The smallest interval of the modern Western tone system; in `equal temperament,' the 1/12 part of an octave, or 100 cents. The notational system allows three types of semitone to be distinguished: the diatonic, which is the same as a minor 2nd (e.g. e-f,c#-d); the chromatic, which is the difference between a major 2nd and a minor 2nd (hence an augmented unison, e.g. f-f#, db[=d flat]-d);and the enharmonic, which is a doubly diminished 3rd (e.g. gbb[=g double flat]-e, b#-db, b-d').

In Pythagorean intonation there are two kinds of semitone, each a different size. The diatonic semitone, often called the'Limma', is the difference between three octaves and five perfect 5ths (reckoned from C, the interval b'-c''), a ratio of 256 : 243, or 90.2 cents.

The Pythagorean chromatic semitone, called the 'apotomç,' is the difference between seven pure 5ths and four octaves (reckoned from C, the interval c'''-c#'''), a ratio of 2187 : 2048, or 113.7 cents. The sum of an apotomç and a limma, then, is equal to a Pythagorean whole tone (ratio 9 : 8), and their difference amounts to a Pythagorean `Comma' about a quarter of an equal-tempered semitone.

In `Just Intonation,' where the pure major 3rd (ratio 5 : 4)amounts to 386.3 cents (almost 1/7 of a semitone smaller than in equal temperament), the relative size of the diatonic and the chrosemitone is reversed. The diatonic semitone, the difference between an octave and a perfect 5th plus a major 3rd, gives a ratio of 16:15, or 111.7 cents. There are two sizes of chromatic semitone that are commonly derived, the smaller (and more frequently used) being the excess of a 4th plus two major 3rds over an octave, a ratio of 25 : 24, or 70.7 cents, the larger being the excess of three 5ths plus a major 3rd over two octaves, a ratio of 135 : 128, or 92.2 cents.In regular mean-tone temperaments as well, the diatonic semitone is larger than the chromatic semitone.

In tonal music, the notation of a pitch that does not belong to the scale of the prevailing key depends largely on considerations of part-writing, often on the resolution by step to or from that pitch. Since such a resolution normally requires adjacent letters in the musical alphabet (e.g. F to E, G to A), the resolution a semitone up, say, from F is Gb, not F#, and the resolution a semitone down from G is F#, not Gb. Sometimes a series of semitone resolutions cannot be notated without some compromise of this principle. Compare, for instance, the notation of f#' versus gb.

In the 20th century the semitone took on a special significance as the generating unit of the chromatic scale: in most modern theories of 12-note music, intervals are reckoned by the semitone: unison = 0, semitone = 1, whole tone = 2, and so on.

The history of diatonic semitones in performing practice is of special interest. If Pythagorean intonation was favoured as much in medieval practice as in theory, then the semitones of late medieval plainchant and early polyphony were about 10% smaller than those now familiar in equal temperament. In the early 14th century Marchetto da Padova gave explicit preference to high leading notes. But many 15th century keyboard instruments provided rather large diatonic semitones, at first between any Dorian final and its leading note and then, with the adoption of mean-tone temperaments, elsewhere as well. The large diatonic semitones of mean-tone temperament became so familiar during the Renaissance and early Baroque periods that Mersenne described them in 1637 as one of the greatest sources of beauty and variety in music, and Doni in 1639 asserted that singers at Rome disliked being accompanied by an instrument tuned in equal temperament because of its small semitones. In the 18th century a certain theoretical prestige was enjoyed by 1/6-comma mean-tone temperament and by the corresponding theoretical division of the octave into 55 equal parts, five of which constituted a diatonic semitone and four a chromatic one. References to this division of the whole tone by Sauveur, P.F. Tosi, Nassare, Sorge (who attributed it to Telemann), Romieu, Quantz, Leopold Mozart and others suggest that equal-tempered diatonic semitones were still regarded as smaller than ideal. Neidhardt said so explicitly in 1732. Yet in the 1670s Christiaan Huygens had expressed preference melodically for a high leading note in the key of E minor (in terms of mean-tone temperament, E-Eb-E) while acknowledging that the lower form (D#), when available on a keyboard instrument, was more resonant with B in the bass. In an 18th-century `good' unequal temperament, or 'tempérament ordinaire,' the semitones varied in size; those among the diatonic notes (E-F, B-C) were larger than those in the `remote' keys (C-Db, F-Gb or E#-F#, A#-B). Rousseau, for example, said in his "Dissertation sur la musique moderne" (1743) that in `l'accord ordinaire du clavecin' the key of C minor was more tender than D minor partly because the semitone Ab-G was smaller than Bb-A; he also remarked that singers would duplicate such shadings only when so obliged by their accompanying instruments. Conflicting 19th-century accounts of intonation among singers and violinists (summarized in Ellis's translation of Helmholtz) leave some doubt whether the preference for small diatonic semitones expressed by such modern artists as Casals and Menuhin was characteristic of musicians throughout the 19th century.

Equal-tempered semitones on the piano may, as E.J. Dent is reported to have demonstrated to the Royal Musical Association (1944) and as Siegmund Levarie and Ernst Levy showed in "Tone: a Study in Musical Acoustics" (Kent, OH, 1968), seem to the ear to vary in size according to their context. Hence for example the a ' [in the example not shown] may readily seem lower than the g#', and thus by implication the semitone A -A larger than G#-A.<<

Bradley Lehman wrote (June 16, 2004):
[To Anna Vriend] Nope, although they're somewhat related.

"Temperament ordinaire" is a process (or a strategy) more than any specific outcome. One starts from a regular meantone (whether it's 1/4, 1/5, or 1/6...to taste) and then raises the sharps somewhat and/or lowers the flats somewhat (to taste) so more of the tonalities can be used in music. This is done usually by the simple expedient of tuning one or more pure fifths (instead of the normally tempered fifths of the regular meantone) during this fudging process. Sometimes, some of the resulting fifths even end up a bit wide--tempered in the "wrong" direction--for the greater goal of better playability in more keys.

The strategy, all around, is to get the circle to meet itself relatively closely somewhere in the extreme sharps and flats, such that the notes can be used (albeit roughly) for one another. For example, playing an E-flat where the music says D#. In regular meantone, only one of those two would be available on the keyboard. In "temperament ordinaire" such pairs of notes become roughly interchangeable, although still not identical.

Hope that helps,

John Pike
wrote (June 17, 2004):
[To Thomas Braatz] Very helpful and clear. Thank you.


The Keyboard Temperament of J. S. Bach: Article | Music Examples | Feedback: Part 1 | Part 2 | Part 3
Discussions of Temperament / Key Character / Tuning:
Part 1 | Part 2 | Part 3 | Part 4 | Part 5 | Part 6 | Meantone

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Last update: July 17, 2004 18:59:55