Sep. 26, 2015

Temperaments & Tuning, Part III

(continued from Part II)

We have seen that in the natural tuning system based upon the acoustical phenomena of harmonic sounds, 2 important musical intervals (major 3rd and perfect 5th) cannot be made to coexist in the pure state, i.e. beat free.  Over the centuries a variety of compromise solutions known as termperaments have been invented and realized.  These give priority to one or the other interval by modifying them in various ways.

In the ancient world and the MIddle Ages the Pythagorean tuning system, in which the 5ths were retained perfectly pure, was used.  The resulting major 3rd was particularly unattractive in sound and therefore treated as a dissonance.  The music of the time however was mainly monadic, and the early forms of vocal and instrumental polyphony made a great deal of use of the interval of a 5th.  It was not until the start of the great flowering of vocal polyphony in the early Renaissance that the interval of a major 3rd gradually came to be heard as consonant instead of dissonant.  Instruments with fixed tuning like the organ gradually adapted to this situation by adopting a system of temperament known as "meantone" which gave the major 3rd priority over 5ths.

The system used in the big Gothic Blockwerk organs of the Middle Ages was therefore Pythagorean temperament, sometimes also called Just intonation, the medieval compromise in general use in European organ building from earliest times up through the 15th century.  In this temperament the goal is to arrive at all perfect 5ths in the Circle, except for one, to be tuned pure.  The entire Pythagorean comma was "dumped" on one unlucky perfect 5th in the Circle, between Ab and Eb where it was least likely to be crossed in a piece of music.  This created 11 pure perfect 5ths and one nasty, greatly diminished "wolf 5th" at Ab-Eb so named for the "howling" sound made by its rapid beating.  This sytem, while it was easy to use, left a lot of the notes of the diatonic scale (the so-called "Pythagorean scale") in quite odd positions, but it was entirely appropriate for organum, a music written in the old ecclesiastical modes in which perfect 5ths and perfect 4ths moving in parallel harmony were the overwhelming dominant sonority, the pitches of C#, F#, and G# hardly appeared at all, and where there was no modulation whatever (photo).  In this sytem major 3rds were extremely sharp at around 408 cents, and minor 3rds were extremely flat at around 294 cents.  Therefore, though used at times, 3rds were avoided at final cadences.

During the 13th century the French Academy of Notre-Dame actually decreed that only a series of perfect 5ths in the "just" or perfect ratio of 3:2 could make up a musical scale used in Christian worship.  In calculating their scale by superimposing a series of 4 of these pure perfect 5ths above a given starting point they inevitably bumped into a new twist (the Syntonic comma) on an old problem (the Pythagorean comma).  Using bass C2 as the starting note, after 4 consecutive perfect 5ths (C2 - G2 - D3 - A3 - E4) they arrived at a middle E4 which was considerably sharper in pitch than it needed to be to make a tolerable major 3rd with the super octave note C4 in tune with the starting note.  The Pythagorean major 3rd thus produced rendered their Pythagorean scale unusuable for contrapuntal music.

During the 15th century meantone temperament was developed as the solution, became firmly established during the early 16th century, and actually maintained its existence in European organ building right up through and including the mid-19th century.  The aesthetic motivation here was that composers during the late 15th century had fallen in love with the sweetness of the major 3rd and were trying to get away from the medieval austerity of bare perfect 5ths.  There was no single, invariable "type" of meantone tuning.  Each tuner had his own method according to his own taste.  But every elegant tuning has a generating principle, and the generating principle behind meantone was that it was better to preserve the purity of the major 3rds than the perfect 5ths.  A major 3rd and perfect 5th built upon the same root, of course, makes up a major triad, the most common chord in European music after 1500.  Since major triads incorporating pure major 3rds sounded so sweet, consonant, and attractive, the perfect 5ths of the entire Circle were sacrificed, flattened slightly, to get most of the major 3rds in the Circle pure.  The major 2nds were then tuned as the arithmetic mean, or average, of the pure major 3rds, hence the name "meantone."

Meantone had 8 pure major 3rds (Eb-G, Bb-D, F-A, C-E, G-B, D-F#, A-C#, E-G#) and 4 unusuable major 3rds or diminished 4ths (B-D#, F#-A#, C#-E#, Ab-C) all about 427 cents wide and of terrible howling quality.  All the perfect 5ths wound up being about 696 cents wide except for one extended 5th (Ab-Eb) which was around 737 cents and sounded really nasty.  The perfect 5ths in meantone would have sounded a little better at 702 cents, but at 696 cents or thereabouts the ear didn't notice them a great deal, especially if the chords were filled in with pure major 3rds to smooth over the discrepancies.  The perfect 5ths here were not quite consonant, and that fact became more obvious if the 3rd of the chord was absent,  thus originating the practice of never having a perfect 5th sounding by itself without having a 3rd filling it in.  The 8 minor triads built upon C, C#, D, E, F#, G, A, and B (those in the relative minor keys of all 8 agreeable major keys) also were agreeable in meantone.  The minor triad on G#, despite its okay minor 3rd, was no good because of its wildly beating 5th.  This allowed the composer, for the very first time in history, the freedom to incorporate these 16 triads into the harmonic vocabulary of his music and to modulate among major and minor keys with 3 or less accidentals in the key signature.

This limited writing to the best keys (C, G, D, A, F, Bb major, and their relative minors) and avoiding bare 5ths without 3rds.  If composers wanted to write tonic, dominant, and subdominant triads, they were limited to key signatures having no more than 2 flats or 3 sharps and just never got around to writing much in any other keys.  What they did write, however, sounds seductively sweet, colorful, and attractive when played in meantone, and its highly irregular chromatic scale gave a very distinctive voice to chromatic writing.

This was the old system known to musicians in the time of J.S. Bach simply as "the method of tuning"  -- and it is why, in order to make his 2- and 3-Part Inventions playable on instruments in general use, Bach confined his selection of keys in these works to those compatible with the meantone system, sticking with the useable keys which were more harmonious while at the same time very nearly going beyond this system's limits.

(continued in Part IV)