Bach's Extraordinary Temperament: Our Rosetta Stone--1 Bradley Lehman Early Music, Volume 33, Number 1, February 2005, pp. 3-23 (Article) Published by Oxford University Press For additional information about this article http://muse.jhu.edu/journals/emu/summary/v033/33.1lehman.html Access provided by University of British Columbia Library (30 Nov 2013 02:52 GMT) Bradley Lehman Bach’s extraordinary temperament: our Rosetta Stone—1 OHANN Sebastian Bach never required equal progression of key differences, and it yields convin- Jtemperament for keyboards. Music historians cing performances of the WTC. The book in its have hitherto failed to notice that he actually wrote entirety shows that Bach was expert in tuning, down and made full use of a specific unequal alterna- expected a precise intonation on his keyboards, and tive whose resources are much richer. Its layout is: allowed the sound to influence his creative imagina- five ¹⁄₆ comma 5ths F–C–G–D–A–E, then three pure tion. As a guide to musicianship it teaches players to 5ths E–B–F –C , and finally three ¹⁄ comma 5ths listen closely to the intervals produced in melody C –G –D –A . Although this may not look like any- and harmony, as details of intonation can help to thing special on paper, its sound is marvellous for the determine the Affekt, phrasing, articulation, tempo complex requirements of tonal music. and accentuation of the music. The ears, the phys- Bach ‘was no lover of dry, mathematical stuff’.1 ical process of tuning, and the feel of the instrument Despite his aversion to computation in numbers during performance all support this. The music and theoretical speculation in words, the 37-year-old sounds so euphonious and pure that it seems other- composer notated that specific temperament on the worldly, especially after a lifetime of hearing the title-page of Das wohltemperirte Clavier (‘The well- same music tuned differently.4 tuned keyboard’, hereafter WTC).2 His presentation This article in its two parts examines the tempera- of the required tuning method is the sinuous spiral ment thoroughly from many angles, including some he drew at the top of the title-page (illus.1). It of the ‘dry, mathematical stuff’. For best direct per- provides a set of tempering instructions for harpsi- ception of the material, however, it is important to chords, clavichords or organs, completely describing set up this pattern of intonation on a keyboard and the distinctive character he expected for every key, play with it! every scale, every interval and every moment. The WTC is explicitly about tuning, the exploration of Background: the commas and other problems to all possible tonalities (and not simply a transposition be solved of a single major key sound, and a single minor key The choice of historically and musically appropriate sound, to all possible positions). keyboard temperaments always depends on the music Until now music history had lost three crucial to be played. What did the composer expect as a pieces of information: (1) that Bach ever wrote down norm? What features of the music are to be high- any required keyboard-tuning method at all; (2) that lighted? What is the appropriate balance of melody this drawing is his precise schematic; and (3) the versus harmony? What key areas are visited during proper derivation of a set of instructions, arising the composition? How many sharps and flats are in from normal 17th-century temperament ordinaire use, given that they might be tuned differently from practice.3 one another? Does the composition have any special Bach’s method is a practical one. It is easy to set expressive effects to be brought out by the choice quickly by ear, usable in all keys, with a pleasing of temperament? What is the player/tuner’s skill Early Music, Vol. , No. © Oxford University Press 2005; all rights reserved doi:10.1093/em/cah037, available online at www.em.oupjournals.org 3 1 The top of the title-page of Bach’s Das wohltemperirte Clavier (1722). The complete page is reproduced in the Grove dictionary, vol. i, p. , and in many modern editions of the music. (Staatsbibliothek zu Berlin Preußischer Kulturbesitz) (then or now!) setting up different selections on there are twelve major 3rds, some or all of them different occasions? What other instruments and must be tuned too wide by some portion(s) of the voices are to be used with the keyboard? What is the SC. On average the major 3rds must be ⁷⁄₁₁ SC too role of musical taste and experience? Are the instru- sharp. (This is a large and easily audible amount: ments to be left in a particular temperament for approximately ¹⁄₇ of an equal-tempered semitone.) years, needing only minor maintenance, or to be retuned afresh on every occasion? To understand the Lesser diesis (hereafter simply ‘diesis’): an octave, less three pure major 3rds; the interval ¹²⁸⁄₁₂₅. For exam- nature of these problems of keyboard temperament ple: if we tune three pure 3rds C–E, E–G , G –B we along with Bach’s solution, a brief review of the basic tuning issues and nomenclature is required. get the interval C–B which is much too flat to be an All octaves are pure, as a rule. Within that octave. As octaves must be pure, the diesis must be premise, the tuning of 12-note keyboards presents distributed to one or more of the three major 3rds three competing problems. The tuner must find a making some or all of them wider than pure. This musically appropriate and tasteful distribution of must be handled four times in any temperament: in three different error intervals. These intervals are the major 3rd stacks starting on F, C, G and D. as follows: The geometric difference between the PC and the SC Pythagorean comma (hereafter ‘PC’): twelve pure is called the schisma. It is very nearly the same size as ¹⁄₁₂ ¹⁄ 5ths, less seven octaves; the interval ⁵³¹⁴⁴¹⁄₅₂₄₂₈₈. either PC or SC. A cycle of twelve consecutive 5ths must remove a Temperaments usually split either the PC or the ¹⁄₁₂ ¹⁄₆ ¹⁄₅ total of one PC, to avoid overshooting the octave. SC into small, manageable portions of , , or ¹⁄₄ Since there are twelve 5ths, some or all of them must . These are distributed among some or all of the be tuned too narrow by some portion(s) of the PC. 5ths. The 5ths are made deliberately out of tune by On average the 5ths must be ¹⁄₁₂ PC too flat. these subtle amounts, such that the resulting 3rds and steps will have a pleasing balance in their own Syntonic comma (hereafter ‘SC’): four pure 5ths, less musical functions. For all practical purposes the ¹⁄₁₂ two octaves and a pure major 3rd; the interval ⁸¹⁄₈₀. PC temperaments are ¹⁄₁₁ SC temperaments, as A cycle of four consecutive 5ths must remove one those two small portions have very nearly the SC if the resulting major 3rd is to be pure. Since same size. 4 early music february 2005 The phrase ‘regular 5ths’ refers to two or more 5ths For analysis of 3rds and 5ths I prefer the that are exactly the same size, geometrically. If they ‘Temperament Units’ (hereafter TU) method devel- are tempered 5ths rather than pure 5ths, their temper- oped by the organ builder John Brombaugh, a much ing (not their beat rate!)5 is the same. They are slightly finer-grained logarithmic scale assigning 720 micro- out of tune, each by the same amount. ‘Regular’ tem- scopic portions to the PC. The SC then works out to peraments, also called ‘mean-tone’ temperaments,6 660 TU, the schisma 60, and the diesis 1260. All these are those that have eleven 5ths the same size, and one numbers are easy to handle on paper or mentally, ‘wolf’ 5th (truly a diminished 6th) allowed to be divisible by 3, 4, 5, 6 and 12: the fractions used in much too wide. The wolf 5th is a garbage area, most descriptions of historical temperaments. The absorbing all the accumulated amount by which the TU system therefore allows a theorist or player to tempered 5ths overcompensated one PC. work almost exclusively with integers, visualizing a The regular temperaments tend to emphasize the temperament by merely adding and subtracting easy quality of major 3rds, providing eight major 3rds portions of commas. The SC error of any major 3rd that are pure or nearly so, and four misspelled major is found by adding the four TU values of the inter- 3rds (actually diminished 4ths) that absorb most of vening 5ths, and dividing by 660. Three major 3rds the SC and diesis errors. The composers then stacked up must distribute a total of 1,260 somehow. avoided these forbidden 3rds and wolf 5th, or used All twelve 5ths in a temperament must add up to them very carefully as a special effect. In the regular Ϫ720 TU. Temperaments can therefore be dia- temperaments the enharmonic pairs of notes (such grammed easily on paper, with simple numbers, and as D and E) are absolutely different from one competing temperaments can be compared directly. another. Any major 3rds resulting from four consecutive Bach’s diagram pure 5ths are called ‘Pythagorean’ 3rds, being one SC Bach’s WTC title-page (shown in illus.1) is a precise too wide.7 The still wider diminished 4ths of regular abstraction of his preferred tuning method, in the temperaments are often called ‘wolves’, since they form of a pen stroke with 11 large loops. are scarcely more usable than the wolf 5th is. ‘Good’ A keyboard temperament can be described com- temperaments (for the Germans) are those in pletely by the following parameters: (1) the relative which none of the twelve major 3rds are worse than sizes of eleven 5ths,10 (2) the orientation11 and (3) the Pythagorean.