Comparing temperaments

The fol­low­ing are Bol Processor + Csound inter­pre­ta­tions of J.-S. Bach’s Prelude 1 in C major (1722) and François Couperin’s Les Ombres Errantes (1730) — both near the end of the Baroque peri­od — using tem­pera­ment scales (Asselin 2000). The names and tun­ing pro­ce­dures fol­low Asselin’s instruc­tions (p. 67-126). Images of the scales have been cre­at­ed using the Bol Processor.

The con­struc­tion of these scales with the Bol Processor is explained in detail on the Microtonality page. The com­plete set of scale images is avail­able on this page.

➡ We hope to be able to release bet­ter sound demos upon receipt of a set of well-designed C-sound instru­ments. (“orc” files). My apolo­gies to harp­si­chord play­ers, tuners and designers!

Bach’s Prelude 1 in C major (1722)

This is the first pre­lude in a series called Well-Tempered Clavier by Johann Sebastian Bach. Well-tempered, ok… But which temperament?

Let us start lis­ten­ing to the piece in equal tem­pera­ment, the pop­u­lar tun­ing of instru­ments in elec­tron­ic times. Unlearned musi­cians believe that “well-tempered” is the equiv­a­lent of “equal-tempered.”

➡ Don’t hes­i­tate to click on the “Image” links to dis­play cir­cu­lar graph­i­cal rep­re­sen­ta­tions of scale inter­vals high­light­ing con­so­nance and dissonance.

The fol­low­ing are tra­di­tion­al tem­pera­ments, each of which has been designed at a par­tic­u­lar time to meet the speci­fici­ties of musi­cal reper­toires en vogue (Asselin 2000 p. 139-180).

The pre­ced­ing exam­ple was Zarlino’s tem­pera­ment, which should not be con­fused with the pop­u­lar “nat­ur­al scale” of Zarlino, an instance of just into­na­tion:

J.S. Bach’s Well-Tempered Clavier (BWV 846–893) is a col­lec­tion of two sets of pre­ludes and fugues in all 24 major and minor keys, dat­ed 1722. To assess the valid­i­ty of a tun­ing scheme it would be nec­es­sary to lis­ten to all pieces. Readers impa­tient to know more may be inter­est­ed in a “com­pu­ta­tion­al” approach to the sub­ject, read Bach well-tempered tonal analy­sis and lis­ten to the results on the page The Well-tempered Clavier.

Fortunately, there are his­tor­i­cal clues to the opti­mal choic­es: Friedrich Wilhelm Marpurg received infor­ma­tion from Bach’s sons and pupils and Johann Kirnberger, one of these pupils, designed tun­ings which he claimed rep­re­sent­ed his mas­ter’s idea of “well-tempered”.

On the page Tonal analy­sis of musi­cal items we show that the analy­sis of tonal inter­vals tends to sug­gest the choice of Kirnberger III rather than Kirnberger II. However, the tem­pera­ment devised by French physi­cian Joseph Sauveur in 1701 also seemed to be a bet­ter fit in terms of melod­ic inter­vals — and indeed it sounds beau­ti­ful… This, in turn, may be chal­lenged by the sys­tem­at­ic match­ing of all works in books I and II against tun­ing schemes imple­ment­ed on the Bol Processor — see page Bach well-tempered tonal analy­sis.

François Couperin’s Les Ombres Errantes (1730)

Again, my apolo­gies to harp­si­chord play­ers, tuners and manufacturers!

This piece is part of François Couperin’s Quatrième livre pub­lished in 1730 ➡ read the full score (Creative Commons licence CC0 1.0 Universal). We used it to illus­trate the inter­pre­ta­tion of mor­dents when import­ing MusicXML files.

First, lis­ten to an (excel­lent) inter­pre­ta­tion of this work by the harp­si­chord play­er Iddo Bar-Shaï (source: https://youtu.be/DCwkMSTFV_E).

Despite its artis­tic qual­i­ty, this per­for­mance dis­plays a few dis­so­nant effects, which are par­tial­ly masked by the abun­dance of melod­ic orna­men­ta­tion: mor­dents, trills, etc. Such a dis­so­ci­a­tion from the theme of the ‘Ombres errantes’ (wan­der­ing shad­ows) can­not be attrib­uted to either the com­pos­er or the per­former. As a result, it is legit­i­mate to ques­tion the tun­ing of the instru­ment. To do this, we need to focus our atten­tion on the tonal­i­ty, even if the sound syn­the­sis appears arti­fi­cial to lis­ten­ers whose atten­tion is focused on tem­po­ral­i­ty, orna­men­ta­tion and sound quality.

As some of the fol­low­ing tem­pera­ments were devised (or described?) after 1730, it is unlike­ly that the com­pos­er used them. Let us try them all any­way, and find the winner!

The best tem­pera­ment for this piece might be Rameau en si_b, which was devised by Couperin’s con­tem­po­rary Jean-Philippe Rameau for musi­cal works with flats in their key sig­na­ture (Asselin, 2000 p. 149) — such as the present one. See the Tonal analy­sis of musi­cal items page for a descrip­tion of a sys­tem­at­ic (auto­mat­ed) analy­sis that con­firms this choice.

We may end up lis­ten­ing to François Couperin’s Le Petit Rien (Ordre 14^e de clavecin in D major, 1722) with two sharps in the key sig­na­ture sug­gest­ing the use of a Rameau en do temperament.

This choice is also con­firmed by the method described on the page Tonal analy­sis of musi­cal items.

Bernard Bel — 2022

Work in progress

Chapter VIII of Pierre-Yves Asselin’s book (2000 p. 139-180) con­tains exam­ples of musi­cal works that illus­trate the rel­e­vance of spe­cif­ic tem­pera­ments. As the scores of many baroque and clas­si­cal mas­ter­pieces are avail­able in the dig­i­tal for­mat MusicXML, we hope to use Bol Processor’s Importing MusicXML scores to transcode them and play these frag­ments with the sug­gest­ed temperaments.

Despite the lim­i­ta­tions of com­par­ing tem­pera­ments on only two musi­cal exam­ples, the aim of this page is to illus­trate the notion of “per­fec­tion” in sets of tonal inter­vals — and in music at large. Read the dis­cus­sion: Just into­na­tion: a gen­er­al frame­work. At the very least, we hope to per­suade those read­ing this that equal tem­pera­ment is not the “per­fect” solution!

Musicians inter­est­ed in con­tin­u­ing this research and relat­ed devel­op­ment can use the beta ver­sion of Bol Processor BP3 to process musi­cal works and cre­ate new tun­ing pro­ce­dures. Follow the instruc­tions on the Bol Processor ‘BP3’ and its PHP inter­face page to install BP3 and learn its basic oper­a­tion. Download and install Csound from its dis­tri­b­u­tion page.

References

Asselin, P.-Y. Musique et tem­péra­ment. Paris, 1985, repub­lished in 2000: Jobert. Soon avail­able in English.