Comparing temperaments

Images of tem­pered scales cre­at­ed by the Bol Processor

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 scales con­struct­ed with tem­pera­ments (Asselin 2000). The names and tun­ing pro­ce­dures fol­low Asselin’s instruc­tions (p. 67-126). Images of the scales have been cre­at­ed by the Bol Processor.

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

We hope to pub­lish bet­ter sound demos after receiv­ing a set of well-designed Csound instru­ments (“orc” files). 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. Uneducated musi­cians believe that “well-tempered” equates “equal-tempered”…

Equal tem­pera­ment (p. 123) ➡ Image

Don’t hes­i­tate to click “Image” links to dis­play cir­cu­lar graph­ic 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).

H.A. Kellner’s BACH in 1975 (p. 101) ➡ Image
Barca in 1786 (p. 106) ➡ Image
Bethisy in 1764 (p. 121) ➡ Image
Chaumont in 1696 (p. 109) ➡ Image
Corrette in 1753 (p. 111) ➡ Image
D’Alambert-Rousseau 1752-1767 (p. 119) ➡ Image
Kirnberger II in 1771 (p. 90) ➡ Image
Kirnberger III in 1779 (p. 93) ➡ Image
Marpurg in 1756 (p. 117) ➡ Image
Pure minor thirds in 16th cen­tu­ry (p. 82) ➡ Image
Rameau en do in 1726 (p. 113) ➡ Image
Rameau en sib in 1726 (p. 115) ➡ Image
Sauveur in 1701 (p. 80) ➡ Image
Tartini-Vallotti in mid. 18th cen­tu­ry (p. 104) ➡ Image
Werckmeister III in 1691 (p. 194) ➡ Image
Werckmeister IV in 1691 (p. 96) ➡ Image
Werckmeister V in 1691 (p. 199) ➡ Image
Zarlino in 1558 (p. 85) ➡ Image

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

Zarlino’s “nat­ur­al scale” ➡ Image
J.S. Bach’s dis­ci­ple Johann Kirnberg (1721-1783) - (source)

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 of this sub­ject, read Bach well-tempered tonal analy­sis and lis­ten to results on page The Well-tempered Clavier.

Fortunately, there are his­tor­i­cal clues to 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 to rep­re­sent his mas­ter’s idea of “well-tempered”.

On 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 designed in 1701 by French physi­cian Joseph Sauveur also seemed to be a bet­ter fit in terms on melod­ic inter­vals — and indeed it sounds beaut­fi­ful… This may in turn be chal­lenged by the sys­tem­at­ic match­ing of all works in books I and II against tun­ing schemes imple­ment­ed on the Bol Processor — read page Bach well-tempered tonal analy­sis.

François Couperin’s Les Ombres Errantes (1730)

Again, 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.

Since a few of the fol­low­ing tem­pera­ments have been designed (or described?) after 1730, these were unlike­ly to be used by the com­pos­er. Let us try them all any­way, and find the winner!

Equal tem­pera­ment ➡ Image
H.A. Kellner’s BACH (p. 101) ➡ Image
Barca in 1786 (p. 106) ➡ Image
Bethisy in 1764 (p. 121) ➡ Image
Chaumont in 1696 (p. 109) ➡ Image
Corrette in 1753 (p. 111) ➡ Image
D’Alambert-Rousseau 1752-1767 (p. 119) ➡ Image
Kirnberger II in 1771 (p. 90) ➡ Image
Kirnberger III in 1779 (p. 93) ➡ Image
Marpurg in 1756 (p. 117) ➡ Image
Pure minor thirds in 16th cen­tu­ry (p. 82) ➡ Image
Rameau en do in 1726 (p. 113) ➡ Image
Rameau en sib in 1726 (p. 115) ➡ Image
Sauveur in 1701 (p. 80) ➡ Image
Tartini-Vallotti in mid. 18th cen­tu­ry (p. 104) ➡ Image
Werckmeister III in 1691 (p. 194) ➡ Image
Werckmeister IV in 1691 (p. 96) ➡ Image
Werckmeister V in 1691 (p. 199) ➡ Image
Zarlino in 1558 (p. 85) ➡ Image
Zarlino’s “nat­ur­al scale” ➡ Image
Matching har­mon­ic inter­vals of “Le Petit Rien” with “Rameau en do” temperament

The best tem­pera­ment for this piece might be Rameau en sib designed 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. Read page Tonal analy­sis of musi­cal items for the descrip­tion of a sys­tem­at­ic (auto­mat­ed) analy­sis con­firm­ing this choice.

We may end up lis­ten­ing to François Couperin’s Le Petit Rien (Ordre 14e 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 page Tonal analy­sis of musi­cal items.

François Couperin’s “Le Petit Rien” (1722), mm = 80, with a “Rameau en do” tem­pera­ment ➡ Image
Source: MusicXML score by Yvan43

Work in progress

Chapter VIII of Pierre-Yves Asselin’s book (2000 p. 139-180) con­tains exam­ples of musi­cal works high­light­ing the rel­e­vance of spe­cif­ic tem­pera­ments. Given that the scores of many Baroque and clas­si­cal mas­ter­pieces are avail­able in 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 lim­i­ta­tion 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 music at large. Read the dis­cus­sion: Just into­na­tion: a gen­er­al frame­work. At least, we hope to con­vince read­ers that equal-temperament is not the “per­fect” solution!

Musicians inter­est­ed in con­tin­u­ing this research and relat­ed devel­op­ment may use Bol Processor BP3’s beta ver­sion to process musi­cal works and cre­ate new tun­ing pro­ce­dures. Follow instruc­tions on page Bol Processor ‘BP3’ and its PHP inter­face 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.

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