https://en.wikipedia.org/wiki/Werckmeister_temperament Jump to content [ ] Main menu Main menu move to sidebar hide Navigation * Main page * Contents * Current events * Random article * About Wikipedia * Contact us Contribute * Help * Learn to edit * Community portal * Recent changes * Upload file [wikipe] Wikipedia The Free Encyclopedia Search [ ] Search * Donate [ ] Appearance * Create account * Log in [ ] Personal tools * Create account * Log in Pages for logged out editors learn more * Contributions * Talk Contents move to sidebar hide * (Top) * 1 Werckmeister I (III): "correct temperament" based on 1/4 comma divisions * 2 Werckmeister II (IV): another temperament included in the Orgelprobe, divided up through 1/3 comma * 3 Werckmeister III (V): an additional temperament divided up through 1/4 comma * 4 Werckmeister IV (VI): the Septenarius tunings * 5 External sources * 6 References [ ] Toggle the table of contents Werckmeister temperament [ ] 6 languages * Cestina * Deutsch * Esperanto * Italiano * Ri Ben Yu * Zhong Wen Edit links * Article * Talk [ ] English * Read * Edit * View history [ ] Tools Tools move to sidebar hide Actions * Read * Edit * View history General * What links here * Related changes * Upload file * Special pages * Permanent link * Page information * Cite this page * Get shortened URL * Download QR code * Wikidata item Print/export * Download as PDF * Printable version In other projects * Wikimedia Commons Appearance move to sidebar hide From Wikipedia, the free encyclopedia Tuning system described by Andreas Werckmeister Werckmeister temperaments are the tuning systems described by Andreas Werckmeister in his writings.^[1]^[2]^[3] The tuning systems are numbered in two different ways: the first refers to the order in which they were presented as "good temperaments" in Werckmeister's 1691 treatise, the second to their labelling on his monochord. The monochord labels start from III since just intonation is labelled I and quarter-comma meantone is labelled II. The temperament commonly known as "Werckmeister III" is referred to in this article as "Werckmeister I (III)".^[4] The tunings I (III), II (IV) and III (V) were presented graphically by a cycle of fifths and a list of major thirds, giving the temperament of each in fractions of a comma. Werckmeister used the organbuilder's notation of ^ for a downwards tempered or narrowed interval and v for an upward tempered or widened one. (This appears counterintuitive - it is based on the use of a conical tuning tool which would reshape the ends of the pipes.) A pure fifth is simply a dash. Werckmeister was not explicit about whether the syntonic comma or Pythagorean comma was meant: the difference between them, the so-called schisma, is almost inaudible and he stated that it could be divided up among the fifths. The last "Septenarius" tuning was not conceived in terms of fractions of a comma, despite some modern authors' attempts to approximate it by some such method. Instead, Werckmeister gave the string lengths on the monochord directly, and from that calculated how each fifth ought to be tempered. Werckmeister I (III): "correct temperament" based on 1/4 comma divisions [edit] This tuning uses mostly pure (perfect) fifths, as in Pythagorean tuning, but each of the fifths C-G, G-D, D-A and B-F# is made smaller, i.e. tempered by 1/4 of the comma. No matter if the Pythagorean comma or the syntonic comma is used, the resulting tempered fifths are for all practical purposes the same as meantone temperament fifths. All major thirds are reasonably close to 400 cents and, because not all fifths are tempered, there is no wolf fifth and all 12 notes can be used as the tonic. Werckmeister designated this tuning as particularly suited for playing chromatic music ("ficte"), which may have led to its popularity as a tuning for J. S. Bach's music in recent years. Fifth Tempering Third Tempering C-G ^ C-E 1 v G-D ^ C#-F 4 v D-A ^ D-F# 2 v A-E - D#-G 3 v E-B - E-G# 3 v B-F# ^ F-A 1 v F#-C# - F#-B 4 v C#-G# - G-B 2 v G#-D# - G#-C 4 v D#-B - A-C# 3 v B-F - B-D 2 v F-C - B-D# 3 v Play major tonic chord^i Because a quarter of the Pythagorean comma is 531441 524288 4 {\ displaystyle {\sqrt[{4}]{\frac {531441}{524288}}}} {\displaystyle {\ sqrt[{4}]{\frac {531441}{524288}}}}, or 27 32 2 4 {\displaystyle {\ frac {27}{32}}{\sqrt[{4}]{2}}} {\displaystyle {\frac {27}{32}}{\sqrt [{4}]{2}}}, it is possible to calculate exact mathematical values for the frequency relationships and intervals: Note Exact frequency relation Value in cents C 1 1 {\displaystyle {\frac {1}{1}}} {\displaystyle {\ 0 frac {1}{1}}} C# 256 243 {\displaystyle {\frac {256}{243}}} {\ 90 displaystyle {\frac {256}{243}}} D 64 81 2 {\displaystyle {\frac {64}{81}}{\sqrt {2}}} {\ 192 displaystyle {\frac {64}{81}}{\sqrt {2}}} D# 32 27 {\displaystyle {\frac {32}{27}}} {\displaystyle 294 {\frac {32}{27}}} 256 243 2 4 {\displaystyle {\frac {256}{243}}{\sqrt E [{4}]{2}}} {\displaystyle {\frac {256}{243}}{\sqrt[{4}] 390 {2}}} F 4 3 {\displaystyle {\frac {4}{3}}} {\displaystyle {\ 498 frac {4}{3}}} F# 1024 729 {\displaystyle {\frac {1024}{729}}} {\ 588 displaystyle {\frac {1024}{729}}} 8 9 2 3 4 {\displaystyle {\frac {8}{9}}{\sqrt[{4}]{2^ G {3}}}} {\displaystyle {\frac {8}{9}}{\sqrt[{4}]{2^ 696 {3}}}} G# 128 81 {\displaystyle {\frac {128}{81}}} {\displaystyle 792 {\frac {128}{81}}} 1024 729 2 4 {\displaystyle {\frac {1024}{729}}{\sqrt A [{4}]{2}}} {\displaystyle {\frac {1024}{729}}{\sqrt 888 [{4}]{2}}} B 16 9 {\displaystyle {\frac {16}{9}}} {\displaystyle {\ 996 frac {16}{9}}} B 128 81 2 4 {\displaystyle {\frac {128}{81}}{\sqrt[{4}] 1092 {2}}} {\displaystyle {\frac {128}{81}}{\sqrt[{4}]{2}}} Werckmeister II (IV): another temperament included in the Orgelprobe, divided up through 1/3 comma [edit] In Werckmeister II the fifths C-G, D-A, E-B, F#-C#, and B-F are tempered narrow by 1/3 comma, and the fifths G#-D# and E-B are widened by 1/3 comma. The other fifths are pure. Werckmeister designed this tuning for playing mainly diatonic music (i.e. rarely using the "black notes"). Most of its intervals are close to sixth-comma meantone. Werckmeister also gave a table of monochord lengths for this tuning, setting C=120 units, a practical approximation to the exact theoretical values^[citation needed]. Following the monochord numbers the G and D are somewhat lower than their theoretical values but other notes are somewhat higher. Fifth Tempering Third Tempering C-G ^ C-E 1 v G-D - C#-F 4 v D-A ^ D-F# 1 v A-E - D#-G 2 v E-B ^ E-G# 1 v B-F# - F-A 1 v F#-C# ^ F#-B 4 v C#-G# - G-B 1 v G#-D# v G#-C 4 v D#-B v A-C# 1 v B-F ^ B-D 1 v F-C - B-D# 3 v Exact frequency Value Approximate monochord Value Note relation in length in cents cents 1 1 {\displaystyle {\ C frac {1}{1}}} {\ 0 120 {\displaystyle 120} {\ 0 displaystyle {\frac {1} displaystyle 120} {1}}} 16384 19683 2 3 {\ 114 1 5 {\displaystyle 114 displaystyle {\frac {\frac {1}{5}}} {\ {16384}{19683}}{\sqrt displaystyle 114{\frac {1} C# [{3}]{2}}} {\ 82 {5}}} (misprinted as 114 1 85.8 displaystyle {\frac 2 {\displaystyle 114{\frac {16384}{19683}}{\sqrt {1}{2}}} {\displaystyle 114 [{3}]{2}}} {\frac {1}{2}}}) 8 9 2 3 {\displaystyle 107 1 5 {\displaystyle 107 {\frac {8}{9}}{\sqrt {\frac {1}{5}}} {\ D [{3}]{2}}} {\ 196 displaystyle 107{\frac {1} 195.3 displaystyle {\frac {8} {5}}} {9}}{\sqrt[{3}]{2}}} 32 27 {\displaystyle {\ 101 1 5 {\displaystyle 101 D# frac {32}{27}}} {\ 294 {\frac {1}{5}}} {\ 295.0 displaystyle {\frac displaystyle 101{\frac {1} {32}{27}}} {5}}} 64 81 4 3 {\ displaystyle {\frac 95 3 5 {\displaystyle 95{\ E {64}{81}}{\sqrt[{3}] 392 frac {3}{5}}} {\ 393.5 {4}}} {\displaystyle {\ displaystyle 95{\frac {3} frac {64}{81}}{\sqrt {5}}} [{3}]{4}}} 4 3 {\displaystyle {\ F frac {4}{3}}} {\ 498 90 {\displaystyle 90} {\ 498.0 displaystyle {\frac {4} displaystyle 90} {3}}} 1024 729 {\displaystyle 85 1 3 {\displaystyle 85{\ F# {\frac {1024}{729}}} {\ 588 frac {1}{3}}} {\ 590.2 displaystyle {\frac displaystyle 85{\frac {1} {1024}{729}}} {3}}} 32 27 2 3 {\ displaystyle {\frac 80 1 5 {\displaystyle 80{\ G {32}{27}}{\sqrt[{3}] 694 frac {1}{5}}} {\ 693.3 {2}}} {\displaystyle {\ displaystyle 80{\frac {1} frac {32}{27}}{\sqrt {5}}} [{3}]{2}}} 8192 6561 2 3 {\ displaystyle {\frac 76 2 15 {\displaystyle 76{\ {8192}{6561}}{\sqrt frac {2}{15}}} {\ G# [{3}]{2}}} {\ 784 displaystyle 76{\frac {2} 787.7 displaystyle {\frac {15}}} {8192}{6561}}{\sqrt [{3}]{2}}} 256 243 4 3 {\ displaystyle {\frac 71 7 10 {\displaystyle 71{\ A {256}{243}}{\sqrt[{3}] 890 frac {7}{10}}} {\ 891.6 {4}}} {\displaystyle {\ displaystyle 71{\frac {7} frac {256}{243}}{\sqrt {10}}} [{3}]{4}}} 9 4 2 3 {\displaystyle 67 1 5 {\displaystyle 67{\ {\frac {9}{4{\sqrt[{3}] frac {1}{5}}} {\ B {2}}}}} {\displaystyle 1004 displaystyle 67{\frac {1} 1003.8 {\frac {9}{4{\sqrt[{3}] {5}}} {2}}}}} 4096 2187 {\ displaystyle {\frac 64 {\displaystyle 64} {\ B {4096}{2187}}} {\ 1086 displaystyle 64} 1088.3 displaystyle {\frac {4096}{2187}}} Werckmeister III (V): an additional temperament divided up through 1/ 4 comma [edit] In Werckmeister III the fifths D-A, A-E, F#-C#, C#-G#, and F-C are narrowed by 1/4, and the fifth G#-D# is widened by 1/4 comma. The other fifths are pure. This temperament is closer to equal temperament than the previous two. Fifth Tempering Third Tempering C-G - C-E 2 v G-D - C#-F 4 v D-A ^ D-F# 2 v A-E ^ D#-G 3 v E-B - E-G# 2 v B-F# - F-A 2 v F#-C# ^ F#-B 3 v C#-G# ^ G-B 2 v G#-D# v G#-C 4 v D#-B - A-C# 2 v B-F - B-D 3 v F-C ^ B-D# 3 v Note Exact frequency relation Value in cents C 1 1 {\displaystyle {\frac {1}{1}}} {\displaystyle {\ 0 frac {1}{1}}} C# 8 9 2 4 {\displaystyle {\frac {8}{9}}{\sqrt[{4}]{2}}} 96 {\displaystyle {\frac {8}{9}}{\sqrt[{4}]{2}}} D 9 8 {\displaystyle {\frac {9}{8}}} {\displaystyle {\ 204 frac {9}{8}}} D# 2 4 {\displaystyle {\sqrt[{4}]{2}}} {\displaystyle {\ 300 sqrt[{4}]{2}}} E 8 9 2 {\displaystyle {\frac {8}{9}}{\sqrt {2}}} {\ 396 displaystyle {\frac {8}{9}}{\sqrt {2}}} F 9 8 2 4 {\displaystyle {\frac {9}{8}}{\sqrt[{4}]{2}}} 504 {\displaystyle {\frac {9}{8}}{\sqrt[{4}]{2}}} F# 2 {\displaystyle {\sqrt {2}}} {\displaystyle {\sqrt 600 {2}}} G 3 2 {\displaystyle {\frac {3}{2}}} {\displaystyle {\ 702 frac {3}{2}}} G# 128 81 {\displaystyle {\frac {128}{81}}} {\displaystyle 792 {\frac {128}{81}}} A 8 4 {\displaystyle {\sqrt[{4}]{8}}} {\displaystyle {\ 900 sqrt[{4}]{8}}} B 3 8 4 {\displaystyle {\frac {3}{\sqrt[{4}]{8}}}} {\ 1002 displaystyle {\frac {3}{\sqrt[{4}]{8}}}} B 4 3 2 {\displaystyle {\frac {4}{3}}{\sqrt {2}}} {\ 1098 displaystyle {\frac {4}{3}}{\sqrt {2}}} Werckmeister IV (VI): the Septenarius tunings [edit] This tuning is based on a division of the monochord length into 196 = 7 x 7 x 4 {\displaystyle 196=7\times 7\times 4} {\displaystyle 196=7\ times 7\times 4} parts. The various notes are then defined by which 196-division one should place the bridge on in order to produce their pitches. The resulting scale has rational frequency relationships, so it is mathematically distinct from the irrational tempered values above; however in practice, both involve pure and impure sounding fifths. Werckmeister also gave a version where the total length is divided into 147 parts, which is simply a transposition of the intervals of the 196-tuning. He described the Septenarius as "an additional temperament which has nothing at all to do with the divisions of the comma, nevertheless in practice so correct that one can be really satisfied with it". One apparent problem with these tunings is the value given to D (or A in the transposed version): Werckmeister writes it as 176. However this produces a musically bad effect because the fifth G-D would then be very flat (more than half a comma); the third B-D would be pure, but D-F# would be more than a comma too sharp - all of which contradict the rest of Werckmeister's writings on temperament. In the illustration of the monochord division, the number "176" is written one place too far to the right, where 175 should be. Therefore it is conceivable that the number 176 is a mistake for 175, which gives a musically much more consistent result. Both values are given in the table below. In the tuning with D=175, the fifths C-G, G-D, D-A, B-F#, F#-C#, and B-F are tempered narrow, while the fifth G#-D# is tempered wider than pure; the other fifths are pure. Note Monochord length Exact frequency relation Value in cents C 196 1/1 0 C# 186 98/93 91 D 176(175) 49/44(28/25) 186(196) D# 165 196/165 298 E 156 49/39 395 F 147 4/3 498 F# 139 196/139 595 G 131 196/131 698 G# 124 49/31 793 A 117 196/117 893 B 110 98/55 1000 B 104 49/26 1097 External sources [edit] * 196-EDL & 1568-EDL and Septenarius tunings * "Well Tempering based on the Werckmeister Definition" * Well Tempered based on Werckmeisters last book Musikalische Paradoxal-Discourse (1707) is Equal Temperament. See: https:// www.academia.edu/5210832/ 18th_Century_Quotes_on_J.S._Bachs_Temperament References [edit] 1. ^ Andreas Werckmeister: Orgel-Probe (Frankfurt & Leipzig 1681), excerpts in Mark Lindley, "Stimmung und Temperatur", in Horen, messen und rechnen in der fruhen Neuzeit pp. 109-331, Frieder Zaminer (ed.), vol. 6 of Geschichte der Musiktheorie, Wissenschaftliche Buchgesellschaft (Darmstadt 1987). 2. ^ A. Werckmeister: Musicae mathematicae hodegus curiosus oder Richtiger Musicalischer Weg-Weiser (Quedlinburg 1686, Frankfurt & Leipzig 1687) ISBN 3-487-04080-8 3. ^ A. Werckmeister: Musicalische Temperatur (Quedlinburg 1691), reprint edited by Rudolf Rasch ISBN 90-70907-02-X 4. ^ "A handy guide to choosing temperaments for the practical musician, note 2". Retrieved 12 November 2023. * v * t * e Musical tunings * Pitch * Cent * Millioctave * Savart * Interval * Interval ratio Measurement * Pitch class * Consonance and dissonance * List of musical intervals * List of intervals in 5-limit just intonation * List of meantone intervals * Microtone * Euler-Fokker genus * Harmonic scale * Harry Partch's 43-tone scale * Hexany * Limit + 5-limit Just + 7-limit intonation * List of compositions * Otonality * Ptolemy's intense diatonic scale * Pythagorean tuning * Scale of harmonics * Tonality diamond * Tonality flux * 6-tone * 12-tone * 15-tone * 17-tone * 19-tone * 22-tone * 23-tone Equal * 24-tone (pieces) * 31-tone * 34-tone * 41-tone * 53-tone Temperaments * 58-tone * 72-tone * 96-tone * Meantone (quarter-comma, septimal) * Schismatic Linear * Miracle * Magic * Regular diatonic Irregular Well temperament/Temperament ordinaire ( Kirnberger, Vallotti, Werckmeister, Young) * Chinese musicology * Shi-er-lu * Dastgah * Maqam + Arabic maqam + Turkish makam Traditional + Mugham non-Western + Muqam * Octoechos * Pelog * Raga (Carnatic raga) * Slendro * Tetrachord * 833 cents scale * A12 scale * Alpha scale Non-octave * Beta scale * Gamma scale * Delta scale * Lambda scale (Bohlen-Pierce) * Retrieved from "https://en.wikipedia.org/w/index.php?title= Werckmeister_temperament&oldid=1195152195" Category: * Musical temperaments Hidden categories: * Pages using the Phonos extension * Articles with short description * Short description matches Wikidata * All articles with unsourced statements * Articles with unsourced statements from November 2016 * This page was last edited on 12 January 2024, at 15:11 (UTC). * Text is available under the Creative Commons Attribution-ShareAlike 4.0 License; additional terms may apply. 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