Nineteen-step mood

Depending on the use - for example as C sharp or as D flat - the black keys on the harpsichord universale as well as B and Ces or Eis and F differ by 41 cents.

Since only a limited number of keys can be played on mid-tone instruments with 12 keys per octave, further keys were added in Western Europe between around 1450 and 1700. In addition to the 12 keys for C, C sharp, D, Eb, E, F, F sharp, G, G sharp, A, B, B, the keys Db, Db, Gb, As and Ais, as well as Eis and His were added. The tones C sharp and D flat, D flat and Eb, ice and F etc. differ by a small diesis , i.e. by 41 cents , which corresponds to almost half a semitone. Therefore these tones could not be confused enharmonically . With this nineteen-step tuning , all keys in the circle of fifths from G major (E flat minor) to F sharp major (D flat minor) can be played.

As an alternative to equal tuning , in which the octave is divided into 12 equal intervals, a nineteen-step tuning was also investigated in the 19th century , in which the octave is divided into 19 equal intervals. Here the neighboring tones differ by 63 cents.

First approaches in the 16th century

Keyboard of a 19-step harpsichord from Le istituzioni harmoniche (edition 1573) by Gioseffo Zarlino

In the 16th century several theorists (with reference to the ancient music theory of Greece , the keys of which they tried to reproduce) tried to balance the compromise between pure intervals and the shifts through additional tones within the octave and enharmonic "variants" to the existing twelve tones with additional ones Realize keys (with a keyboard). The attempt to find more and more precise differentiations for enharmonic scales led to proposals for 19-, but also for 24- and 36-step scales and keyboards, for which instruments were also built. 19-step harpsichords were apparently quite common in the 16th century. This made it possible to have more intervals sound relatively pure and thus to be able to play more keys with a harmonious sound. In all of this, however, the mathematical representation was less problematic than the vocal practice and the construction of corresponding instruments.

As early as 1558, the Italian composer and theorist Gioseffo Zarlino mentioned in his work Le istituzioni harmoniche a tuning that referred to nineteen tone steps within the octave without going into detail. This is evidently the ¹ ⁄ ₃ -decimal tone tuning proposed by the theoretician Francisco de Salinas in 1577 , which contains the twelve tones of the scale customary at the time - C, C sharp, D, Eb , E, F, F sharp, G, G sharp , A, B and H - seven more enharmonic variants - His, Des, Dis, Eis, Ges, As and Ais - added. According to contemporary evidence, the blind Salinas was able to play very skillfully at this temperature on a 19-step instrument constructed according to his plans. Klaus Lang writes about this:

“In this tuning the fifths and major thirds are reduced by 1/3 (syntonic) commas, while the major sixths remain pure. Zarlino himself says that this method doesn't sound as good as the other two methods. An interesting property of this tempering method, however, is that if you tune one of the instruments that were relatively widespread in the 16th century with 19 steps per octave with its help, the circle of fifths can be closed, i.e. the wolf fifth is eliminated. "

Reconstruction of the harpsichord universale after Praetorius

According to the description in the Syntagma musicum by Michael Praetorius , the 19-note Cimbalo cromatico had five divided upper keys for the enharmonic subsemitonies and the additional notes ice and his. With a temperament in the usual ¹ ⁄ _4-point mean-tone tuning, the instrument allows you to play with 15 instead of just 8 pure thirds (on Gb, Db, A-flat, E-flat, Bb, F, C, G, D, A, E, H, F sharp, C sharp, G sharp). Depending on the use - for example as Cis or Des - the upper keys as well as His and C or Eis and F differ by 41 cents. See: The 19-step, mid-tone keyboard .

The chanson Seigneur Dieu ta pitié by the French composer Guillaume Costeley is composed for a 19-step tone system; Because Costeley reported in 1570 that he had composed this chromatic-enharmonic chanson spiritually "a good twelve years ago" ("il ya bien douze ans") , i.e. around 1557, as an exercise in the use of a 19-step scale. He also explained in great detail how to build 19-step keyboard instruments and thought of an equal division of the octave.

Original works for cimbalo cromatico were written by Giovanni Maria Trabaci , Ascanio Mayone , Gioanpietro del Buono , Adriano Banchieri , and the Englishman John Bull .

Equal division of the octave since the 19th century

In the 19th century, research began on alternatives to 12-note equal-scale tuning . In order to generate purer intervals, in addition to the 31, 43, 50 and 53-step divisions of the octave, the 19-step division was especially examined for pragmatic reasons. In his Essay on Musical Intervals, Harmonics, and the Temperament of the Musical Scale (1835), the theorist Wesley Woolhouse propagated, among others, an equally tuned tone system which divides the octave (contrary to the conventional trend) into 19 (instead of 12) equal intervals . The remarkable thing about it is that for major and minor thirds and sixths frequency ratios arise that are a lot closer to the pure interval than those in the usual equal pitch. All other intervals are, however, further removed from their pure equivalents.

For the even nineteen-step tuning there is a whole series of compositions both with classical standards and in the rock and pop sector. The development in the field of electronic musical instruments and computer-aided systems for sound synthesis give compositions in this and other alternative moods a considerable boost.

The sound material

The mathematical rule for determining the frequency of a tone of the 19-step equal tempering is

${\displaystyle f(i)=f(0)\cdot 2^{i/19}\,,}$

where f (0) is the frequency of any reference tone, f ( i ) is the frequency of the tone that is i 19th octave steps higher.

The smallest representable tone difference of the system has the frequency ratio

${\displaystyle {\frac {f_{1}}{f_{0}}}={\frac {f_{2}}{f_{1}}}=\ldots ={\sqrt[{19}]{2}}\approx 1{,}037155\approx 63{,}16\,\mathrm {Cent} \,.}$

All tempered nineteen- note tunings have the same specific Enharmonics , which differs significantly from that used in 12-note temperaments. This is how z. For example, the notes F sharp and G flat actually have different heights, and with a 19-step circle of fifths one could, for example, place the seam between A sharp and Fez. This results in modulation paths that differ from the classic 12-step design .

As in the twelve-level equally tempered mood, interval sizes can also be described as multiples of the smallest representable interval for the nineteen-level equally tempered mood. The following values ​​are obtained for step intervals:

Interval name	example	purely	19-step tuning	12-step tuning
Diatonic whole tone	C-D D-E	204 cents 182 cents	3 steps (189 cents)	2 steps (200 cents)
Diatonic semitone	E-F	112 cents	2 steps (126 cents)	1 step (100 cents)
Chromatic halftone	F-F sharp	92 cents	1 step (63 cents)	1 step (100 cents)

Properties of selected intervals

The intervals of the pure tuning, which are understood as consonances in classical music , are sometimes better (thirds and sixths), sometimes less well (fourths and fifths) reproduced by the 19-step tuning. Here is a tabular comparison (the differences are given in cents , the better approximation is highlighted) :

interval	Prime	kl. third	major third	Fourth	Fifth	kl. Sixth	gr. sixth	octave
Diff. 19 levels	0	0.15	−7.37	7.22	−7.22	7.37	−0.15	0
Diff. 12 levels	0	−15.64	13.69	1.96	−1.96	−13.69	15.64	0

The following table shows the values ​​of all intervals, in equal and pure tuning as well as their deviation from each other in cents :

interval	Equally tempered interval	Pure interval	Difference in cents 2)	Difference in cents between the twelve-level, equal-level tuning and the pure interval 2)
Prime	${\displaystyle {\sqrt[{19}]{2^{0}}}=1=0\,\mathrm {Cent} }$	${\displaystyle {\tfrac {1}{1}}=1=0\,\mathrm {Cent} }$	0 cents	0 cents
Excessive prime and diminished second	${\displaystyle {\sqrt[{19}]{2^{1}}}\approx 1{,}037155\approx 63{,}16\,\mathrm {Cent} }$
Small second	${\displaystyle {\sqrt[{19}]{2^{2}}}\approx 1{,}075691\approx 126{,}32\,\mathrm {Cent} }$	${\displaystyle {\tfrac {16}{15}}=1{,}0{\overline {6}}\approx 111{,}73\,\mathrm {Cent} }$	14.58 cents	−11.73 cents
Big second	${\displaystyle {\sqrt[{19}]{2^{3}}}\approx 1{,}115658\approx 189{,}47\,\mathrm {Cent} }$	${\displaystyle {\tfrac {9}{8}}=1{,}125\approx 203{,}91\,\mathrm {Cent} }$	−14.44 cents	−3.91 cents
Excessive second and diminished third	${\displaystyle {\sqrt[{19}]{2^{4}}}\approx 1{,}157110\approx 252{,}63\,\mathrm {Cent} }$
Minor third	${\displaystyle {\sqrt[{19}]{2^{5}}}\approx 1{,}200103\approx 315{,}79\,\mathrm {Cent} }$	${\displaystyle {\tfrac {6}{5}}=1{,}2\approx 315{,}64\,\mathrm {Cent} }$	0.15 cents	−15.64 cents
Major third	${\displaystyle {\sqrt[{19}]{2^{6}}}\approx 1{,}244693\approx 378{,}95\,\mathrm {Cent} }$	${\displaystyle {\tfrac {5}{4}}=1{,}25\approx 386{,}31\,\mathrm {Cent} }$	−7.37 cents	13.69 cents
Excessive third and diminished fourth	${\displaystyle {\sqrt[{19}]{2^{7}}}\approx 1{,}290939\approx 442{,}11\,\mathrm {Cent} }$
Fourth	${\displaystyle {\sqrt[{19}]{2^{8}}}\approx 1{,}338904\approx 505{,}26\,\mathrm {Cent} }$	${\displaystyle {\tfrac {4}{3}}=1{,}{\overline {3}}\approx 498{,}04\,\mathrm {Cent} }$	7.22 cents	1.96 cents
Excessive fourths 1)	${\displaystyle {\sqrt[{19}]{2^{9}}}\approx 1{,}388651\approx 568{,}42\,\mathrm {Cent} }$	${\displaystyle {\tfrac {45}{32}}=1{,}40625\approx 590{,}22\,\mathrm {Cent} }$	−21.8 cents	9.78 cents
Diminished fifth	${\displaystyle {\sqrt[{19}]{2^{10}}}\approx 1{,}440247\approx 631{,}58\,\mathrm {Cent} }$	${\displaystyle {\tfrac {64}{45}}=1{,}4{\overline {2}}\approx 609{,}78\,\mathrm {Cent} }$	21.8 cents	−9.78 cents
Fifth	${\displaystyle {\sqrt[{19}]{2^{11}}}\approx 1{,}493759\approx 694{,}74\,\mathrm {Cent} }$	${\displaystyle {\tfrac {3}{2}}=1{,}5\approx 701{,}96\,\mathrm {Cent} }$	−7.22 cents	−1.96 cents
Excessive fifth and diminished sixth	${\displaystyle {\sqrt[{19}]{2^{12}}}\approx 1{,}549260\approx 757,89\,\mathrm {Cent} }$
Small sixth	${\displaystyle {\sqrt[{19}]{2^{13}}}\approx 1{,}606822\approx 821{,}05\,\mathrm {Cent} }$	${\displaystyle {\tfrac {8}{5}}=1{,}6\approx 813{,}69\,\mathrm {Cent} }$	7.37 cents	−13.69 cents
Major sixth	${\displaystyle {\sqrt[{19}]{2^{14}}}\approx 1{,}666524\approx 884{,}21\,\mathrm {Cent} }$	${\displaystyle {\tfrac {5}{3}}=1{,}{\overline {6}}\approx 884{,}36\,\mathrm {Cent} }$	−0.15 cents	15.64 cents
Excessive sixth and diminished seventh	${\displaystyle {\sqrt[{19}]{2^{15}}}\approx 1{,}728444\approx 947{,}37\,\mathrm {Cent} }$
Minor seventh	${\displaystyle {\sqrt[{19}]{2^{16}}}\approx 1{,}792664\approx 1010{,}53\,\mathrm {Cent} }$	${\displaystyle {\tfrac {16}{9}}=1{,}{\overline {7}}\approx 996{,}09\,\mathrm {Cent} }$	14.44 cents	3.91 cents
Major seventh	${\displaystyle {\sqrt[{19}]{2^{17}}}\approx 1{,}859270\approx 1073{,}68\,\mathrm {Cent} }$	${\displaystyle {\tfrac {15}{8}}=1{,}875\approx 1088{,}27\,\mathrm {Cent} }$	−14.58 cents	11.73 cents
Excessive seventh and diminished octave	${\displaystyle {\sqrt[{19}]{2^{18}}}\approx 1{,}928352\approx 1136{,}84\,\mathrm {Cent} }$
octave	${\displaystyle {\sqrt[{19}]{2^{19}}}=2=1200\,\mathrm {Cent} }$	${\displaystyle {\tfrac {2}{1}}=2=1200\,\mathrm {Cent} }$	0 cents	0 cents
Notes: 1) Excessive fourth , sometimes referred to as tritone refers defined as: Major third ( 5 / 4 ) plus Large second ( 9 / 8 ). This is equivalent to: fifth ( 3 / 2 ) minus diatonic halftone ( 16 / 15 ). 2) If the difference is negative, the interval with the same temperature is narrower than the pure one.

See also

• Sound system
• Circle of fifths
• cent
• The division of the octave into 53 equal pitches
• Quarter-tone music
• Archicembalo

literature

• Edward L. Kottick: Harpsichords with more than twelve notes to the Octave. In: A History of the Harpsichord. Indiana University Press, Bloomington (Indiana) 2003, pp. 88-89, 487 (footnotes). (engl.)
• Klaus Lang : On harmonious waves through the sounds of the sea. Temperatures and moods between the 11th and 19th centuries (= contributions to electronic music. 10, ZDB -ID 1415612-x ). Institute for Electronic Music, Graz 1999 ( PDF file ( Memento from March 12, 2007 in the Internet Archive )).
• Mark Lindley : Mood and Temperature. In: Frieder Zaminer (ed.): History of music theory. Volume 6: Hearing, Measuring and Arithmetic in the Early Modern Era. Wissenschaftliche Buchgesellschaft, Darmstadt 1987, ISBN 3-534-01206-2 , pp. 109-332.
• Christopher Stembridge: Music for the “Cimbalo cromatico” and other Split-Keyed Instruments in Seventeenth-Century Italy. In: Performance Practice Review 5, no. 1, 1992, pp. 5-43.
• Christopher Stembridge: The “Cimbalo cromatico” and other italian Keyboard Instruments with nineteen or more divisions to the Octave… In: Performance Practice Review 6, no. 1, 1993, pp. 33-59.
• Denzil Wraight, Christopher Stembridge: Italian Split-Keyed Instruments with fewer than Nineteen Divisions to the Octave. In: Performance Practice Review 7, no. 2, 1994, pp. 150-181.

Web links

To the 16th century

• Encyclopedia of Microtonal Music Theory , from it:
  • 19-tone equal-temperament and 1/3-comma meantone
  • 2/7 comma meantone
• Guillaume Costeley: Seigneur dieu ta pitié :
  • Sheet music (without text) and sound recording (with electronically simulated organ)
  • Essay (English, see the postscript)

To the 20th and 21st centuries

(all web links in English)

• Saku Bucht, rkki Huovinen: Perceived consonance of harmonic intervals in 19-tone equal temperament . (PDF)
• Ivor Darreg: A Case for Nineteen .
• Hubert S. Howe, Jr .: 19-Tone Theory and Applications
• William A. Sethares: Tunings for 19 Tone Equal Tempered Guitar .

Sound samples

• Jacob Barton: Foum (MP3)
• Jeff Harrington: Prelude 1 for 19ET Piano (MP3; 5.5 MB)
• Jeff Harrington: Prelude 2 for 19ET Piano (MP3; 4.1 MB)
• Jeff Harrington: Prelude 3 for 19ET Piano (MP3; 6.2 MB)
• Aaron Kristler: http://www.akjmusic.com/audio/juggler.ogg ( Memento from February 5, 2012 in the Internet Archive ) Juggler (ogg; 3.4 MB)

Individual evidence

1. ↑ Klaus Lang: On harmonious waves through the sounds of the sea. 1999, p. 62.
2. ^ Syntagma musicum. Volume 2: De Organographia , 1619 ( online , accessed May 9, 2017).
3. ↑ Edward L. Kottick: A History of the Harpsichord. Indiana University Press, Bloomington (Indiana) 2003, p. 89.
4. ↑ Bull's "chromatic" Ut Re Mi Fa Sol La; in: The Fitzwilliam Virginal Book (revised Dover Edition, 2 volumes). Edited by JA Fuller Maitland et al. W. Barclay Squire, corrected et al. ed. by Blanche Winogron. Dover Publications, New York 1979/1980, Vol. 1, p. 183 (No. LI).
5. ↑ In music practice, intervals are sometimes differentiated according to their absolute size: Everything that is less than or equal to the second is called a step , larger intervals a jump ; this distinction is made e.g. B. observed in the rules of counterpoint .