Pure tuning with keyboard instruments

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A pure tuning of keyboard instruments with 12 keys per octave is only possible for a single key , because in pure tuning the pitches octave, fifth , fourth and major third of the diatonic scale used must exactly have the frequency ratios 2: 1, 3: 2, 4 : Form 3 or 5: 4 to the root note. For example, if the C major scale can be played in pure tuning, F major and G major are no longer pure. In order to be able to use more keys, either more than 12 keys or compromises in tuning, i.e. deviations from the purity, are necessary.

A total of six major chords with absolutely pure intervals can be displayed on a 12-step keyboard.

Limits of the pure tuning in keyboard instruments

Problem due to the partial tones

The pure intervals can be derived from the partial tone series . From the 8th partial on, the partial series contains two major seconds . However, these two intervals likewise named vary in size: the larger whole step has a frequency ratio 9 / 8 and the smaller a frequency ratio 10 / 9 . A purely tuned major third consists of a major and a minor whole step. In C major, for example, there is a large whole step between c and d and a small whole step between g and a, in F major there is a small whole step between c and d and in G major between g and a there is a large whole step. This difference cannot be represented on a normal keyboard instrument.

Problem of the syntonic comma

Four perfect fifths give a different tone than a perfectly tuned major third: starting from C, the fifths are called g, d¹, a¹ and e². This e² is higher than the E as a purely tuned major third above the C by the amount of the syntonic comma . But there is only one e-key on a conventional keyboard instrument.

Problem presentation according to the presentation in the tonneau network

With the representation in Euler's tone network , the tones, which are determined by the pure fifth series ... -bfcgda- ..., are differentiated from those ..., b-, f-, c-, g-, d-, a- ... (low point b, Low point f ...), which are given by the relationship of the third to the tonic. In the notation, for example, the pure C major scale cd, efg, a, hc, the pure F major scale fg, abc, d, ef and the pure G major scale ga- , hcd-, e-, f sharp-g it becomes clear that different keys are necessary for the tones d and, d or, a and a, since, d is a syntonic comma lower than d and a is a syntonic comma higher than, a is to be voted.

Problems with modulations

In the case of a modulation with pure tuning in a neighboring key (for example from C major to G major or A minor to D minor), two tones change, one of which is recognizable with a change of sign, the other slightly by a syntonic comma . (That's about 1 / 5 semitone.) This flexibility in adjusting the pitch is only the human voice or comparable instruments reserved. With keyboard instruments with only twelve keys within an octave, pure intonation in all keys is not possible.

Approaches to the pure tuning on the keyboard instrument

Extension of the keyboard

→ Main article: Archicembalo

In order to expand the stock of keys of the usual mean-tone tuning, keyboard instruments were often equipped with additional upper keys (subsemitonies, broken keys , English split keys) at places of professional music care in Western Europe between approx. 1450 and 1700 . Such instruments are related to the so-called enharmonic instruments. Instruments with up to four subsemitonies are known. The development apparently began in Italy and quickly gained some popularity. North of the Alps, it was only Gottfried Fritzsche who built the first organ with subsemitonies in Germany in 1612 (in the electoral palace chapel in Dresden). The most common division was d / es, the second most common was g sharp / a flat.

Gioseffo Zarlino built an instrument with 19 keys per octave in 1558, Vido di Trasuntino created the Clavemusicum omnitonum in 1606 , a harpsichord with 31 keys per octave and Nicola Vicentino in 1555 the Archicembalo, a keyboard instrument with 36 keys per octave on two manuals. However, these instruments proved impractical for playing piano literature.

The historical instruments with 19 keys were designed for a mid-tone tuning. From a mathematical point of view, at least 26 keys per octave are necessary to be able to play purely tuned scales with up to 7 accidentals.

Orthotonophonium (pure harmonium) - Arthur von Oettingen system, Schiedmayer Pianofortefabrik, Stuttgart, 1914

The following authors of tone systems with more than 26 steps (tones) per octave have been handed down:

The pure harmonium

Hermann von Helmholtz , an ardent advocate of pure tuning, describes in detail (p. 516 ff.) About his studies on the purely tuned harmonium with two differently tuned manuals.

As far as the musical effects of the pure tuning are concerned, the difference between this and the equally floating or even the Greek tuning after pure fifths is very noticeable. The pure chords, especially the dura chords in their favorable registers, have, in spite of the rather sharp timbre of the tongue tones, a very full and at the same time saturated echo sound; they flow very calmly in full flow, without trembling or floating. If you put equal or Pythagorean chords next to them, they appear rough, cloudy, trembling and restless ...

The greatest and most unpleasant difference is the difference between natural and tempered chords in the higher octaves of the scale, because here the wrong combination tones of the tempered tuning are more noticeable and because the number of beats increases with the same tone difference, and the roughness is much more intensified, than in a lower position ...

The modulations are therefore much more expressive ... some fine shades can be felt that otherwise almost disappear ...

Description of the pure harmonium

Here the names of Euler's Tonnetz , which Helmholtz also used, are useful.

Designations
c - g - d - a - etc .:
a chain of perfect fifths 702 cents (3/2)
, c -, g -, d -, a - etc. ( deep point c - deep point g - deep point d - etc. ):
the same chain of perfect fifths, but set a syntonic comma lower.

If you go down a series of perfect fifths from b (1110 cents) to ces (1086 cents), it is well known that the last note - octaved - is a Pythagorean comma (23.5 cents) lower than b. On the other hand, if you go down a syntonic comma from h, you get the tone, h ( low point h , 1088 cents), which differs from Ces only by the schism (2 cents). This difference is at the "limit of the perceptible tone differences" (, h = 495,000 Hz, ces = 494,442 Hz). Helmholtz therefore uses, h = ces, also fes =, e, ces =, h, ges =, fis, des =, cis, as =, gis, es =, dis, b =, ais and f =, eis.

Helmholtz further noted that the error can be minimized from 2 cents to an eighth note by increasing the fifths of the chain gcfb-es-as-des-ges-ces by 1/4 cent, then ces =, h. In practice, however, this was no longer used. The builders of the pure harmonium with two manuals (J. and P. Schiedmayer in Stuttgart) tuned the pure harmonic purely by ear with pure fifths and thirds. Here is the bill in cents (and the comparison with the pure mood).

Designations (all in cents )
Octave o = 1200
5th q = 1200 x log 2 (3/2) = 701.955
3rd octave t = 1200 log 2 (5/4) = 386.314
syntonic comma s = 1200 log 2 (81/80) = 21.506
x + q means: x is tuned up by a perfect fifth
x + t means: x is tuned up by a pure third
Lower manual: Comparison with the pure mood Upper manual: Comparison with the pure mood
c = 0 c = 0 e = a + qo = 407.82 e = 4q-2o = 407.82
g = c + q = 701.955 g = q = 701.955 , g sharp = e + t = 794.134 , g sharp = 8q-4o-s = 794.134
d = g + qo = 203.91 d = 2q-o = 203.91 h = e + q = 1109.775 h = 5q-2o = 1109.775
a = d + q = 905.865 a = 3q-o = 905.865 dis1 = h + to = 296.089 , dis = 9q-5o-s = 296.089
, e = c + t = 386.314 , e = 4q-2o-s = 386.314 fis = h + qo = 611.73 fis = 6q-3o = 611.73
, h = g + t = 1088.269 , h = 5q-2o-s = 1088.269 , a sharp = f sharp + t = 998.044 , ais = 10q-50-s = 998.044
, f sharp = d + t = 590.224 , f sharp = 6q-3o-s = 590.224 cis = fis + qo = 113.685 cis = 7q-4o = 113.685
, cis = fis1 + qo = 92.179 , cis = 7q-4o-s = 92.179 , eis = ais1 + qo = 499.999 , eis = 11q-6o-s = 499.999
fes =, e = 386.314 fes = -8q + 5o = 384.36, e = 4q-2o-s = 386.314 as =, g sharp = 794.134 as = -4q + 3o = 792.18, g sharp = 8q-4o-s = 794.134
ces =, h = 1088.269 ces = -7q + 5o = 1086.315, h = 5q-2o-s = 1088.269 es =, dis = 296.089 es = -3q + 2o = 294.135, dis = 9q-5o-s = 296.089
gb =, f # = 590.224 total = -6q + 4o = 588.27, f sharp = 6q-3o-s = 590.224 b =, ais = 998.044 b = -2q + 2o = 996.09, ais = 10q-5o-s = 998.044
des =, cis = 92.179 des = -5q + 3o = 90.225, cis = 7q-4o-s = 92.179 f =, ice = 499.999 f = -q + ​​o = 498.045, cis = 11q-6o-s = 499.999
, as = fes + t = 772.627 , as = -4q + 3o-s = 770.674 , c = as + to = -19.553 , c = -s = -21.506
, es = ces + to = 274,582 , es = -3q + 2o-s = 272.629 , g = es + t = 682,402 , g = qs = 680.449
, b = tot + t = 976.537 , b = -2q + 2o-s = 974.584 , d = b + to = 184.357 , d = 2q-os = 182.404
, f =, b + qo = 478.492 , f = -q + ​​os = 476.539 , a = d1 + q = 886.312 , a = 3q-os = 884.359
The two manuals of the Reinharmonium after Hermann von Helmholtz

The following are playable: F sharp, B, E, A, D, G, C, F, B, E flat, A flat, D flat, G flat and C flat major.

Pure major chords: Pure minor chords:
fes-, as-ces
, as-ces-, es
ces-, es-ges
, es-ges-, b
ges-, b-des
, b-des-, f
des-, f-as
, f-as-, c
as-, c-es
, c-es-, g
es-, gb
, gb-, d
b-, df
, df-, a
f-, ac
, ac-, e
c-, eg
, eg-, h
g-, hd
, hd-, fis
d-, f sharp-a
, f sharp a, c sharp
a-, cis-e
, c sharp-e, g sharp
e-, gis-h
, g sharp-b-, dis
h-, dis-f sharp
, dis-f sharp, a sharp
f sharp, a sharp c sharp
, ais-cis-, ice
It is also possible to use the major dominants of the following

To play minor keys in pure, there (accurate to 2 cents)

,, g sharp =, as; ,, dis =, it; u. s. w.

key Major dominant chord Playable as
,A minor , e ,, g sharp, h , e, as h
, E minor (1 #) , h ,, dis, f sharp , h, es, f sharp
, B minor (2 #) , f sharp, a sharp, c sharp , f sharp, b flat, c sharp
, F sharp minor (3 #) , cis ,, ice, gis , c sharp, f, g sharp
, c sharp minor (4 #) , gis ,, his, dis , g sharp, c, dis
, G sharp minor (5 #) , dis ,, fisis, ais , dis, g, ais
, D flat minor (6 #) , ais ,, cisis, ice , ais, df
=
E flat minor (6 b) b, df b, df
B flat minor (5 b) f, ac f, ac
   

"For the remaining minor keys the series of tones is not quite as sufficient as for the major keys". The dominant major chord can only be played with a Pythagorean third.

key Major chord with pyth. third Not playable
, D minor (1 b) , a, cis, e ,, cis
, G minor (2 b) , d, f sharp, s ,, f sharp
, C minor (3 b) g, hd , g ,, h, d
, F minor (4 b) , c, e, g ,, e
, B flat minor (5 b) , f, a, c ,, a
,, E flat minor (6 b) , b, d, f ,, d

Temperatures

Since keyboard instruments with more than 12 keys pose both structural and technical difficulties, a number of different tuning systems have been developed as a compromise in order to select the pitches assigned to the conventional 12 keys so that as many chords as possible are in the "purest possible tuning" "are playable. These can be classified into three categories:

However, none of these mid-tone or tempered tuning systems expressly do not belong to pure tuning .

Historical sequence of mood compromises

The mean- tone tunings that have prevailed for a long time with many pure thirds approximate the pure tuning very well, but only (in the 1/4-point mean-tone tuning) in the keys of Bb, F, C, G, D and A- Major, as well as G, D, A, E, B and F sharp minor. In these keys the chords of the tonic, the subdominant and the dominant sound in the thirds pure and in the fifths with beats. Listen to: mean fifths .

In order to make the keys of the entire circle of fifths playable, the mean-tone tunings were extended to well-tempered tunings so that the keys of the entire circle of fifths became playable. However, this was only made possible by approximating ( sharpening ) the pure thirds more or less to the Pythagorean thirds, depending on the key . Especially with the piano, the equal tuning finally prevailed, in which there is no longer any “ key character ”.

With this historical development, however, the number of precisely tuned intervals on the keyboard instrument with 12 keys per octave decreased.

The cellist Pablo Casals comments on this problem (“The Way They Play” 1972): “Don't be alarmed if you have a different intonation than the piano. That is due to the piano, which is out of tune. ”The difference in intonation between pure tuning and equal tuning is negligible in the fifths (pure: 702 cents , equal 700 cents) but can be heard - and this is often overlooked - in the thirds (major third pure : 386 cents, equal: 400 cents. Pure minor third: 316 cents, equal 300 cents).

Nowadays, harpsichords are usually tuned to medium-tone or well-tempered music when performing music from the 16th to 18th centuries. It can be observed that historically tuned organs are regaining importance.

Diatonic harmonica

The diatonic harmonica with one to three rows was usually tuned almost purely in its possible tone supply in many cases, or the tuning was approximated to one of the medium-tone tunings. It was only in the last few decades that equal tuning was used in many cases. Even today, individual manufacturers prefer traditional tunings for the Styrian harmonica. Single-row instruments for Cajun music or some harmonica are also at least close to the pure tuning. If the pure tuning is used on multi-row instruments, alterations with slight pitch differences are available for some tones, for example two D s, but how the tones are tangible to one another depends on the respective playing row . The notes that occur twice are tuned in a single row or with thirds that have only a very slight beat . Thus, the quarters from row to row result in a tonnage network . The possibilities are not as dramatic as with a pure tone instrument, but they are considerable if a musician knows how to use these peculiarities.

Electronic pitch change device

Adriaan Daniël Fokker developed an electric piano with pure tuning.

Martin Vogel had a 72-note harmonium built with four manuals and developed an automatic circuit through which the "correct" fifths, thirds and sevenths are automatically set when the keys are lowered.

With the appropriate computer software, pitch changes can be programmed on suitable keyboards so that pure intonation is possible with 12 keys per octave. This was the goal of the Tonalizer , which was presented in 1987 at the Musica fair in Hamburg. and also Mutabor , a computer program that was developed at the Technical University of Dresden .

References and comments

  1. For example the cadenza chords ceg, fac, ghd of C major, the chord of the sub- median a -c-es, the counter-sound (= major parallel ) e-flat -gb and the Neapolitan sixth chord f-a -flat -des of C minor. See: Chromatic scale with pure tuning .
  2. ^ Ibo Ortgies : Pipe Organs with Subsemitones, 1468-1721. and Historical Organs with Subsemitones, 1468–1721. Appendix B. In: Ján Haluska: The Mathematical Theory of Tone Systems (= Pure and Applied Mathematics. 262). Marcel Dekker et al., New York NY et al. 2004, ISBN 0-8247-4714-3 , pp. 141-146 and pp. 369-374.
  3. ^ Ibo Ortgies: Historical organs with subsemitonies. Chronological overview. (As of April 2009).
  4. ^ Mathematics and Music (formerly published on the website of the Kantonsschule Zürcher Unterland) http://www.gwick.ch/MaMu/down/Tasten.pdf
  5. Hermann von Helmholtz : The theory of tone sensations as a physiological basis for the theory of music . Vieweg, Braunschweig 1863, (Unchanged reprint: Minerva-Verlag, Frankfurt am Main 1981, ISBN 3-8102-0715-2 , excerpt ( memento of the original from July 13, 2015 in the Internet Archive ) Info: The archive link was automatically inserted and still not checked. Please check the original and archive link according to the instructions and then remove this note. ). @1@ 2Template: Webachiv / IABot / kilchb.de
  6. Instruments Huygens-Fokker .
  7. Martin Vogel: Memorandum on the construction of keyboard instruments in pure tuning. Publishing house for systematic musicology, Bonn 1986.
  8. Egino Klepper: Psyche of the keys. Musical vocal systems on the border between math and music. In: Culture & Technology. Vol. 13, No. 4, 1989, ISSN  0344-5690 , pp. 248-253, digitized version (PDF; 5.32 MB) .
  9. Mutabor .

See also

Web links