Euler's clay net

The Euler notation is named after the mathematician Leonhard Euler , who introduced it for the fine analysis of pieces of music in pure tuning . It can be seen from the designation whether the occurring thirds and fifths sound pure. For example, the tonic, subdominant and dominant chords of the pure C major scale have the representation c -, e - g and f-, ac as well as g-, hd (major chords with a pure major and minor third and a pure fifth), where the "low point" before , e and , a as well as , h means that the tones sound a syntonic comma lower than the e and a as well as h in the series of perfect fifths ... fcgdaeh ...

The Euler tone lattice represents the gamut of just intonation is in a two dimensional grid of pure fifth and Terzintervallen. Euler had already in the published in 1739. Tentamen novae theoriae musicae discussed in detail the mathematical relationships of music. In this work he looked for a mathematical justification for the feeling of consonance and dissonance in the art of music and put together an interval relationship system on the basis of prime numbers. In another work, De harmoniae veris principiis per speculum musicum repraesentatis , which was published in 1773, he described the tonal network of fifths and thirds.

The fifth-third scheme

The introduction of the pure major third (with the string length ratio between lower and upper tone of 5: 4) - as a replacement and simplification of the Pythagorean ditone (81:64) - goes back to Didymos' enharmonic tetrachord division (around 100 years after Pythagoras ). However, the tone system in ancient Greece cannot be compared with the tone system to which Leonhard Euler refers. In today's tonal system, the pure major third was first mentioned around 1300 by Walter Odington in his De Speculatione Musices .

In western music, the major third (frequency ratio: 81:64) of the Pythagorean system was perceived as a dissonance. With the emergence of polyphony in the 15th century, the pure major third (5: 4) became more and more important as part of the triad.

The C major chord in the overtone series?

This results in a tone system based on the intervals octave ² ⁄ ₁ , fifth ³ ⁄ ₂ and third ⁵ ⁄ ₄ . The other intervals of the fifth-third system can be represented as multiples of these intervals.

The large number of combinations of these intervals results in a (theoretically) infinite pitch space. This pitch space is often represented graphically by means of a pitch network as follows:

Representation in the barrel network

Since the pure third could not be represented with fifths, Leonhard Euler represented the network of relationships of the pure tuning with the help of rows of fifths, each of which differed by a syntonic comma. In the following network of relationships, the fifths are lined up in the horizontal direction and the thirds in the vertical direction.

The notation, x ("deep comma x") or 'x ("apostrophe x") - the comma in front of the tone designation - etc. means that the tone, x or' x is a syntonic comma lower or higher than the tone x is.

This graphical representation of the fifth-third scheme is understood as a network of relationships between pitch classes without fixed octave (also: "Chroma", "tonal character" ; English pitch class ), so that for the calculation of specific interval ratios nor the corresponding multiple of the octave ² / ₁ added - or has to be taken away.

The pure scales always have the same appearance in this graphic representation. Interval relationships are always the same for every scale:

scale	Scale tones listed in a table
F major	f	G	, a	b	c	, d	, e	f	, D minor	, d	, e	f	,G	, a	b	c	, d
C major	c	d	, e	f	G	, a	,H	c	,A minor	, a	,H	c	, d	, e	f	G	, a
G major	G	a	,H	c	d	, e	, fis	G	, E minor	, e	, fis	G	, a	,H	c	d	, e
x major	x	x	, x	x	x	, x	, x	x	x minor	x	x	'x	x	x	'x	'x	x

(For a more extensive table, see The scales in the circle of fifths )

The calculation of the corresponding cent values using the example of the tones c - e -, e with octave = 1200 cents, fifth = 701.955 cents and syntonic comma = 21.506 cents results, for example, with c = 0 cents rounded:

e = c + 4 fifths - 2 octaves = 408 cents and, e = e - syntonic comma = 386 cents (= c + pure major third)

The corresponding frequency values are calculated with c = 264 Hz as follows:

{\ displaystyle e = {\ frac {264 \ cdot ({\ frac {3} {2}}) ^ {4}} {2 ^ {2}}} = 334 {,} 125 \, \ mathrm {Hz} \ quad {\ text {and}} \ quad, e = e \ cdot {\ frac {80} {81}} = 330 \, \ mathrm {Hz} \ left (= c \ cdot {\ frac {5} { 4}} \ right)}

.

The cent values of the tones result in:

volume	calculation	Cent value
c	0	0
d	2q - o	204
, e	4q - 2o - k	386
f	−q + o	498
G	q	702
, a	3q - o - k	884
,H	5q - 2o - k	1088
c ′	O	1200

(For a more extensive table, see The scales in the circle of fifths )

It shows the tone relationships of the harmonious-pure mood . For example, the notes, e and, a and, h of the C major scale sound a syntonic comma lower than in the Pythagorean chain of fifths . The chords c-, eg and f-, ac as well as g-, hd consist of pure thirds ( ⁵ ⁄ ₄ and ⁶ ⁄ ₅ ) and perfect fifths ( ³ ⁄ ₂ ).

The actual "Tonnetz" was introduced in 1773 by Leonhard Euler as speculum musicum ("image of music") in his eponymous text De harmoniae veris principiis per speculum musicum repraesentatis , and from then on - together with the designations introduced by Moritz Hauptmann for previously eponymous Tones that differ by a syntonic comma - modified by numerous theorists for various purposes (including by Hermann v. Helmholtz , Arthur v. Oettingen and Hugo Riemann ). The different characters of tones with the same name but different positions in the (infinite) pitch space result in a harmonic-pure tuning not only from a different tone environment and harmonization (e.g. the, e in the C major chord c-, eg and the e in the E major chord e-, g sharp-b), but also from a (minimal) pitch difference between the respective pitches (, e and e):

In the C major scale cd, efg, a, hc, the interval c-, e is a major third with the frequency ratio ⁵ ⁄ ₄ . In the series of fifths cgdae, the interval (octave back) ce is a Pythagorean third with the frequency ratio ⁸¹ ⁄ ₆₄ . These two intervals differ by the syntonic comma with the frequency ratio ⁸¹ ⁄ ₈₀ .

The C major scale in a harmonious, pure fifth-third tuning

The pure C major scale can be understood as the selection of those seven pitches from the fifth-third scheme that are used for the intonation of the three main functions subdominant (S), tonic (T) and dominant (D) - i.e. for the "authentic" Cadence needed:

C major tonality

The actual scale is created by transposing these pitches into the corresponding octave - for example between c ¹ and c ² . In contrast to the Pythagorean scale, it no longer consists of two, but three interval steps of different sizes, the large whole tone ⁹ ⁄ ₈ , the small whole tone ¹⁰ ⁄ ₉ and the diatonic semitone ¹⁶ ⁄ ₁₅ :

pure C major scale

The thesis (for example with Sigfrid Karg-Elert ) that with this seven-step scale, which allows the harmonic-pure intonation of the main functions T, S and D, melodically unclean, because the respective thirds, e /, a and, h in melodic context can be perceived as too low is controversial. Ross W. Duffin proves that the "expressive intonation", i. H. the use of sharpened leading tones, especially since Pablo Casals (1876–1973) has been a thoroughly modern phenomenon. He mentions, for example: Hermann von Helmholtz , who physically examined Joseph Joachim's playing , found that Joachim intoned the thirds (almost) purely, which is also confirmed in his recordings from 1903 - despite technical defects.

Use of Euler's sound network for fine musical analysis

Example 1: triad on the second level in C major

Comparing the two triads

Triad on the 2nd degree , d - f -, a (minor chord with a pure minor and major third) and
Triad on the 5th degree g -, b - d (major chord with a pure major and minor third)

you can see immediately that the tones d and d differ by a syntonic comma.

Example 2: Difference G sharp and A flat

	Chords (with frequencies in Hertz) C major tonic: c (132) c ′ (264), e ′ (330) g ′ (396) c ″ (528) C major subdominant in minor with sixte ajoutée : f (176) f ′ (352) 'as ′ (422.4) c ″ (528) d ″ (594) A minor dominant seventh chord:, e (165), e ′ (330) ,, g sharp ′ (412.5), h ′ (495) d ″ (586.7) A minor tonic:, a (220), e ′ (330), a ′ (440) c ″ (528) As is 9.9 Hz higher than G sharp with the frequency ratio ^AS / _{G sharp} = ¹²⁸ ⁄ ₁₂₅ (41 cents). This interval is the little Diësis .

Example 3: The "comma trap"

Choirs that listen to each other particularly well can detonate. This is often due to the different intonation of tones of the same name. This has been known for a long time. The detailed musical analysis about it, which is easier to understand with the help of the names of Euler's sound network, is not part of the musical education.

A classic example of a "comma trap" is the occurrence of the second degree chord:

The third chord f-, ad sounds impure in C major. A choir in which the voices listen to each other sings the appropriate d a syntonic comma lower, here denoted by, d. It is about the minor parallel to the F major chord and the F major scale is fg-, bc- , d- , ef. In the following chord g-, hd, however, the “correct” d of C major must be sung again.

If this is not observed, you fall into the comma trap, as shown in the following audio example. After repeating it four times, the set sounds almost half a tone lower. Repeating the comma trap four times results in a detonation of almost a semitone

Tonic	, e	G	c
Subdominant	f	, a	c
Subdominant parallel	f	, a	, d
Dominant	,G	,,H	, d	With the same, d of the soprano a (syntonic) comma too deep
Tonic	,, e	,G	, c	One point too deep
Subdominant	, f	,, a	, c	One point too deep
Subdominant parallel	, f	,, a	,, d	One point too deep
Dominant	,,G	,,,H	,, d	With the same ,, d of the soprano now two commas too low
Tonic	,,, e	,,G	,, c	Two commas too deep
Subdominant	,, f	,,, a	,, c	Two commas too deep
Subdominant parallel	,, f	,,, a	,,, d	Two commas too deep
Dominant	,,,G	,,,,H	,,, d	With the same ,,, d of the soprano now three commas too low
Tonic	,,,, e	,,,G	,,, c	Three commas too deep
Subdominant	,,, f	,,,, a	,,, c	Three commas too deep
Subdominant parallel	,,, f	,,,, a	,,,, d	Three commas too deep
Dominant	,,,,G	,,,,,H	,,,, d	With the same ,,, d of the soprano now four commas too deep
Tonic	,,,,, e	,,,,G	,,,, c	Four commas (86 cents ) too deep

The circle of fifths in Euler's tone network

Two tones change to the next key. The chromatic semitone with 92.179 cents (frequency ratio: ¹³⁵ ⁄ ₁₂₈ ) can be seen in the note image, the change by a syntonic comma with 21.506 cents (frequency ratio ⁸¹ ⁄ ₈₀ ) can be read here from the changed cents.

The ♯ keys in Euler's tone network

The circle of fifths shown in pure tuning and Eulerian spelling on a keyboard.

( Cent values in brackets)

C major: c (0) d (204), e (386) f (498) g (702), a (884), h (1088) c
G major: ga (906), hcd, e, f sharp (590) g
D major: de (408), f sharp ga, b, c sharp (92) d
A major: ah (1110), c sharp de, f sharp, g sharp (794) a
E major: e f sharp (612), g sharp ah, c sharp, d sharp (296) e
B major: b c sharp (114), dis e f sharp, g sharp, a sharp (998) h
FIS major: f sharp g sharp (816), a sharp b c sharp, d flat, ice (500) f sharp
CIS major: cis dis (318), ice fis gis, ais, his (2) cis

The ♭ keys in Euler's Tonnetz ( cent values in brackets)

C major: c (0) d (204), e (386) f (498) g (702), a (884), h (1088) c
F major: fg, from (996) c, d (182), ef
B flat major: bc, d es (294) f, g (680), ab
Eb major: es f, g a flat (792) b, c (-22), d es
A flat major: as b, c des (90) es, f (477), g as
D flat major: des es, f gb (588) a flat, b (975), c des
G flat major: ges as, b ces (1086) des, es (273), f tot
C flat major: ces des, es fes (384) gb, a flat (771), b ces

Andreas Werckmeister already stated that the enharmonic equation is possible with an accuracy of a schism of 2 cents:

, h (1088) = ces (1086)
, his (2) = c (0)
, cis (92) = des (90)
, dis (296) = es (294)
, e (386) = fes (384)
, ice (500) = f (498)
, f sharp (590) = total (588)
, g sharp (794) = a flat (792)
, ais (998) = b (996)

literature

Euler, Leonhard . In: Music in Past and Present , Volume 3, 1616
Hermann von Helmholtz : The theory of tone sensations as a physiological basis for the theory of music. Vieweg, Braunschweig 1863 (reprint: Minerva-Verlag, Frankfurt am Main 1981, ISBN 3-8102-0715-2 , excerpt ).
Ludwig Riemann: Popular representation of acoustics in relation to music. Following Hermann von Helmholtz's "Theory of Sound Sensations". Vieweg, Braunschweig 1896.

Web links

Joachim Mohr: The Euler spelling .
Try it on the PC keyboard. Free Java applet

Individual evidence

^ Leonhard Euler: Tentamen novae theoriae musicae ex certissimis harmoniae principiis dilucide expositae. St. Petersburg 1739.

^ Leonhard Euler: De harmoniae veris principiis per speculum musicum repraesentatis. In: Novi Commentarii academiae scientiarum Petropolitanae , 18, St. Petersburg 1774.

↑ Walter Odington

^ The Harvard dictionary of music. Don Michael Randel, 2003, ISBN 0-674-01163-5 , p. 56, heading: Arithmetic and harmonic mean, section 2 ( online ).

↑ The following notation is often used: ("underlined x") instead of ("deep point x") and ("overlined x") instead of 'x ("single point x")

{\ displaystyle {\ underline {x}}}

{\ displaystyle, x}

{\ displaystyle {\ overline {x}}}

↑ Jacques Handschin: The tone character. An introduction to tone psychology. Zurich 1948.

↑ Carl Dahlhaus : Investigations into the origin of harmonic tonality . Kassel 1965. Renate Imig: Systems of functional designation in harmony teachings since Hugo Riemann . Düsseldorf 1970.

↑ Moritz Hauptmann: The nature of harmonics and metrics. Leipzig 1853.

↑ Hermann v. Helmholtz: The theory of tone sensations as a physiological basis for the theory of music. Brunswick 1863.

↑ Arthur von Oettingen: Harmoniesystem in dual development. Studies on the theory of music. Dorpat / Leipzig 1866; Revised second edition as Das duale Harmoniesystem , Leipzig 1913.

↑ z. B .: Hugo Riemann: Ideas for a “doctrine of sound ideas”. In: Jahrbuch Peters 21/22, 1914/15

↑ Martin Vogel: The theory of the tone relationships. Bonn - Bad Godesberg 1975, p. 103f.

^ Ross W. Duffin, How Equal Temperament Ruined Harmony (And Why You Should Care). WW Norton & Company, New York NY 2007, ISBN 978-0-393-06227-4 ( excerpt ).

↑ For the expert the calculation of the frequencies with c '= 264 Hz:
,, g sharp ' = Hz = 422.4 Hz, 'as' = Hz = 412.5 Hz, calculation with octave = 1200 · lb (2), fifth = 1200 lb ( ³ ⁄ ₂ ), comma = 1200 lb ( ⁸¹ / _8o ) and c = 0 cents: ,, g sharp = 8 fifths - 4 octaves - 2 decimal points = 772.627 cents, 'as = −4 fifths + 3 Octaves + comma = 813.686 cents, difference = as ′ - ,, gis = 41.059 cents ${\ displaystyle 264 \ cdot \ left ({\ tfrac {3} {2}} \ right) ^ {8} \ cdot \ left ({\ tfrac {1} {2}} \ right) ^ {4} \ cdot \ left ({\ tfrac {80} {81}} \ right) ^ {2}}$
${\ displaystyle 264 \ cdot \ left ({\ tfrac {2} {3}} \ right) ^ {4} \ cdot 2 ^ {3} \ cdot \ left ({\ tfrac {81} {80}} \ right )}$

↑ Bettina Gratzki p. 76: The pure intonation in choral singing (= Orpheus series of publications on fundamental questions in music 70). Verlag für systematisches Musikwissenschaft GmbH, Bonn 1993, ISBN 3-922626-70-X ( Excerpt ( memento of the original from February 19, 2014 in the Internet Archive ) Info: The archive link has been inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. ). @1@ 2Template: Webachiv / IABot / kilchb.de

↑ With q = 701.955 (fifth) and t = 386.314 (third) the following is calculated (modulo octave = 1200): c = 0, g = q, d = 2q, a = 3q, e = 4q etc. f = −q, b = −2q, es = −3q etc. With s = 21.506 (syntonic comma):, c = c - s,, g = g - s,, d = d - s, ..., f = f - s,, b =, b - s,, es = es −s etc.

↑ Andreas Werkmeister: Musical temperature. Quedlinburg 1691. He neglects this interval of 2 cents because it makes "hardly a circular line" on the monochord.

[1] Leonhard Euler: Tentamen novae theoriae musicae ex certissimis harmoniae principiis dilucide expositae. St. Petersburg 1739.

[speculum_musicae-2] Leonhard Euler: De harmoniae veris principiis per speculum musicum repraesentatis. In: Novi Commentarii academiae scientiarum Petropolitanae , 18, St. Petersburg 1774.

[3] Walter Odington

[4] The Harvard dictionary of music. Don Michael Randel, 2003, ISBN 0-674-01163-5 , p. 56, heading: Arithmetic and harmonic mean, section 2 ( online ).

[5] The following notation is often used: ("underlined x") instead of ("deep point x") and ("overlined x") instead of 'x ("single point x") ${\ displaystyle {\ underline {x}}}$ ${\ displaystyle, x}$ ${\ displaystyle {\ overline {x}}}$

[Toncharakter-6] Jacques Handschin: The tone character. An introduction to tone psychology. Zurich 1948.

[7] Carl Dahlhaus : Investigations into the origin of harmonic tonality . Kassel 1965. Renate Imig: Systems of functional designation in harmony teachings since Hugo Riemann . Düsseldorf 1970.

[Mauptmann-8] Moritz Hauptmann: The nature of harmonics and metrics. Leipzig 1853.

[Tonempfindungen-9] Hermann v. Helmholtz: The theory of tone sensations as a physiological basis for the theory of music. Brunswick 1863.

[Dualismus-10] Arthur von Oettingen: Harmoniesystem in dual development. Studies on the theory of music. Dorpat / Leipzig 1866; Revised second edition as Das duale Harmoniesystem , Leipzig 1913.

[Tonvorstellungen-11] z. B .: Hugo Riemann: Ideas for a “doctrine of sound ideas”. In: Jahrbuch Peters 21/22, 1914/15

[Vogel,_S._103f-12] Martin Vogel: The theory of the tone relationships. Bonn - Bad Godesberg 1975, p. 103f.

[13] Ross W. Duffin, How Equal Temperament Ruined Harmony (And Why You Should Care). WW Norton & Company, New York NY 2007, ISBN 978-0-393-06227-4 ( excerpt ).