Tonnetz

The Euler tone lattice is the representation of the tonal range of the pure sentiment in a two-dimensional grid of pure fifths and Terzintervallen, which goes back to the mathematician Leonhard Euler. Euler had already in 1739 published work Tentamen novae theoriae musicae examined in detail the mathematical relationships of the music. In this work he sought a mathematical justification for the consonance and dissonance in music perception and set an interval kinship system based on prime number together. In another work, De veris Harmoniae principiis by speculum musicum repraesentatis, which was published in 1773, he described the tone lattice of fifths and thirds.

The fifth-third scheme

The introduction of the pure major third ( with the string length ratio between upper and lower tone of 5:4 ) - as a substitute and simplify the Pythagorean Ditonus ( 81:64 ) - goes back to the enharmonic Tetrachordteilung Didymus ' ( about 100 years after Pythagoras ). However, the sound system in ancient Greece is not to be compared with the sound system, to which Leonhard Euler relates. In today's sound system was first mentioned in the pure major third in 1300 by Walter Odington in his De Speculatione Musices.

In Western music, the major third ( frequency ratio: 81:64 ) was the Pythagorean system perceived as a dissonance. With the advent of polyphony emancipated in the 15th century, the pure major third ( 5:4 ), which became ever more musical importance as part of the triad.

C major chord in the harmonic series? / I

This gives a sound system that is based on the intervals octave 2/1, fifth 3/2 and third 5/4. The other intervals of the fifth-third system can thus be represented as multiples of these intervals.

From the variety of combinations of these intervals results in a (theoretically) infinite tonal space. This pitch space is often represented graphically by means of a Tonnetzes as follows:

Representation in tone lattice

Since the pure third was not viewed with Quinten, presented Leonhard Euler represents the relationships of pure mood with the help of fifth series that differ by a syntonic comma. The following relationships, the fifths in the horizontal direction and the thirds rows in the vertical direction.

The notation x ( " low point x " ) or ' x ( " apostrophe x " ) - the comma before the Tonbezeichnung - etc means that the sound, x or ' x is a syntonic comma lower or higher than the tone x is.

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

The pure scales have always the same look in this graphic illustration. Interval relationships are for each scale is always the same:

( More extensive table, see the scales in the circle of fifths )

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

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

The cent values ​​of the notes played to himself:

( More extensive table, see the scales in the circle of fifths )

It shows the tonal relationships of the harmonic just intonation. For example, sound the tones, and e, a and h, the C- major scale a syntonic comma lower than in the Pythagorean fifths chain. The chords c, eg and f -, and g - ac, hd consist of pure thirds ( 5/4 and 6/5) and pure fifths (3 /2).

The actual " tone lattice " was introduced in 1773 by Leonhard Euler as a speculum Musicum ( " image of music" ) in one of his signature " De Harmoniae veris principiis by speculum musicum repraesentatis ", and from then on - together with the introduced of Moritz Hauptmann designations for same tones so far, which differ by a syntonic comma, modified by numerous theorists for various purposes ( inter alia by Hermann von Helmholtz, Arthur von Oettingen and Hugo Riemann ). The different characters of tones of the same name but different location in the ( infinite) pitch space results in harmonious and pure mood not only in connection with other audio environment and harmonization (such as the e in C major chord c, eg and e in e- major chord e, g # -h), but also from a ( minimal ) difference in pitch between the respective pitches (, e and s):

In the C major scale cd, efg, a, hc the interval c, e is a major third with the frequency for money 5 / fourth In the bottom row CGDAE the interval ( zurückoktaviert ) ce 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 harmonic- pure fifth-third mood

The pure C major scale can be understood as a selection of those seven pitches of the fifth-third scheme, the intonation of the three main functions of subdominant (S ), tonic ( T) and dominant ( D ) - thus for the "authentic" cadence are required:

C- major tonality? / I

The actual scale is created by transposition of these pitches in the appropriate octave range - for example, between c1 and c2. It consists now - in contrast to the Pythagorean scale - no longer two, but of three interval steps of different size, the large whole tone 9/8, the small whole tone 10/9 and the diatonic semitone 16/15:

Pure C- major scale? / i

This seven-step scale now allows indeed the harmonic intonation of the main functions T, S and D, but it acts melodic improper because the respective thirds, e /, a and h in melodic context as too low to be felt. In particular, the leading tone, hc is problematic with its 111.73 cents, since it contradicts the tendency to increase the strut effect of leading tones through the closest possible intonation ( for example, by the Pythagorean Limma hc 256/243 to 90.22 cents). Many of the intonation difficulties - such as strings, which often play sharpened thirds or leading tones, and wind instruments - can be traced back to the incompatibility of harmonic and quasi- Pythagorean- melodic purity ..

The two different sized whole tones 9/8 and 10 /9 - the triggers this " conflict " - arise (inevitably ) from the arithmetic division of the ( pure ) major third 5/4. Their difference is equal to the so-called syntonic comma or didymischem 81/80 with about 21.51 cents; that is, the pitches that are in the row above the ( same ) pitches in tone lattice, tenth - sound are about deeper than this intones (e: e = 80:81 ). One solution to this problem provides additional Meantone with the geometric division of the major third, resulting in two equal whole tones (each 193.156 cents), without compromising the purity of the third question.

Application of the Euler Tonnetztes the musical fine analysis

Example 1: Difference Gis and As

Listen? / I

C major tonic: c (132 ) c '( 264), e' (330 ) g '( 396 ) c'' ( 528)

C major subdominant in minor ajoutée with sixte: f ( 176) f '( 352) ' as' ( 422.4 ) c'' ( 528) d'' ( 594 )

A minor tonic: a ( 220), e ' (330 ), a' ( 440) c'' ( 528)

As is here 9.9 Hz higher than G # with the frequency ratio AS / G # = 128/125 (41 cents). In this interval, there is the small DIESIS.

Example 2: The " comma trap"

Just choirs, which are particularly well listen to one another, can detonate. This then is often due to the different intonation sounds the same name. This has long been known. The musical fine analysis on which is easier to understand with the help of the terms of the Euler Tonnetztes not heard, but unfortunately for musical educational heritage.

A classic example of a " comma trap" is the occurrence of stage II chord:

Listen? / I

The third chord f, a, d sounds unclean in C major. A chorus in which the voices listen to each other, the appropriate d a syntonic comma sings deeper, here referred to d. It is after all the minor parallel to the F- major chord and the F major scale is fg-, bc -, d-, ef. The following chord g, hd must then, however, again the "right" d from C major to be sung.

If this is ignored you fall into the trap comma, as shown in the following Sample. After four repetitions of music theory sounds a semitone lower.

The four-time repeating decimal case results in a detonation by a semitone? / I

The circle of fifths in the Euler tone lattice

To the next each key change two tones. The chromatic semitone 92.179 cents ( frequency ratio: 135/128 ) is evident in the score, the change to a syntonic comma with 21.506 cents ( frequency ratio of 81/80 ) is here in the amended specification cents to read.

The ♯ - keys in the Euler tone lattice ( cent values ​​in parentheses)

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

The ♭ - keys in the Euler tone lattice ( cent values ​​in parentheses)

• C major: c ( 0) d ( 204), e ( 386) f ( 498 ) g ( 702), A ( 884 ), h ( 1088) c
• F major: f g a b (996) c, d ( 182 ), e f
• B flat major: b c d it (294 ) f, g (680 ), a b
• E-flat major: it f, g as ( 792 ) b, c (-22 ), it d
• A flat major: as b, c of (90 ) to f ( 477 ), g as
• D flat major: of it, f tot ( 588 ) as, b ( 975 ), c the
• G Flat Major: ges as, b ces ( 1086 ) of it (273 ), f tot
• Cb major: ces of it fes ( 384 ) total, as ( 771 ), b ces

Even Andreas Werckmeister found that up to an accuracy of a schism of 2 cents enharmonic equivalence are possible:

• , His (2) = c (0)
• Cis (92 ) = the (90)
• , dis ( 296 ) = it (294 )
• Ice (500 ) = f (498)
• , fis (590 ) = tot ( 588 )
• , gis ( 794 ) = as ( 792 )
• , ais ( 998 ) = b ( 996 )