Just intonation
The pure voice (also natural or harmonious mood ) originated in Europe with the advent of polyphony in the second half of the 15th century. Sound formats in just intonation consist exclusively of intervals whose frequency ratios are the ratios of small integers. In contrast to the Pythagorean in the Middle Ages with perfectly tuned octaves ( frequency ratio 2/1) and fifths ( frequency ratio 3/2) and the major thirds ( frequency ratio 5/4) are additionally voted purely in just intonation.
Use of pure mood
The first mention of the use of pure mood at Ramis de Pareja in a musical treatise of 1472. Through Lodovico Foglianos ' font Musica Theorica of 1529 the idea of pure sentiment gained a higher profile. In polyphonic music, the ideal of pure mood to the present day in vocal ensembles, choirs and ensembles with wind and string instruments is a wide distribution. In choral singing schools is paid for five centuries to a hearing beyond the Halbtonschranke to bring pure chords sounded. As can be intoned correctly - not only " slightly higher " or " a little deeper " - is explained precisely by means of the theory of pure mood.
For modulations not only tones with change of sign, but each also yet another sound to a syntonic comma change. For instance, requires G major, compared to C major instead of the sound the tone F # F, but also the tone A is a syntonic comma higher. (See modulations in just intonation ). Therefore can only be realized when key instruments just intonation when significantly more than 12 keys per octave are present, such as in Archi harpsichord. Such instruments are virtually unplayable and could not prevail.
On keyboard instruments with fixed 12 pitches per octave had a compromise - a temperature - be found. One must keep in mind that there is a deviation from the ideal of pure mood at each temperature.
The mean tone in the 17th and 18th centuries includes purely tuned major thirds as in the pure atmosphere, but has the limitation that not all keys are playable. The well-tempered tunings and the equal tempered mood to let the play in all keys, but move away from the ideal of pure mood.
After the advent of pure sentiment continued to be valid in many music theorists to the 17th century Pythagorean than ideal. Towards the end of the 19th century string players began to intone the leading notes higher and therefore closer to back of the Pythagorean.
Not only in the western musical tradition, even in non-European musical cultures one encounters the concept of " just intonation " when intoned harmonic and pure. In some cases, other intervals of the harmonic series can be included such as the natural seventh ( 7th partial ) or the alphorn - Fa ( 11 partials ).
Keys in just intonation
Net tuned keys play a crucial role in the performance practice of music of the Renaissance and Baroque periods of a cappella choirs, string quartets and orchestras. For pure intonation and good acoustics obtained emphasized lows (because of the difference tones) and crystal clear sound (because of common overtones ). In the Mean moods the mood is best achieved with the pure thirds in keyboard instruments - but only for a limited number of keys.
Listen? / I
( No beats. )
Listen? / I
( Low beats, by the slightly " detuned " Quinten conditionally. See: meantone fifths ).
Listen? / I
( Violent beats - about ten times faster than mean- mainly by the " detuned " third conditional. )
Octave, fifth and major third are the basic intervals of just intonation. All other intervals can be assembled from these basic intervals. Why they are called this system fifth-third system.
Frequency ratios of the major and minor scale in just intonation
Pure scales based on the following frequency ratios:
The chords of the tonic CEG and C- It -G, the subdominant FAc and F -As- c and the dominant GHD or GBd ( in minor) consist of pure fifths and thirds. The fifth, however, D -A is unclean. Therefore, a chord on the second stage is already an (interim) modulation toward subdominant. When the scales in just intonation is important to note that there are two types of whole tones, for example C to D with the frequency ratio 9/8 and D to E with the frequency money 10 / 9th
This makes the scales soft overheard on the scale in equal temperament tuning, in which each semitone exactly makes up one-twelfth of an octave and a whole tone corresponds to exactly two semitones.
The major third
Fundamental is the characteristic pure third with the frequency for money 5 / fourth The mean tone with its many pure thirds realized almost perfectly just intonation of keyboard instruments - but only for a limited number of keys.
The first mention of the pure major third in 1300 by Walter Odington in his De Speculatione Musices. Earlier descriptions of this interval are in reference to the ancient Greek sound system.
Listen clean? / I
Listen meantone? / I
Listen Pythagorean? / I
Listen equally-? / I
In pure and meantone tuning is heard no beat in the pure third ( 386 cents). In the mean tone you can hear the slightly tempered fifth in the second chord in a minor beating. The " sharpened " in gleichstufiger third (400 cents ) or even Pythagorean (408 cents) atmosphere with a strong beat is perceived as friction. (See also the example of the major third with a reinforced differential tone ).
Note: Pure intervals are characterized by integer frequency ratios, tempered intervals usually, however, have an irrational frequency ratio. Therefore, the size comparison is made with the unit cents.
Listen clean? / I
For comparison, mean- Listen / i Listen equally- / i?
The chord at the second stage
In the pure mood of the C- major scale with the D of the dominant chord of GHD and the A of Subdominantenakkordes FAC results in a fifth DA, which is a syntonic comma too tight and thus appears dissonant.
Dissonant D minor in C major? / i
The chord on the second stage is an intermediate modulation toward subdominant. With the D of A minor (or F major ) results in a pure minor chord DFA. In the following cadence then the D chord in the Sp of the second stage is lower by a syntonic comma as in the chord D the dominant.
Two d in C major? / i
If this is not observed, it may come to a lowering of the mood of a choir. (see " comma trap". )
However, the level II chord can also - more rarely discussed in the literature - as a double dominant - often clarifies be interpreted as D- F # -A. In this case - in the direction of the dominant modulation - increases the A by a syntonic comma.
Mathematical Description
The interval space of pure mood is the fifth-third system.
All intervals can be displayed ( major third ) as multiples of the three basic intervals Ok ( octave ), Q ( bottom ) and T.
Problems with keyboard instruments
For modulations tones change not only by a semitone, but also some tones to a syntonic comma ( see modulation in just intonation ). This can not be achieved at a keyboard with twelve tones per octave. They were forced to use tempered tunings. First:
- The mean- moods, then
- The well-tempered tunings, and finally
- The equal temperament tuning.
Modulations require an adjustment of the pitch
For example humbles himself at a modulation from C major to F major, not only the H by a semitone to B, but also the D to a syntonic comma. At a modulation from C minor to F minor, the B decreased by a syntonic comma and the D by a semitone to Des.
Accordingly, not only the F increases at a modulation from C major to G major by a semitone to F #, but also the A by a syntonic comma. At a modulation from C minor to G minor, the F increased by a syntonic comma and the ace by a semitone to A.
Comparison of extended just intonation with the equal temperament tuning
Referring to the sounds of major and minor scales, the FIS of G major and the OF of F minor to obtain the 12-level chromatic scale in just intonation.
Chromatic scale in just intonation of C major and C minor supplemented by FIS and DES:
Note: When comparing intervals using the unit cents. Where: 1 octave = 1200 cents.
The following triads contain pure fifths and pure major thirds and small.
In this mood can only C major and C minor and A flat major and E minor play pure ( with minor dominant HD - Fis ). ( If the display in the major dominant chord H -Dis - Fis replaced by E Minor by its enharmonic change, the result is a time that is too high by 41 cents. ) In order to play in all keys, are at the same stage mood matched the halftones. The fact that this will not produce more pure triad, is accepted as a compromise.
Chromatic scale of equal temperament tuning:
The equal temperament can also be obtained by distributing the Pythagorean comma in the circle of fifths FCGDAEH - Fis - Cis - Gis Dis -Ais - ice -His = (C? ) On the twelve fifths. The perfect fifth ( 702 cents) differs only slightly from the equal temperament (700 cents). The major third ( 386 cents) is at least " sharpened " in the equal temperament tuning ( 400 cents) to 14 cents.
Frequency ratios of the extended scale
Referring to C major and C minor scale nor the F sharp of G major ( dominant table ) and the Des of F minor ( subdominantisch ) is added, you get twelve steps.
Here, B is selected as the pure minor third to G (belonging to C minor ). In some ( quint stressed ) representations of the B is selected as the fourth to F (given in parenthesis) with the advantage that the interval AB is a diatonic semitone. Then this B heard but to F major.
As can be seen, a tuned instrument in this mood for C major and C minor ( with suitable B) and A flat major and E minor is useful and is not realized in practice.
Small and big halftone
In just intonation, there is the large, the diatonic semitone with the frequency ratio and the small, the chromatic semitones with the frequency ratios and
In the extended chromatic C-Dur/c-Moll-Tonleiter:
The Euler tone lattice
The two main intervals in the triad formation are the fifth and the third. Since the pure -minded third can not be represented with perfectly tuned fifths, Leonhard Euler developed the appearance of a fifth-third - interwoven relationship, which consists of various bottom rows at a distance of a syntonic comma. This tone lattice was further developed by various music theorists like Moritz Hauptmann, Hermann von Helmholtz and Arthur von Oettingen and led to new musical instruments such as the harmonium clean.
The pure scales have in this representation for consistent representation:
The calculation of associated cent values with octave = 1200 cents, fifth = 701.955 cents and syntonic comma = 21.506 cents results in, for example, with c = 0 cents rounded:
- E = c 4 Quinten - 2 octave = 408 cents, e = e - syntonic comma = 386 cents ( = c pure major third )
The corresponding frequency values are calculated as c = 264 Hz:
- And.
Sound sample: comparison pure, meantone and equal temperament
1st line? / I 2nd line? / I
1st line? / I 2nd line? / I
1st line? / I 2nd line? / I
1st line? / I 2nd line? / I
Here are purely the cadence chords GBD, C -Es- G and D- F #- A in just intonation in G Minor. In modulation to B- flat major ( bar 6 and bar 13 ) with the pure cadence chords BDF, Es- GB and FAC increased the pitch C is a syntonic comma.
More sound examples
For the aural representation of pure moods a known, though not historically correct example was chosen, which makes it possible to clearly hear the diffizilen differences. It was designed by Johann Sebastian Bach for one of the (many) well-tempered tunings; exactly for what, today can not be reconstructed with certainty.
Johann Sebastian Bach: Prelude in C Major from the first volume of the Well -Tempered Clavier, BWV 846
Example 1: bars 1 to 5
- Although described in the text 7-point (pure ) C major scale leads to a melodic meaningful greater whole tone ( 9/8) cd [ in a) ], but is the fifth since [ at b ) ] defiled (40 / 27 instead of 3/ 2; she has about 680.448 cents a syntonic comma too small). Listen? / I
- By expanding the scale to a debased d [ in a) ] the fifth is there now, although adjusted [ at b ) ], but arises, first, a unmelodischer small whole tone (10 /9) and on the other a clearly audible ( syntonic ) comma difference ( 81/80 ) between the d's in the second and the third clock [ wherein c) ]. Listen? / I
- Assuming the natural seventh (7/ 4 instead of 9/5 ) in the dominant [ in d) ] is formed, an additional " septimales point " (64/ 63, about 27.264 cents) f between the second and third cycle [ e ) ]. The sliding- fe is called " septimaler semitone " (21 /20, about 84.467 cents) closely intones [ at f ) ]. Listen? / I
Example 2: bars 5-11
- The 12-level chromatic scale contains (only) the impure fifth since (40 /27) which will play here twice as part of the double dominant seventh chord [ in a) ]. Listen? / I
- The balance of this fifth one Pythagorean eingestimmtes now requires a (27/ 16 instead of 5/3), which is higher than in the a as the root of a syntonic comma to Tonikaparallele [ at b ) ]. As the decimal differences heard here in one of the middle voices, they are comparably small - with a quite acceptable sonic result. Listen? / I
- In contrast, the intoned as natural seventh tone do the double dominant in turn leads to clearly audible " septimal comma" [ in c ) ]. Listen? / I