Meantone temperament

In meantone tunings is a series of tempered moods that were common for keyboard instruments, mainly in the Renaissance, Baroque and often in later times ( up to the 19th century). The mean tone with its many pure thirds realized almost perfectly just intonation of keyboard instruments - but only for a limited number of keys.

As with just intonation using the characteristic pure major thirds ( with the frequency ratio ) is fundamental for the pure fifths of the Pythagorean be slightly narrowed. C major tonal center around which in the strict meantone eight pure thirds are located (on E, A, D, G, C, F, B and Es). The pure third resembles the syntonic comma ( frequency ratio: ) by the fact that four consecutive fifths are lowered by each of the syntonic comma. In the just intonation major third () is divided into a greater whole tone () and a small whole tone () ( with rational frequency ratios ), in mean tone - hence the name - but in two equal whole tones with the irrational frequency ratio.

To reduce or avoid the fifth Wolf strict meantone modified in many tests, but at the same time increases the pure thirds ( sharpened ) to be.

Construction

In the mean tone eleven fifths circle of fifths are each reduced by so much that the resulting four- fifths of these major thirds are pure or nearly pure. In the most common and most commonly described variant, the major third is pure. The four fifths therefore be reduced by the per syntonic commas. In other words, we reduce the 11 fifths to the syntonic comma, so the usable thirds are pure exact. The resulting mood is the - ( syntonic ) comma meantone.

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.

For example, in just intonation the pure third is divided into a large and a small whole tone, in contrast, the mean tone into two equal whole tones.

In the given keys we get because of the pure thirds in the tonic, subdominant and dominant a remarkably good sound quality, which is, however, clouded by the fact that in other scales unusable intervals arise. The twelfth " fifth ", which closes the circle of fifths, is in truth a diminished sixth ( usually Gis - It ), which differs greatly from that of the pure fifth, and generally musically useless. It is often called wolf fifth. Four putative major thirds, fifths their chain contains the wolf fifth, are diminished fourths ( Cis -F, F # B, G # -C, H- Es), which can be used as major thirds also usually not. Therefore, there remain eight pure major thirds.

(!) The intervals marked with * are only in the enharmonic change thirds and fifths and one sees that only the following triads are playable:

E-flat, B- flat major, F major, C major, G major, D major, A major and E major and C minor, G minor, D minor, A minor, E- minor, B minor, F-sharp minor and C-sharp minor.

These chords were in the Lasso Palestrina Lechner Cavelieri time ( 1600) fully exhausted, but very rarely, for example, the A-flat major or F minor triad.

To play the A flat major triad, you would need higher next to the button for Gis a button for As 41 cents; in order to play the B major triad, you would need next to the button for it is a button for Dis 41 cents lower, etc

See under the heading Cent ( music) the tables of the fifths and thirds in the mean-, well-tempered, equal temperament and Pythagorean tuning.

Cadence in F major (almost pure) and A flat major ( with "Wolf" and "wrong" thirds )

To further keys were to make playable - developed well-tempered tunings, which led ultimately to the same stage mood of our keyboard - at the expense of pure third.

Other well-known, but historically in vocal practice rarely hardly to be detected meantone tunings are the -, -, -, and - comma meantone, in which the 11 fifths are reduced by the corresponding fraction of the syntonic comma. The reduction of the discord of the wolf fifth at the same time, however, leads to a decrease in the purity of the "good" major thirds.

One generally speaks of the mean tone tuning, as is usually the comma - Meantone meant. Only for her major thirds are pure exact. The Comma Meantone can be implemented relatively easily if you learn the four -tempered fifths to tune exactly. The other sounds are then obtained via the attunement pure large thirds.

The pure major third is characteristic of the meantone

The mean tone with its many pure thirds approaches the pure atmosphere with a loud pure thirds in cadences at best.

→ Main article: The major third in just intonation.

Small and big halftone

→ Main article: small and large halftone in just intonation

As with the pure mood a distinction is made between the mean tone diatonic, large halftone with 117.108 cents and the chromatic, small halftone with 76.049 cents.

Intervals of the Comma mean tone

This is also the rule of the White Burger cantor Maternus Beringer applies (1610 ):

For the musical practice of the exchange of large and small semitones in meantone temperament is momentous. Thus, the use of chromatic sections with different semitone has an expressive effect. Leading tones to the top ( cis, dis, e, f #, G # and h) are deep and leading tones down ( the, it, f, as and b ) intones high, and the Mollterzen most commonly occurring are quite small. Cadences get in meantone tuning therefore a special, particularly deviating from the equal temperament tuning character (example in the box at right).

Interval table

See: Intervals of - comma mean tone

History

While the large ( Pythagorean ) third was usually perceived in the Middle Ages as a dissonance, it formed ( as a pure interval ) from the Renaissance an important consonance.

Even if one can view as a practical description of the mean tone scattered sources of the 15th and early 16th century already, it was first described in 1571 correctly and unambiguously by Gioseffo Zarlino. In the German -speaking world it was Michael Praetorius, who described it in 1619 in his " Organographia " (Syntagma Musicum, Volume 2 ) as a common practice and three types stated how they could be put in practice (in addition to a non- significant modification, but no key addition allows ). Due Praetorius ' description Meantone until the 18th century was often referred to as " Praetorianisch ". In organ building was used in Germany until well into the 18th century as standard tuning - in some regions even beyond - which is why in organ building contracts and audit reports ( reports of acceptance ), the mood did not need to be called.

In Northern Germany Meantone for example for all organs Hamburg 1729 is in printed sources, and also the newly built by Arp Schnitger organ of the Bremen Cathedral stood in the mean tone still to retuning 1775-1776. Recent research has again made ​​plausible that the organs that were Dieterich Buxtehude in Lübeck available, were in this Standardtemperierung. There is indeed no comments Buxtehude to mood issues - his dedication poem for Andreas Werckmeister Harmonologia Musica from 1702, a counterpoint and improvisation theory, also does not take up questions regarding mood and can not be interpreted as support Werckmeisterscher mood designs.

The wolf fifth and the four diminished fourths were considered in the 17th and 18th centuries as completely useless. Guesswork out recently that they have been used compositionally ( ie about H- It - Fis as a supposed B major, F - G # -C as a putative F minor, etc.), but are by expressions of the sources of the 17th and 18. century refuted regularly.

To extend the key set of the ordinary mean tone, were equipped at sites of professional musical life in Western Europe between about 1450 and 1700 not rare keyboard instruments with additional upper keys (usually 1-2 rare 4, harpsichord universal even 7 " Subsemitonien ", English split keys). Such instruments are related to the so-called enharmonic instruments. The development apparently began in Italy and quickly gained a certain distribution. North of the Alps, it was not until Gottfried Fritzsche, 1612 the first organ with Subsemitonien built in Germany ( in the Electoral Palace Orchestra Dresden ). Praetorius describes a " harpsichord universal " ( " Cimbalo cromatico "), which has 19 tones per octave: In addition to the five split upper keys, there are additional narrow black keys for the ice and His.

On stringed keyboard instruments slowly but increasingly well-tempered tunings sat down since the end of the 17th century, and practical approaches to the equal temperament, that is, those moods, which allowed the use of all keys. The well-tempered tunings were not the day to the electronic instruments and mostly on pianos to hearing equal temperament, but those in which the individual keys sometimes more, sometimes less " charged" sounded ( key characteristics, which was also in the 18th century understood as a subjective moment ).

Long could only assume that Bach has in the transposition of earlier works and partial recomposition of preludes and fugues of the two volumes of the " Well-Tempered Clavier " (!) Thought to the then quite new unequal -stage well-tempered tunings, even if the equal temperament, practical placed in his later life can not be excluded. Note also that Friedrich soupy in 1722 described in a manuscript that all pianos are tuned meantone in Dresden - in the same year as Bach compiled the first volume of the Well-Tempered Clavier and provided with a dated title page. According to a new, but still controversial interpretation around the year 2000, the garland can be interpreted on the cover of Well-Tempered Clavier as the mood statement.

The history of the mean tone is indeed quite well known in their theoretical ramifications, but the practical application, dissemination and apparently often much later than previously thought successful transition towards newer moods (often directly to approximations to the equal temperament ) in many regions explored only beginning, as you too often assumed that theoretical mood designs soon penetrated into practice. However, as Werckmeister and others, the new moods designed complained, the organ builders did not follow their designs and even stayed late at the mean- vocal practice.

The mean tone presented the best approximation to the power of perfect fifths and pure thirds in just intonation dar. For the accompaniment of vocal, instrumental and mixed vocal- instrumental music offered it for a long time the best condition. In addition, in worship were chorales and to monitor their auditions in church modes with ease meantone. It was from the church musical practice out for a long time no need for a wave of tunings. However, certain problems in the company of ensembles resulted from the existence of different vocal pitch Standards: In Germany, about organs were around 1700 commonly in the ( common ) Chorton (a ' = approximately 465 Hertz) or occasionally in the High choir pitch (a' = approximately 495 Hertz) while most of the instruments and singers in concert pitch (a ' = about 415 Hz ) played music. (For comparison: In gleichstufiger mood with a '= 440 Hertz gis ' = 415.3 Hz, b ' = 466.2 Hz and h' = 493.9 Hz ). From the organist was required to transpose here, where there was light, that the limits of meantone were achieved or exceeded. Unless this is constantly happening, the companion " Wolf " could omit tones, maybe play around or provided with an ornament (which, however, the sound can also be highlighted ), by suitable choice of register conceal the ugly sound. Towards the end of the 17th century, the musical development of ensemble music was so advanced that Meantone often no longer appeared to be suitable. Here began the development of new moods. So you did not spring from receivables, in solo keyboard works " remote " to use keys.

This insight has implications for the view of alleged organ repertoire, which stood in mean- not - playable keys ( ie use of chords made ​​that were not available in the mean tone ). For example, neither the tonal ranges nor the sentiments of most organs in Bach's time suitable for its " Clavier - practice ", Part 3, making them " over written " at an assumed market for performance material. There must be other explanations for the origin and purpose of such music, such as playing on stringed instruments pedal (pedal harpsichord, pedal clavichord ), educational foundations for complex contrapuntal improvisation et cetera. This problem shows that the question of temperature, not only on the mean, central to the performance practice.

Vocal practice

The old organ builder who voted their instruments without tuner. As physical devices they were only the Monocord and the pendulum as well as her own heartbeat available.

Pure fifths, octaves and thirds could agree without further notice. The fifths in - mean tone but had to be placed close to a comma. There were instructions for the observation of beats. However, you had to note that the number of beats per unit time is greater, the higher are the fifths. After tempering of four somewhat narrower Quinten you could check the tuning by a pure third. The other sounds were easily vote by pure thirds. Had, for example, CG, GD, DA and AE tempered, the other tones could be achieved by pure thirds: D- F #, Eb G, E G #, FA, GH, A -C and BD. Were tuned within an octave all twelve tones, you completed the entire tonal range of the instrument by pure octaves.

Calculations of the beats

Explanation: If the fundamental frequency, then the pure fifth above the frequency has.

The mean- fifth the frequency is 1/4 comma below:

For pure fifths for the 3rd partial ( octave fifth) of the fundamental tone is identical to the second partial ( octave ) of the fifth. The frequency of the beat with tempered fifth is then calculated from the difference of these overtones:

In our example, is calculated from a '= 440 Hz, the frequencies of E' forward and d ', g, and c' backward as follows:

The pure third has no beats, the Pythagorean third c'e ' (c' = 260.74 Hz, e ' = 330 Hz), however, ( beats per second ), or about ten times as much as in the mean- fifths, and was therefore perceived as a dissonance.

Structure of the scale

Root: C, the beginning of the circle of fifths in it.

The frequency ratio of the syntonic comma is that of the fifth.

Each of the 11 mean- fifths Qm is a by - syntonic comma reduced perfect fifth.

Your frequency ratio is accordingly.

Abbreviations: Ok = octave, Qm = the - comma meantone fifth

The circle of fifths the comma meantone fifths does not open. The twelfth fifth Dis differs from the beginning of the circle of fifths it by an interval - called small diesis - with the frequency ratio (about 1/5 of gleichstufiger whole tone ).

All possible intervals of the mean tone can be found in section clay structure.

Thus we obtain the following intervals:

• Eight pure major thirds: Eb G, BD, FA, CE, GH, D- F #, A -C, E- G #
• Eleven meantone fifths: Es- B, BF, FC, CG, GD, DA, AE, EH, H- Fis, Fis - Cis Cis - Gis
• A too large wolf fifth: Gis It 7OK = - 11 sqm with the frequency ratio

• Four to major thirds ( diminished fourths ): H - It, F # -B, cis- F, G # C

In the mean tone, not all sharp or flat tones are available. In the above example only It, B, F sharp, C sharp and G sharp, but not their enharmonic change tones Dis, Ais, Gifts, In and As. The same is true for the enharmonic change tones of other sounds; for example, are also Fes and ice is not available.

The enharmonic change tones - for example, and D sharp - differ by the small DIESIS of 41 cents. At the same interval of three superposed pure major thirds less than an octave. Even with the variations of the mean tone, in which the usable major thirds are only approximately pure, a big difference between the enharmonic change tones remain.

One can therefore acceptable only play in keys in which the missing tones are not needed.