Mid-tone mood

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Under mean-moods is a series of tempered tunings , in the Renaissance , the Baroque primarily for and in many cases also in later times (until the 19th century) keyboard instruments were in use. The mid-tone tuning with its many pure thirds comes pretty close to the pure tuning for keyboard instruments - but only for a limited number of keys.

As with the pure tuning, the use of the characteristic pure major thirds (with the frequency ratio ) is fundamental, for which the pure fifths of the Pythagorean tuning are slightly narrowed. The Pythagorean major third (= 4 perfect fifths - 2 octaves) becomes a pure third when the perfect fifths are lowered by the syntonic comma. In the case of the pure tuning, the pure third is divided into a large whole tone ( ) and a small whole tone ( ) (with rational frequency ratios), in the middle tone tuning - hence the name - into two equal whole tones with the irrational frequency ratio .

In order to reduce or avoid the wolf fifth , the strict midrange was modified in many experiments, but at the same time the pure thirds are increased (sharpened) . Ultimately, this led to the development of well-tempered moods and equal moods .


Principle of the -comma-mean-tone tuning: eleven slightly tempered fifths result in eight pure thirds. The "thirds" (diminished fourths) B-D flat (B-E flat), F-sharp A sharp (F sharp-B flat), C sharp-ice (C sharp-F) and G sharp-His (G sharp-C) are useless.

In the mean-tone tuning, eleven fifths of the circle of fifths are reduced by so much that the major thirds resulting from four of these fifths become pure or almost pure. In the most common and most frequently described variant, the major third is pure. The four fifths are therefore reduced by the syntonic comma. In other words: If you reduce the 11 fifths by the syntonic comma, the usable thirds become exactly pure. The resulting mood is the - (syntonic) - comma - mean-tone mood .

interval CD de ef fg ga Ah hc
(in cents)







(in cents)
193 193 117 193 193 193 117
(in cents)
200 200 100 200 200 200 100
Like the C major scale here, the scales are also structured in B, F, G, D and A major.

An audio sample of the slightly out of tune mean-tone fifths can be found here .

Note: Pure intervals are characterized by integer frequency ratios, while tempered intervals usually have an irrational frequency ratio. Therefore the size comparison is made with the unit cent .

Example: In the pure tuning, the pure third is divided into a major and a minor whole tone, whereas in the middle tone tuning it is divided into two equally large whole tones.

pure: . In cents: 204 cents + 182 cents = 386 cents = pure third.
meantone: . In cents: 193 cents + 193 cents = 386 cents = pure third.
equally-: . In cents: 200 cents + 200 cents = 400 cents = equal tempered third.

In the specified keys we get a remarkably good sound quality because of the pure thirds in the tonic, subdominant and dominant; On the other hand, there is the disadvantage that in scales far from C major, unusable intervals arise. The twelfth “fifth”, which closes the circle of fifths, is actually a diminished sixth (usually G sharp-E flat), which deviates strongly from the perfect fifth and is usually musically useless. It is often called the " wolf fifth ". Four supposedly major thirds, the chain of fifths of which contain the wolf fifth, are diminished fourths (C sharp-F, F sharp-B, G sharp-C, B-flat), which also usually cannot be used as major thirds. There are therefore eight pure major thirds.

Fifths and thirds in the -comma-mean tone tuning. For comparison: pure major third = 386 cents (frequency ratio ) , pure fifth = 702 cents (frequency ratio ) .
Triad ceg c sharp-f-g sharp (!) d-f sharp-a es-gb e-gis-h fac fis-b-cis (!) ghd g sharp-c-es (!) a-cis-e bdf h-es-fis (!) ceg
Major third in cents












Fifth to cents 697 697 697 697 697 697 697 697 738
( wolf fifth )
697 697 697 697

The intervals marked with (!) Are only in the enharmonic confusion of thirds and fifths; you can see that only the following triads are playable: E flat major, B flat major, F major, C major, G major, D major, A major and E major as well as C minor, G minor , D minor, A minor, E minor, B minor, F sharp minor and C sharp minor.

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

In order to be able to play the A flat major triad, one would need a key for A flat 41 cents higher in addition to the key for G sharp; In order to be able to play the B major triad, one would need a key for D flat 41 cents lower in addition to the key for Eb, etc.

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

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

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

In order to make other keys playable, well-tempered tunings were developed - at the expense of the pure third - which ultimately led to the tuning of our keyboard instruments.

Other well-known, but historically in vocal practice seldom to barely verifiable mean-tone tunings are the -, -, -, and -comma mean-tone tuning, in which the 11 fifths are reduced by the corresponding fraction of the syntonic comma. The reduction in the dissonance of the wolf fifth leads to a reduction in the purity of the “good” major thirds.

If one usually speaks of a mean-tone mood, then the -comma-mean-tone mood is usually meant. Only with her are the major thirds exactly pure. The -comma-mean-tone tuning can be achieved relatively easily if you learn to tune the four tempered fifths precisely. The other notes are then obtained by tuning in pure major thirds.

The pure major third is characteristic of the mid-tone tuning

The mid-tone tuning with its many pure thirds best approximates the pure tuning with pure thirds in cadences.

Small and large halftone

The diatonic and chromatic semitones in the -comma-mean tone tuning

As with the pure tuning, a distinction is made between the diatonic , large semitone with 117.108  cents and the chromatic , small semitone with 76.049 cents.

Intervals of the -dot-mean-tone mood

interval c-cis cis-d of it-e ef f-f sharp f sharp-g g-g sharp g sharp-a from bra hc
in cents 76 117 117 76 117 76 117 76 117 117 76 117

The rule of the Weißenburg cantor Maternus Beringer (1610) also applies here:

Semitones on the same line in the staff (the chromatic ones) are to be intoned as a small semitone (semitus minor). Semitones on adjacent lines (the diatonic ones) but as a major semitone (semitonus major).
Four-part Pavane by Thoinot Arbeau in a mid-tone tuning on a concert grand

For musical practice, the change from large and small semitones in the mid-tone tuning is momentous. The use of chromatic sections with different semitone steps has an expressive effect. Leading tones upwards (c sharp, d flat, e, f sharp, g sharp and b) are voiced low and leading tones downwards (des, Eb, f, a flat and b) are voiced high, and the most common minor thirds are quite small. With a mid-tone tuning, cadences therefore have a special character, especially one that deviates from the tuning of the same level (example in the box on the right).

Interval table

Mid-tone keyboard

Comparison of equal, mid-range and pure tuning (in cents )

interval example equal mid-tone purely
chromatic halftone c-cis 100 76 c-c sharp: 92 or d-flat: 71
diatonic semitone cis-d 100 117 112
Whole tone CD 200 193 cd: 204 or de: 182
minor third c-it 300 310 316
major third ce 400 386 386
Fourth cf. 500 503 498
Tritone c-f sharp 600 579 590
Fifth cg 700 697 702
little sext ec 800 814 814
Sext approx 900 890 884
minor seventh cb 1000 1007 cb: 1018 or d-c ': 996
major seventh ch 1100 1083 1088

Extensive table: Intervals of -decimal-mean-tone tuning

-Comma-mean-tone tuning in chromatic order:

Frequency relationships to the fundamental C with the mean-tone fifth

F sharp
G sharp

Euler's notation, extended for mid-tone tuning

Mid-tone mood

In Euler tone lattice is preceded by the low point and high point a reduction or increase in a syntonic commas. The change of the fifths by 1/4 point is typical for the 1/4 point mean pitch. This change is marked with preceding points. Four preceding points therefore correspond to a preceding comma.

pure fifths ... as it b f c G d a H f sharp ...
mean fifths ... 'as °°° it °° b ° f c .G ..d ... a ,H ., fis ...

In this context means , x (depth point x) and 'x : (apostrophe x) x a syntonic comma decreased or increased.

.x (low point x) or ° x (high point x): x decreased or increased by 1/4 point.

The mid-tone keyboard with 12 keys then contains the following tones:

Tone designation °°° it °° b ° f c .G ..d ... a , e .,H .., f sharp ..., cis ,, g sharp
in cents 310.26 1006.84 503.42 0 696.58 193.16 889.74 386.31 1082.89 579.47 76.05 772.63

Example of size calculation : From c you reach .., f sharp over 6 fifths reduced by 3 octaves and 1 comma and two 1/4 commas.

.., f sharp = 6 fifths - 3 octaves - 1.5 commas = 1200 ⋅ (6 log 2 (3/2) - 3 log 2 (2) - 1.5 log 2 (81/80)) cents = 579, 47 cents.

Mid-tone tuning, playable from C flat major, G flat major, etc. to B major, F sharp major and C sharp major

The mid-tone tuning comes very close to the pure tuning, as the major thirds sound pure and the fifths are only slightly diminished. With 12 keys, only the keys from E flat major to E major are playable in mid-tone. For this reason, the keyboard was often increased to 19 keys in the 16th century so that all keys from C flat major to D flat major were playable. See Archicembalo and Nineteen Step Tuning .


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

Even if isolated sources from the 15th and early 16th centuries can already be seen as a practical description of the mean-tone tuning, it was first described correctly and unambiguously by Gioseffo Zarlino in 1571 . 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 indicated three ways in which it could be practically placed (in addition to an insignificant modification, which, however, did not have an additional key enables). Due to Praetorius' description, the mean tone tuning was often referred to as "Praetorian" until the 18th century. In organ building it was used as the standard tuning in large parts of Germany well into the 18th century - in some regions even beyond that - which is why the tuning did not need to be specified in organ building contracts and test reports (acceptance reports).

In northern Germany, for example, the mean-tone tuning is documented in printed sources for all of Hamburg's organs in 1729, and the organ in Bremen Cathedral , which was newly built by Arp Schnitger , was still in the mean-tone tuning until it was retuned from 1775 to 1776. Recent research has also made it plausible again that the organs that were available to Dieterich Buxtehude in Lübeck were at this standard temperature. Incidentally, Buxtehude did not make any comments on questions of mood - his dedication poem for Andreas Werckmeister's Harmonologia Musica from 1702, a counterpoint and improvisation theory, does not refer to questions of mood and cannot be interpreted as support for Werckmeister's draft moods.

In Central Germany, on the other hand, the organ builder Christian Förner , his students ( Zacharias Thayßner , Christoph Junge , Tobias Gottfried Trost) and grandchildren (including Tobias Heinrich Gottfried Trost and Johann Friedrich Wender ) modified the mean-tone tuning or replaced it with a well-tempered tuning in order to use all keys to be able to. This mood is documented for the Förner organ of the castle church in Weißenfels , built between 1668 and 1673 . Zacharias Thayßner built the organ of the collegiate church St. Servatii in Quedlinburg from 1677 to 1682 , where Andreas Werckmeister was officiating. Johann Friedrich Wender built the organ of Divi Blasii in Mühlhausen from 1687 to 1691 and the organ of the Bonifatius Church (today Bach Church ) in Arnstadt from 1699 to 1703 . The young Johann Sebastian Bach officiated on these two organs from 1703 to 1708. From the beginning they enabled him to write organ works that go well beyond the keys that the mean- tone comma tuning allows.

The wolf fifth and the four diminished fourths were considered completely unusable in the 17th and 18th centuries. More recent assumptions that these were used for compositional purposes (e.g. B-Es-F sharp as a supposed B major, F G sharp C as a supposed F minor, etc.) are, however, supported by statements made by sources from the 17th and 18th centuries. Century regularly refuted.

Italian harpsichord with the upper keys C sharp, D flat / E flat, F sharp, G sharp / A flat and B flat (D flat / E flat and G sharp / A flat as "broken" upper keys), which allows playing in more mid-tone tempered triads without changing the setting: from A flat major to B major and F minor to G sharp minor.
Reconstruction of the “harpsichord universale”. The mid-note tuned major triads C-E-flat to C sharp-ice-G sharp and minor triads from A-flat C-sharp to A sharp-C sharp-ice are now possible.
Depending on the use - for example as C sharp or as D flat - the black keys on the harpsichord universale as well as B and Ces or Eis and F differ by 41 cents.

In order to expand the stock of keys of the usual mean-tone tuning, keyboard instruments were often equipped with additional upper keys at places of professional music care in Western Europe between around 1450 and 1700; because only the following keys can be played on a twelve-step scale in a mid-tone tuning:

  • C major (A minor) with C, D, E, F, G, A, H
  • F major (D minor) additionally with B , G major (E minor) additionally with F sharp ,
  • B flat major (G minor) additionally with E flat, D major (B minor) additionally with C sharp
  • E flat major (C minor) also with A flat .

With that the twelve keys are exhausted.

  • For an additional A major (F sharp minor) you have to divide the A flat key into G sharp / A flat ,
  • for additional E major (C sharp minor) the E button in Dis / Es,
  • for additional H-Dur (G sharp minor), the B-key in Ais / B,
  • for additional F sharp major (d flat minor) the F key in ice / F.
  • for additional A flat major (F minor) the C sharp key as C sharp / D flat ,
  • for additional D flat major (B flat minor) the F sharp key as F sharp / G flat and
  • for additional G flat major (E flat minor) the B key as B / C flat .

The difference between G sharp and A flat or D sharp and E flat, etc. is 41  cents (almost half a semitone).

As a rule, instruments were equipped with one to two, more rarely four, and in the case of the universal harpsichord even seven “subsemitonies”, or split keys . Such instruments are related to the so-called enharmonic instruments, such as the orthotonophonium with 72 keys per octave.

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 ). Michael Praetorius describes a "Cembalo universale" ("Cimbalo cromatico") that has 19 tones per octave: In addition to the five divided upper keys, there are additional narrow upper keys for the ice and his.

Since the end of the 17th century, well-tempered moods have slowly but increasingly established themselves on stringed keyboard instruments, as well as practical approximations to equal pitch, that is, moods that allow the use of all keys. The well-tempered moods were not the same mood that can be heard today on electronic instruments and mostly on pianos , but rather those in which the individual keys sounded sometimes more, sometimes less “tense” (key characteristic that was also understood as a subjective moment in the 18th century ).

For a long time, one could only assume that Bach, when transposing older works (!) And partially re-composing the preludes and fugues of the two volumes of the “ Well-tempered Clavier ”, thought of the then completely new unequal well-tempered tunings, even if the equal temperament was practical placed in his later phase of life cannot be ruled out. It should also be noted that Friedrich Suppig described in a manuscript in 1722 that all pianos in Dresden were tuned to the mean tone - in the same year that Bach compiled the first volume of the Well-Tempered Clavier and provided it with the dated title page. According to a new but still controversial interpretation around the year 2000, the garland on the title page of the Well-Tempered Clavier can be interpreted as a tuning instruction .

The theoretical ramifications of the history of the mean-tone tuning are well known, but the practical application, dissemination and the transition to newer tunings, often much later than previously assumed (often directly to approximations to equal tuning), is common in many regions researched only in rudimentary form, since it was all too often assumed that theoretical drafts of mood would soon also be implemented in practice. However, as Werckmeister and others who designed new tunings complained, the organ builders did not follow their designs and stuck with the medium-tone voice practice for a long time, with the exception of central German organ builders who succeeded Christian Förner .

The mid-tone tuning represented the most favorable approximation to the net of pure fifths and pure thirds of the pure tuning. For the accompaniment of vocal, instrumental and mixed vocal-instrumental music it offered the best conditions for a long time. In addition, chorals and their preludes in church modes could be easily accompanied in the mean tone during the service. For a long time there was no need for a wave of change of mind from the practice of church music. However, certain problems in accompanying ensembles arose from the existence of different pitch standards: in Germany, for example, around 1700 organs were usually in the (common) chorus tone (a ′ = approximately 465  Hertz ) or occasionally in the high chorus tone (a ′ = approximately 495 Hertz) , while most of the instruments and singers played in concert pitch (a ′ = approximately 415 Hertz). (For comparison: in equal tuning with a ′ = 440 Hertz, g sharp ′ = 415.3 Hertz, b ′ = 466.2 Hertz and h ′ = 493.9 Hertz). The organist was asked to transpose, which easily showed that the limits of the midrange were reached or exceeded. As long as this did not happen all the time, the accompanist could leave out "Wolf" tones, perhaps play around them or add an ornament (which, however, can also emphasize the tone), and cover up the ugly tone by choosing a suitable register. Towards the end of the 17th century, the musical development of ensemble music was so advanced that the mid-tone tuning often no longer seemed suitable. This is where the development of new moods began. So it did not arise from demands to use "distant" keys in solo works for keyboard instruments.

Vocal practice

The frequency of the beats in mid-tone tempered fifths is just under 1% of the frequency of the fundamental.
Picture and sound with a fundamental tone of 263 Hz

You could easily tune in perfect fifths, octaves and thirds. The fifths in the mid-tone tuning, however, had to be placed closer by commas. For this there were instructions for observing beatings . It was important to note that the higher the fifths, the greater the number of beats per unit of time. After tempering four slightly narrower fifths, the tuning could be checked with a pure third. The other notes could be easily tuned with pure thirds. For example, if you had CG, GD, DA and AE tempered, the other notes could be achieved with pure thirds: D-F sharp, Eb-G, E-G sharp, FA, GH, A-C sharp and BD. If all twelve tones were tuned within one octave, the entire tone spectrum of the instrument was completed with pure octaves. The old organ builders tuned their instruments without a tuner. The only physical devices available to them were the monochord , pitch pipe and pendulum as well as their own pulse rate.

Example with the first four mean-tone fifths and the corresponding third

Four-point- mean-tone fifths and a pure third (a ′ = 440 Hz)

A major cadence Vibration medium-tone fifths.png

Note the different fast beats in the fifths. No beating in the pure third.

One “hears” here: The tempering of the fifths is so low that it is not perceived as a dissonance.

Table of beats

(see notes)
Frequencies Beats
per second
c′g ′ 263.18 393.55 2.45
gd ′ 196.77 294.25 1.83
there' 294.25 440 2.74
ae ′ 220 328.98 2.05
c′e ′ 263.18 328.98 0

The higher the fifths, the higher the number of beats (about 1% of the fundamental frequency).

Structure of the scale

Root note: C, beginning of the circle of fifths in Eb .

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

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

Their frequency ratio is accordingly .

Abbreviations: Ok = octave, Q m = the -comma-mean-tone fifth

Sound label Distance to the keynote in cents Frequency ratio to the fundamental
It −3Q m + 2Ok 310.26
B. −2Q m + 2Ok 1006.84
F. −Q m + Ok 503.42
C. 0 0
G Q m 696.58
D. 2Q m - Ok 193.16
A. 3Q m - Ok 889.74
E. 4Q m - 2Ok (pure third) 386.31
H 5Q m - 2Ok 1082.89
F sharp 6Q m - 3Ok 579.47
Cis 7Q m - 4Ok 76.05
G sharp 8Q m - 4Ok 772.63
( Dis ) (9Q m - 5Ok) (269.21) ( )

The circle of fifths of the -comma-mean-tone fifths does not work. The twelfth fifth D flat differs from the beginning of the circle of fifths E flat by an interval - called a small diesis - with the frequency ratio (approx. 1/5 equal whole tone).

All possible intervals of the mid-tone tuning can be found in the Tone Structure section .

So we get the following intervals:

  • Eight major thirds: Eb-G, BD, FA, CE, GH, D-F sharp, A-C sharp, E-G sharp
  • Eleven mid-tone fifths: Eb-Bb, BF, FC, CG, GD, DA, AE, EH, B-F sharp, F sharp-C sharp, C sharp-G sharp
  • Too large a wolf fifth: G sharp-Eb = 7Ok - 11 Q m with the frequency ratio
  • Four major thirds ( diminished fourths ): B-Eb, F-sharp-B flat, C sharp-F, G sharp-C

In the mid-tone tuning, not all raised or lowered tones are available. In the above example only Eb, Bb, F sharp, C sharp and G sharp, but not their enharmonic alternating tones D flat, A sharp, G flat, D flat and A flat. The same also applies to the enharmonic alternating tones of the other tones; for example fez and ice cream are not available either.

The enharmonic alternating tones - for example Eb and Dis - differ by the small Diësis of 41 cents. Three superimposed pure major thirds are also smaller than the octave by the same interval. Even with the variants of the mean-tone tuning, in which the usable major thirds are only approximately pure, there remains a big difference between the enharmonic alternating tones.

One can therefore only play decently in keys in which the missing notes are not needed.


More moods

Instruments and generators

  • Archicembalo
  • ZynAddSubFX is a microtonal open source software synthesizer, with the help of which the mid-tone tuning can be realized and tried out.
  • With the program TTMusik by Joachim Mohr you can calculate and listen to the frequencies of pure, mid-tone and equal chords in Hz and in cents.

Web links

Individual evidence

  1. ^ Herbert Kelletat: On the musical temperature. I. Johann Sebastian Bach and his time. 2nd Edition. Merseburger, Berlin 1981, ISBN 3-87537-156-9 , pp. 17-25.
  2. The tempering of the fifths is not perceived as a dissonance, on the contrary: each fifth has its own "color tone", caused by the beats, which are perceived as a subliminal vibrato. See: Voice Practice
  3. For example, the Pythagorean third C E is the sum of the four fifths C GDA E octaved . This is the syntonic comma too high compared to the pure third.
  4. The averaging over the division of the comma takes place via the additive perception of the intervals (see interval space: large whole tone: 203.910 cents; small whole tone: 182.404 cents; medium tone whole tone: (203.910 + 182.404) / 2 cents = 193.157 cents)
  5. Michael Praetorius writes about the false thirds ( Syntagma musicum . Volume 2: De Organographia (1619). Reprint: Bärenreiter, Kassel 2001, ISBN 3-7618-1527-1 , p. 155): “... and it's best that the Wulff with his humble howling in the forest stay / and our harmonicas Concordantias not interturbire. "
  6. Maternus Beringer: Musicae, that is the free, lovely singing art . Nuremberg: Georg Leopold Fuhrmann, 1610 (reprint: Bärenreiter, Kassel 1974).
  7. ^ Ibo Ortgies: Temperature. In: Siegbert Rampe (Ed.): The Bach Handbook. Vol. 4: Piano and Organ Music. Laaber-Verlag, Laaber 2007, pp. 623-640.
  8. Roland Eberlein : Tunder, Buxtehude, Bruhns, Lübeck: For which instruments did you write and how were they tuned? (PDF) walcker-stiftung.de. Pp. 5-7. Retrieved April 7, 2016.
  9. ^ Johann Caspar Trost: Detailed description of the new organ work on the Augustus Castle in Weissenfels . Nürnberg 1677, p. 37 (facsimile in: Acta Organologica. 27, 2001, p. 36-108).
  10. The pipe lengths of preserved pipes of the Arnstadt organ show that the tuning could not have contained a wolf fifth .
  11. There is no evidence that the Wolfs Intervals were used by professional musicians to joke, as shock intervals or to convey drastic affects by composers. The contrapuntal correct resolution of leads, dissonances etc. is not affected if, for example, an excessive A-C sharp-F (A 6 3 #) chord arises in D (minor) above the 5th level A, which occurs in a cadence would be resolved as follows: 6 3 # - 6 4 - 5 4 - 5 3 #. C sharp-F is right here a diminished fourth that arises between the major third (C sharp) and the minor sixth (F) above the basic sound (A). It is correctly transferred to the minor third (DF), which forms an sixth fourth chord with the basic sound (A) (which of course can be further resolved).
  12. Of course minor, whether the altered notes - here F sharp and G sharp - are represented in harmonic minor must be checked separately
  13. The assignment of the "black" keys is not uniform. Equivalently, for example, G sharp can be tuned instead of A flat.
  14. ^ Michael Praetorius: Syntagma musicum. Volume 2: De Organographia. (1619). Reprint: Bärenreiter, Kassel 2001, ISBN 3-7618-1527-1 , pp. 63–66. (There is a reconstruction in the Organeum in Weener .)
  15. "as our dear forefathers who, despite their pitch pipes and monochords, could not get rid of the organ wolf [...]" Allgemeine musical newspaper . tape 19 , 1817, pp. 414 ( side view in Google Book search).
  16. Calculation: If the basic frequency is , then the perfect fifth above it has the frequency . The mean-fifth with the frequency is 1 / 4 -point including:
    In the case of perfect fifths, the 3rd partial tone (octave + fifth) of the root is identical to the 2nd partial tone (octave) of the fifth. The frequency of the beat with a tempered fifth is then calculated from the difference between these overtones:
    (a little less than one percent of the base frequency).
    In our example, the frequencies of e 'forward and d', g and c 'backward are calculated from a ′ = 440 Hz as follows:
    , , , And .
    The pure third has no beats , but the Pythagorean third
    c′e ′ (c ′ = 260.74 Hz; e ′ = 330 Hz) ( beats per second), i.e. about ten times as many as with the mean-tone fifths, and was therefore perceived as a dissonance.
  17. The 1/4 point mean fifths are dimensioned in such a way that four octaves result in a pure third and three pure thirds differ from an octave like D-flat and E-flat - corresponds to 12 mean-tone fifths.