# Mood (music)

In music, tuning is understood to be the definition of the pitches ( frequencies ) of sound sources, in particular of musical instruments . In practice, it is often sufficient (for example with some wind instruments ) to fix the absolute pitch by comparing it with a given frequency (e.g. the concert pitch a 1 = 440 Hz). Especially with stringed and keyboard instruments also need the frequency ratios of the strings or the individual sounds are adjusted to each other.

Most string instruments , such as the violin , can in principle produce any pitch within a wide range, in particular z. B. play every interval purely . (A pure tuning is based on the tone material of the overtone series ). In contrast, with keyboard instruments, twelve semitones per octave are generally firmly tuned. There is no such thing as a tuning that is pure in all keys, with only twelve pitches; therefore compromises have to be made (see tempered tuning , well -tempered tuning , pure tuning for keyboard instruments ).

Many wind instruments cannot produce all usable pitches (usually twelve) within the octave equally easily and purely, but have preferred diatonic scales. This, too, is usually simply referred to as the “tuning” of the instrument, sometimes more precisely as the basic tuning for brass instruments . Regardless of this, the above-mentioned adjustment of the absolute pitch to a tuning tone, e.g. B. the concert pitch, also necessary for the interplay with other instruments.

## Concert pitch

A specification like “ tuning 440 Hz ” indicates the frequency of a certain tuning tone, usually the a '(a dashed a , also called concert pitch ). Today's instruments are tuned to a 1 = 440 or 442 Hz , for example . Around 1900, 435 Hz was common. In earlier centuries, depending on the location, different pitches, also significantly higher than 440 Hz, were used (→ Cornett tone ). Today, a concert pitch of 415 Hz is often used on historical instruments, which is almost exactly a semitone lower than 440 Hz (→ chorus ). Listening to 440 Hz sine tone ? / i

## Fundamental scale and transposing instruments

The tuning of an instrument sometimes means its basic scale or its basic note, such as F major or the tone f 1 on the alto recorder ; see woodwind instrument .

Sometimes the tuning of an instrument refers to the note that is played instead of a written C when notes written specifically for that instrument are used (see Transposing Musical Instrument ). This is the case with some instruments, e.g. B. the trumpet , at the same time the keynote (see above), but not with others; z. For example, the clarinet "tuned in Bb" is more likely to be viewed as a key scale in F major.

Accordingly, “C tuning” can mean that notes in real pitch are common for the instrument in question.

## Mood systems

A tuning system is the way in which the exact frequency relationships of the playable tones within an octave , i.e. in most cases the twelve semitone steps of the selected tone system , are tuned in an instrument . Another name for this (especially for keyboard instruments ) is temperature or tempering . The tuning systems only make statements about the frequency relationships between the individual tones . No statement is made about the absolute pitch or the frequency itself. The absolute pitch is determined by specifying the frequency of the starting tone or the tone a 1 .

### Frequency relationships in the sound system

Each tone has a different meaning in each scale , e.g. For example, the E in the E major scale is the root note , in the C major scale the third tone (the third ) and in the A major or A minor scale the fifth tone (the fifth ). For each of the possible positions in the scale space there are different frequency ratios of the tones to one another, but these must be adjusted to one another in order to be able to be played on an instrument in different keys . A series of pure major thirds (frequency ratio 5: 4) cannot be brought into agreement with a series of pure fifths (frequency ratio 3: 2). With the usual limitation of twelve pitches per octave , this means that compromises have to be made: the more purely a certain key is tuned, the more impure other keys sound.

This problem is particularly noticeable with polyphony.

### Overview of the tuning systems

There are a variety of systems for tuning the tone system tones for a 12-step keyboard. The most important tuning systems are:

The choice of mood depends on which music is to be played. The equal tuning predominantly used today is suitable for music after 1800 . Earlier music or non-European music ( world music ) lives very much from the purity of intonation or from different key characters . These demands cannot be met by the equal mood. As part of the historical performance practice of early music , older tuning systems are therefore being increasingly researched again in order to enable adequate reproductions.

### history

#### Antiquity

Pythagoras with various instruments tuned in fifths (indicated interval ratios 16: 12: 9: 8: 6: 4).
From: Franchino Gaffurio: Theorica musicae , 1492 (1480?)

The first theoretical description of a tuning system was ascribed to Pythagoras of Samos in ancient times according to the legend of Pythagoras in the forge . Pythagoras was a philosopher and mathematician who emigrated to southern Italy . He was of the opinion that the entire cosmos was ordered according to certain numerical relationships and that music was an image of the cosmic order.

According to ancient tradition, the credibility of which is disputed in modern research, Pythagoras examined the intervals between string parts with integral length ratios on the monochord . For example, when a string is divided in half, it sounds an octave higher than its full length; the corresponding numerical ratio is therefore 1: 2. He was the first to describe the natural intervals known from the overtone series .

Pythagoras is said to have introduced a seven-note scale based on the perfect fifth (with the simplest number ratio after the octave, 2: 3). The tones are determined from a starting tone by fifth steps and transposed into a common octave . The result is the Pythagorean tuning , which Euclid's music theory later also dealt with. Of Euclid , the first accurate calculation of the proportion of the frequency comes Pythagorean comma .

In the 1st century BC The music theorist Didymos introduced the "natural third" (frequency ratio 5: 4) into his enharmonic tone system .

Another tradition goes back to Aristoxenus , who pointed out the inaccuracies of older investigations with the monochord and established an alternative tone scale calculation.

#### middle Ages

The Pythagorean tone system was adopted by the Romans and medieval Europe . Monochords , bells (see illustration) and organ pipes were tuned in Pythagorean and used in Gregorian music-making practice. The harmony of early polyphony preferred the intervals that are actually pure in Pythagorean tuning (the complementary intervals fifth and fourth as well as prime and octave ).

#### Renaissance

During the Renaissance there were two main developments that were important for the sound system:

• The increasing chromaticism in vowel polyphony finally expanded the tone supply to twelve tones.
• The feeling of dissonance changed. The third, which was still classified as dissonant in the Middle Ages, became the carrier of harmony in the emerging major-minor system. This feeling of dissonance was supported, however, by the fact that the third in the Pythagorean tuning used in the Middle Ages (frequency ratio 81:64) sounds dissonant from today's perspective; the pure major third (frequency ratio 5: 4) only came into the system with the change to pure tuning.

The new orientation towards the third and the need for a chromatic scale led to problems with the quint-based Pythagorean or even the pure tuning:

In accordance with the new ideal of sound, the pure major third (with the next simpler frequency ratio of 5: 4 after the fifth and fourth) was chosen as the new master interval and, since the pure tuning could not be achieved on the usual keyboard instruments, the so-called medium - tone tuning was developed . In order to avoid the syntonic comma , slightly reduced fifths were introduced, four of which, stacked on top of one another, form a pure major third.

Through the sequence of eleven mid-tone fifths

Eb - B - F - C - G - D - A - E - H - F sharp - C sharp - G sharp

one obtained the twelve tones of our occidental tone system.

So you got eight major thirds as desired. (e.g. C - E with four mean-tone fifths C - G - D - A - E); four thirds had to remain impure (e.g. B-D flat, because D-flat is in tune as an Eb and D-flat is therefore not four mean-tone fifths above B, but eight mean-tone fifths below B; see above sequence of fifths).

However, the mid-tone tuning was also a compromise and in some ways unsatisfactory. Theory and practice:

• This system created many intervals that cannot be expressed by whole number fractions, which contradicted the Pythagorean conception of music. The reason for this are the mean-tone fifths introduced in favor of the third purity, for which a string length ratio of applies, which is not a rational number .${\ displaystyle 1: {\ sqrt [{4}] {5}}}$
• Twelve mid-tone fifths stacked on top of one another result in a tone that is lower than the initial tone by the so-called little Diësis (see the problem of Pythagorean tuning ).
• The A-flat-Eb or G sharp-D flat fifth is too big around the small diesis, because the A-flat "alias G sharp" is not tuned as a mean fifth below E flat, but rather eleven middle-tone fifths above E flat (see above sequence of fifths). This so-called wolf fifth sounds very impure. In the mid-tone tuning, keys that contain this fifth (e.g. E flat major or C sharp major) sound extremely dissonant and can be used everywhere to represent certain affects .

Nevertheless, the mid-tone mood prevailed. Modulatory development, as it later became common, was not very common in the Renaissance. Therefore, the good sounding key range was initially sufficient. In order to make other keys playable in the mid-tone tuning, keyboard instruments with z. B. 31 tones built in the octave, but could not prevail.

#### Baroque

In the course of the 17th century this restriction to central keys was increasingly perceived as a nuisance. In order to become more free in the choice of keys, tuning systems began to be developed in which all keys can be played, even if not all of the same quality. For this, compromises in the purity of the thirds had to be accepted. These moods were called “well -tempered moods ” or “good temperatures”, in contrast to the mean-tone mood that was now perceived as “bad”. Examples of this are the tuning by Andreas Werckmeister Werckmeister III-VI , ( cadences in C and D flat major ? / I ) or the tuning by the organ builder Gottfried Silbermann .

However, there was no temperature that prevailed universally like the mean-tone mood previously (which, by the way, did not simply disappear with the new temperature approaches). The example of Werckmeister shows that initially it was not necessarily attempted to establish a uniform mood. In his most important work, Musicalische Temperatur, he describes different temperatures that can be more or less suitable depending on the needs.

Basically, one can distinguish between two approaches (with Werckmeister and others):

• Some systems tried to make the keys with a few accidentals sound as clear as possible, but also to make those with many accidentals playable, albeit with a more cloudy sound. (Example: the Werckmeister II temperature.)
• Other systems tried to make all keys playable as well as possible. (Example: the Werckmeister III temperature). At the end of the development, this approach led to today's common temperature . With this approach, however - in contrast to the central keys in the approach described above - you have to accept a relatively dull sound of all keys.

#### Equal temperature

Already in the Renaissance one looked for methods to tune the lute equally. Since it is not possible to tune each note individually on fret instruments, problems arise. (Because, for example, not all major thirds are the same in the mid-tone tuning, the fourth fret on the A string would have to shorten the string for the major third C sharp to 45 of the length, on the B string D alias "There are not attuned as a major third. the string would have here in the meantone on 25 / 32 the length can be shortened.)

Since the possibilities of calculating the roots were still limited at this time, it was not yet possible to calculate the equal semitone with the ratio . Nevertheless, it was possible to build a fingerboard of the same level, since geometrical methods for the sewing construction of simple root relationships were available. The Venetian musician and music theorist Gioseffo Zarlino described such a method as early as 1558 . ${\ displaystyle 1: {\ sqrt [{12}] {2}}}$

The lutenist Vincenzo Galilei , father of Galileo Galilei , did not give up the simple whole number relationships. He shortened the string per bunch on 17 / 18 length. In theory, the twelfth semitone does not quite reach the octave, but in practice the result is quite useful because the tone is generated by pressing and the distance between fret and string (the string becomes longer) and by finger pressure on the string (string tension increases) is increased a little. Even after the fingerboard has been designed, the string length can be adjusted by repositioning the bridge, so that the result is even better.

Mathematicians and music theorists tried in the following almost 200 years to use different methods to determine more precise numerical values ​​for the same temperature. In the 19th century , the same temperature finally became generally accepted.

Today there are again discussions about how organs should be tuned, for example. Many historical compositions are based on different sound properties of different keys and chords that cannot be reproduced on equally tuned instruments. This is particularly important for historical performance practice .

### Comparison of the mood systems

Graphic comparison of the Pythagorean and medium-tone tuning with the well-tempered tuning according to Werckmeister (III) and the tuning of the same level. The respective pitches of the fundamental tones are indicated in the vertical direction.
The four Pythagorean tones C, F, G and C are practically identical in all tuning systems. With the leading tones upwards (C sharp, D flat, E, F sharp, G sharp and B) the Pythagorean tuning is always high and the middle tone tuning is always low; With the leading tones down (Db, Eb, A flat and Bb), the Pythagorean tuning is always low and the mid-tone tuning is always high. The Werckmeister mood compensates for the differences and approaches the mood of the same level.

Note: Pure intervals are characterized by simple integer frequency ratios, tempered intervals also have irrational frequency ratios. Therefore, the size comparison is made here with the unit cents , where 1 octave = 1200 cents.

The following table shows the amount of pitches a major - scale of different moods in cents (rounded) to:

 Surname Prime great second major third Fourth Fifth major sixth major seventh octave Pure mood 0 204/182 386 498 702 884 1088 1200 Pythagorean mood 0 204 408 498 702 906 1110 1200 1/4 point mean tone tuning 0 193 386 503 697 890 1083 1200 Equal mood 0 200 400 500 700 900 1100 1200

Note: In the pure tuning there is the large whole tone (in C major for example CD) with 204 cents and the small whole tone (in C major for example DE) with 182 cents. Both are referred to as the big second. In the mid-tone tuning, the two whole tones are averaged at 193 cents each. Together they result in the perfect third with 386 cents.

The following tables can be used to estimate how far which fifths and thirds in different tunings deviate from the pure intervals. This shows how strongly "out of tune" corresponding major chords sound in the different keys. (the bold numbers show the wolf fifth or, analogously , the wolf turtle, conspicuous: the four almost pure thirds in the Pythagorean tuning)

 Fifths to cents C-G D-flat C sharp G sharp THERE Eb – A Dis – Ais EH F-C F sharp – C sharp Ges – D flat G-D As – Es G sharp – Dis A-E B-F H-F sharp C-G 1/4 point mean tone tuning 697 697 697 697 697 697 697 697 738 697 697 697 697 Werckmeister III well-tempered 696 702 696 702 702 703 702 696 702 702 702 696 696 equal mood 700 700 700 700 700 700 700 700 700 700 700 700 700 Pythagorean mood 702 702 702 702 702 702 702 702 679 702 702 702 702

For comparison: pure = Pythagorean fifth = 702 cents, equal fifth = 700 cents, 1/4 decimal point mean fifth = 697 cents.

 Major third in cents CE Des – F C sharp ice D-Fis It G E-G sharp FA F sharp-A sharp Gb -B G-H A-C G sharp-His A-Cis B-D H – Dis Ces – Es C-E 1/4 point mean tone tuning 386 427 386 386 386 386 427 386 427 386 386 427 386 Werckmeister III well-tempered 390 408 396 402 402 391 408 396 408 402 397 402 390 equal mood 400 400 400 400 400 400 400 400 400 400 400 400 400 Pythagorean mood 408 384 408 408 408 408 384 408 384 408 408 384 408

For comparison: pure = 1/4 decimal point mean major third = 386 cents, equal major third = 400 cents, Pythagorean major third = 408 cents.

A direct acoustic comparison of the aforementioned and other moods is possible with suitable software such. B. GrandOrgue , Hauptwerk or with appropriately arranged keyboards with their "presets" or through so-called "user scale tuning" possible. There are tuners with pre-programmed historical tunings for use on real instruments .

## Tuning instruments

As voices of instruments is called setting the pitch . A wind instrument is tuned as a whole, with string instruments ( string instruments , guitar , harp ) each string is tuned individually. With pianos and organs , several strings or pipes have to be tuned for each individual note. Some instruments are even tuned in "registers"; this includes, for example, adjusting the octave purity of guitars.

Most instruments can be tuned within certain structural limits, but there are also instruments that cannot be tuned or can only be tuned with great effort due to their construction. These include, above all, instruments from the field of percussion and idiophones , e.g. B. Chimes.

When tuning instruments, one can distinguish between three tasks:

• the tuning of an instrument "in itself" (important for keyboard and string instruments)
• the coordination of several instruments with one another
• the tuning of an instrument to absolute pitches

The manual process of tuning varies from instrument to instrument. With wind instruments this is usually done by changing the tube length, with string instruments by changing the tension of the strings. With keyboard instruments, every single note usually has to be tuned. The tuning of an ensemble is usually based on the most inflexible instrument, mostly the keyboard instruments. In the orchestra , the oboe gives the a 'as the concert pitch due to its overtone-rich sound . If no reference instrument is available, a tuning fork , a pitch pipe or an electronic tuner can be used as an aid. Electronic musical instruments do not have to be tuned “in themselves” due to the tone generation used, but they can often be changed overall in small to transposing steps.

Due to the frequency of the tuning process, many experienced instrumentalists have developed a kind of perfect pitch for their vocal tones (basic tones for wind instruments, open strings for strings).

## literature

• Ernst Kochsiek : Concert moods . Experiences and encounters with famous pianists. With audio CD. Edition Bochinsky, 2001, ISBN 978-3-923639-46-5
• Klaus Lang: On harmonious waves through the sound of the sea - temperatures and moods between the 11th and 19th centuries , edited by Robert Höldrich, Institute for Electronic Music (IEM) at the University of Music and Performing Arts in Graz (1990), PDF- version
• Gottfried Rehm : Introduction to old tuning systems. In: Guitar & Laute 4, 1982, 1, pp. 12-14.