# Equal mood

Equal tuning (also equally tempered tuning ) is the name of a tuning system that divides an octave into twelve equal semitone steps of 100  cents . Other terms are: equal tempered / equal temperament or equal temperament . The term tempered tuning, which is often used colloquially, is too imprecise, as equality is only one possible way of tempering intervals .

The pure tuning of keyboard instruments is afflicted with the problem of the so-called Pythagorean comma, ie that with 12 tones per octave, pure fifths and pure octaves are mutually exclusive and therefore only a limited number of perceived chords are available. Equal tuning solves this problem by distributing the inevitable comma (music) evenly across all pitches so that all intervals except the octave are equal and slightly impure. In earlier tuning systems this was not the case, which limited the possibility of modulation to other keys. Critics of the equal temperament, however, regret that it loses the individual character of individual keys of the earlier medium-tone or well-tempered tunings.

In the practical implementation of equal tuning, especially with stringed pianos, it should be noted that, due to the inharmonicity of the piano strings, an additional stretching of the octaves is necessary.

## Moods with tempered intervals

Musical instruments in which an intonation of pure octaves, fifths, fourths, thirds etc. is not possible in all keys, are predominantly tuned equally in western music - a compromise in intonation. This is particularly important for musical instruments in which the pitch and number of keys or the number of tones per octave are determined by constructive parameters, e.g. B. keyboard instruments such as organ, harpsichord, piano, or stick games and plucked instruments with multiple strings , and therefore a key-related retuning or adjustment of the pitch is not possible during the game.

Depending on the harmonic context in which a note is played, it should actually have a slightly different pitch in order to sound pure ( beat-free ) in a chord . For example, the G sharp note does not correspond to the A flat note, and this problem ultimately exists with all tones of a scale, depending on the harmonic context in which they are used. Tempering was therefore required for keyboard instruments , which was first implemented in the medium-tone tunings and then in the well-tempered tunings. It is characteristic of all these temperaments that they were developed on the basis of musical considerations. The exact position of all twelve semitones is determined in a medium-tone or well-tempered tuning in such a way that some keys or chords sound cleaner, others, mostly the less common ones, sound more impure.

Alone with the mathematically determined equal pitch, all keys sound the same (slightly impure).

Other instruments, such as string or wind instruments , on the other hand, can definitely intone purely, whereby the player can then compensate for the system-related impurities by slightly adjusting the pitch.

When these purely playable instruments play together with the piano, intonation conflicts can arise. The cellist Pablo Casals wrote :

“Don't be shocked if you have a different intonation than the piano. It's because of the piano, which is out of tune. The piano with its equal tuning is a compromise in terms of intonation. "

- The Way They Play

## Intervals in equal mood

In the equally tempered tuning, the octave is divided into twelve identical semitone steps:

Halftone = 1 / 12 · octave = 100 cents (frequency ratio ).${\ displaystyle {\ sqrt [{12}] {2}} \ approx 1 {,} 0594630943593}$

This compensates for the Pythagorean comma that exists between the twelfth perfect fifth above a tone, for example CGDAEH-F sharp-C sharp-G sharp-Dis-A sharp-Eis-His, compared to its seventh octave. These fifths are now all tuned 1/12 of this comma lower, so that the open spiral of fifths closes to form a circle of fifths . Compared to the Pythagorean (perfect fifth) tuning with the perfect fifth of 702 cents, the equal tuning has a slightly reduced fifth of 700 cents; accordingly, the fourth of the equal tuning (500 cents) - which complements the fifth to the octave - is about 2 cents further than a pure fourth (498 cents). The major third of the pure tuning (386 cents) is increased ("sharpened") by around 14 cents in the equal tuning (400 cents), while the minor sixth (pure: 814 cents, equal: 800 cents) is reduced by the same value becomes. The minor third (pure: 316 cents, equal: 300 cents), on the other hand, is tuned too narrow by around 16 cents, while the major sixth (pure: 884 cents, equal: 900 cents) is tuned too far by the same value.

On an equally tuned instrument, apart from the octave, not a single interval is “ideal”, i.e. H. in a simple integer frequency ratio purely voted, and the variations are quite audible. In today's music perception , however, this is generally perceived as acceptable (habituation effect).

## history

Geometric representation of the equal mood from Sopplimenti musicali (1588) by Gioseffo Zarlino

The equal temperament tuning was the first time in 1584 by Chu Tsai-yu nine-digit numbers are fairly accurately calculated (朱載堉) in China with the help of a system. In Europe, however, these calculations only became known in 1799, without Chu Tsai-yü being named. In 1588 Gioseffo Zarlino offered an exact geometric representation. Simon Stevin was the first European to describe the Singconst in Vande Spiegheling (manuscript around or before 1600), using a method he had developed for calculating the roots, but he wrongly believed that natural major thirds were guaranteed.

As practiced by Vincenzo Galilei , lute tunings of the sixteenth century, designated as equal, were mostly based on the semitone with the ratio 18:17 (around 99  cents ).

In the 17th century in particular, the equal mood was not only used by theorists such as B. Pietro Mengoli and Marin Mersenne , but also discussed by composers, instrument makers and performing musicians. This is evidenced, for example, by a dispute about moods between Giovanni Artusi and Claudio Monteverdi shortly after 1600. The music theorist Giovanni Battista Doni (c. 1593 - 1647) alleged anecdotally in a letter that Girolamo Frescobaldi had the same temperature for the organ in the Basilica S. Recommended Lorenzo in Damaso. But there is no evidence of Frescobaldi's support for equality, and it would have been unprecedented in organ building of his time.

In German-speaking countries was used for equally- the concept of equal temperament , so Andreas Werckmeister in 1707 in his posthumously published Musical Paradoxal-Discourse . There Werckmeister suggests distributing the Pythagorean comma evenly over all twelve fifths. He also calls this mood "well tempered" and justifies it with mystical or religious arguments:

“We go on / and know / when the temperature is set up / that all fifths 1/12 commat: the tert: maj: 2/3 the min: 3/4 comm. float, and an accurate ear can also bring it to a standstill / and know how to tune it / so then certainly a well-tempered harmony through which the whole circle and all clavis will be found. Which then can be a role model / how all pious / and well-tempered people will live and celebrate with God in everlasting equal / and eternal harmony "

- Andreas Werckmeister : Musical Paradoxal Discourse, 1707

Werckmeister expressly does not mean that the beat frequencies are the same. The difficulty he addressed, to vote equally, can z. For example, a piano tuner can master the fact that he knows the different beat frequencies of the fifths in the different high registers of the piano and uses them for tuning.

The practical significance, however, initially remained minor. But there were increasing supporters of the equal mood. B. Johann Georg Neidhardt , Friedrich Wilhelm Marpurg and Jean-Philippe Rameau belonged. In 1749 Georg Andreas Sorge wrote Detailed and Clear Instructions on Rational Calculation , in which he dealt with the mathematical questions of equal mood, which he called “rational equal temperature”. Towards the end of the 18th century, equal moods gained the upper hand over unequal moods ; in the 19th century it finally prevailed.

With this, however, the key characters lost their importance for new compositions , because different keys no longer sounded different in this respect. When performing older works on equally tuned instruments, essential artistic aspects of the composition are often lost for the same reason. For example, older composers of their time often used bad-sounding "impossible" keys to make negative facts such as pain or sin tangible.

Today instruments with fixed pitches, such as the piano or the guitar , are tuned equally by default. Many organs and harpsichords, however, are historicized with different, unequal tunings .

## Quantitative Aspects of Equal Tuning

### Frequency calculation

Relationship between frequency, semitone and octave with logarithmic representation

The mathematical rule for determining the tones on the entire scale of equal pitch is

${\ displaystyle f (i) = f_ {0} \ cdot 2 ^ {i / 12} \ ,,}$

where f 0 is the frequency of any output tone (e.g. the frequency of the concert pitch a 'at 440 Hz). i is the semitone step distance to the selected tone with frequency f 0 . Such a mathematical sequence is called a geometric sequence . If you want to plot the frequencies over equidistant pitch names on a straight line, you have to use simple logarithmic paper . It makes sense to use the logarithm of two rather than the tens for labeling.

For example, if you want to determine the frequency of the tone g ', you count the semitone step distance from the concert pitch a' ( i = minus 2, because you count down) and insert the values ​​into the equation:

${\ displaystyle f (-2) = 440 \, \ mathrm {Hz} \ cdot 2 ^ {- 2/12} \ approx 391 {,} 995 \, \ mathrm {Hz} \,}$

for the tone g '' one obtains a semitone distance to f 0 of i = 10:

${\ displaystyle f (10) = 440 \, \ mathrm {Hz} \ cdot 2 ^ {10/12} \ approx 783 {,} 991 \, \ mathrm {Hz} \,}$

As you can see, g '' has twice the frequency of g '. The octave purity is thus preserved, whereas all other intervals are slightly impure.

### Frequencies and cent values

When comparing intervals, the unit of cents is used . The following applies: 1 octave = 1200 cents.

Comparison of the frequencies of equal tuning and pure tuning.

Equal tuning chromatic scale:
Name of the tone c cis / des d dis / it e f f sharp / total G g sharp / as a ais / b H c
Frequency [Hz] 261.6 277.2 293.7 311.1 329.6 349.2 370 392 415.3 440 466.2 493.9 523.3
In cents 0 100 200 300 400 500 600 700 800 900 1000 1100 1200
Extended scale of the pure tuning of C major and C minor supplemented by F sharp and D flat:
Name of the tone c of d it e f f sharp G as a b H c
Frequency [Hz] 264 281.6 297 316.8 330 352 371.25 396 422.4 440 475.2 495 528
In cents 0 112 204 316 386 498 590 702 814 884 1018 1088 1200
interval Equally tempered interval In cents Pure interval In cents Difference in cents
Prime ${\ displaystyle {\ sqrt [{12}] {2 ^ {0}}} = 1}$ 0 cents ${\ displaystyle {\ tfrac {1} {1}} = 1}$ 0 cents 0 cents
Small second ${\ displaystyle {\ sqrt [{12}] {2 ^ {1}}} = {\ sqrt [{12}] {2}} \ approx 1 {,} 059463}$ 100 cents ${\ displaystyle {\ tfrac {16} {15}} \ approx 1 {,} 066667}$ 111.73 cents −11.73 cents
Big second ${\ displaystyle {\ sqrt [{12}] {2 ^ {2}}} = {\ sqrt [{6}] {2}} \ approx 1 {,} 122462}$ 200 cents     ${\ displaystyle {\ tfrac {9} {8}} = 1 {,} 125}$
${\ displaystyle {\ tfrac {10} {9}} \ approx 1 {,} 111111}$
203.91 cents
182.40 cents
−3.91 cents
17.60 cents
Minor third ${\ displaystyle {\ sqrt [{12}] {2 ^ {3}}} = {\ sqrt [{4}] {2}} \ approx 1 {,} 189207}$ 300 cents ${\ displaystyle {\ tfrac {6} {5}} = 1 {,} 2}$ 315.64 cents −15.64 cents
Major third ${\ displaystyle {\ sqrt [{12}] {2 ^ {4}}} = {\ sqrt [{3}] {2}} \ approx 1 {,} 259921}$ 400 cents ${\ displaystyle {\ tfrac {5} {4}} = 1 {,} 25}$ 386.31 cents 13.69 cents
Fourth ${\ displaystyle {\ sqrt [{12}] {2 ^ {5}}} = {\ sqrt [{12}] {32}} \ approx 1 {,} 334840}$ 500 cents ${\ displaystyle {\ tfrac {4} {3}} \ approx 1 {,} 333333}$ 498.04 cents 1.96 cents
excessive fourth
tritone *
${\ displaystyle {\ sqrt [{12}] {2 ^ {6}}} = {\ sqrt {2}} \ approx 1 {,} 414214}$ 600 cents ${\ displaystyle {\ tfrac {45} {32}} = 1 {,} 40625}$ 590.22 cents 9.78 cents
Fifth ${\ displaystyle {\ sqrt [{12}] {2 ^ {7}}} = {\ sqrt [{12}] {128}} \ approx 1 {,} 498307}$ 700 cents ${\ displaystyle {\ tfrac {3} {2}} = 1 {,} 5}$ 701.96 cents −1.96 cents
Small sixth ${\ displaystyle {\ sqrt [{12}] {2 ^ {8}}} = {\ sqrt [{3}] {4}} \ approx 1 {,} 587401}$ 800 cents ${\ displaystyle {\ tfrac {8} {5}} = 1 {,} 6}$ 813.69 cents −13.69 cents
Major sixth ${\ displaystyle {\ sqrt [{12}] {2 ^ {9}}} = {\ sqrt [{4}] {8}} \ approx 1 {,} 681793}$ 900 cents ${\ displaystyle {\ tfrac {5} {3}} \ approx 1 {,} 666667}$ 884.36 cents 15.64 cents
Minor seventh ${\ displaystyle {\ sqrt [{12}] {2 ^ {10}}} = {\ sqrt [{6}] {32}} \ approx 1 {,} 781797}$ 1000 cents ${\ displaystyle {\ tfrac {16} {9}} \ approx 1 {,} 777778}$
${\ displaystyle {\ tfrac {9} {5}} = 1 {,} 8}$
996.09 cents
1017.60 cents
3.91 cents-
17.60 cents
Major seventh ${\ displaystyle {\ sqrt [{12}] {2 ^ {11}}} = {\ sqrt [{12}] {2048}} \ approx 1 {,} 887749}$ 1100 cents ${\ displaystyle {\ tfrac {15} {8}} = 1 {,} 875}$ 1088.27 cents 11.73 cents
octave ${\ displaystyle {\ sqrt [{12}] {2 ^ {12}}} = 2}$ 1200 cents ${\ displaystyle {\ tfrac {2} {1}} = 2}$ 1200 cents 0 cents
Remarks:
• If the difference is negative, the interval with the same temperature is narrower than the pure one.
• * Tritone ( excessive fourth ), defined as: major third (frequency ratio 54 ) plus major second (frequency ratio 98 ) = fifth (frequency ratio 32 ) minus diatonic semitone (frequency ratio 1615 ). The excessive fourth (for example C-F sharp or G flat-C, frequency ratio 4532 corresponding to 590 cents) is smaller in pure tuning than the diminished fifth (for example F sharp-C or C-G flat, frequency ratio 6445 corresponding to 610 cents) . In equal tuning, however, both are equal to half an octave (600 cents).
• Comment on the major second and minor seventh: In pure tuning there are two whole tones with the frequency ratios 98 and 109 . Accordingly, there are two small sevenths with the frequency ratios 2:  98 = 169 and 2:  109 = 95 .

## Special forms

The division of the octave into twelve tones with the same frequency ratio to their neighboring tones is the most common in Western systems today, but not the only way to approach pure intervals . Better approximations can be achieved with more notes per octave. Equal classifications that have actually been used are e.g. B .:

In the new music of the 20th and 21st centuries experimented with numerous equal (and other) tone systems, with the octave being divided into approximately 17, 19, 31, 53, 72 equal steps.

Occasionally, intervals other than the octave are also divided. So z. B. Karlheinz Stockhausen for his electronic study II from 1952 a sound system based on the division of an interval with the frequency ratio 5/1 into 25 equal steps. Since the distance between the steps is slightly larger than the traditional tempered semitone , a tone system is created that is suitable for creating (inharmonic) tone mixtures .

## literature

• Mark Lindley: Mood and Temperature . In: Frieder Zaminer (Ed.): History of Music Theory , Volume 6: Listening, measuring and computing in the early modern times . Darmstadt 1987, pp. 109-332
• Ross W. Duffin: How Equal Temperament Ruined Harmony (And Why You Should Care) . WW Norton & Company, New York / London 2007 ( excerpt )
• Andreas Werckmeister : Musical Paradoxal Discourse. Calvisius, Quedlinburg 1707, Digitale-sammlungen.de

4. When adding (subtracting, multiplying) intervals, the corresponding frequency ratios are multiplied (divided, raised to the power). The octave has the frequency ratio of 2, the halftone = 1 / 12 octave according to the frequency ratio${\ displaystyle 2 ^ {\ frac {1} {12}} = {\ sqrt [{12}] {2}}.}$