Temperamento equabile en
From "Physics, Waves, Music": a website about the physics of waves and sound, acoustics of musical instruments, musical scales, harmony and music.
Why construct an equal temperament scale?
On the pages about the Pythagorean scale and natural scales, we illustrated the problems introduced by these scales; especially concerning the problem of tonality change. From a strictly mathematical point of view, all problems stem from not having divided the octave into "equal parts". This fact makes the scales non invariant when their tonics are shifted. Notwithstanding the apparent simplicity of the solution that leads to equal temperament (i.e. divide the scale in semitones of the same amplitude), the creation of this type of temperament was a historically complex process that was extremely gradual and full of "second thoughts". The need to correct the "disuniformity" of the Pythagorean and natural scales slowly created [1] the need to use temperate scales, in which the (temperament) corrections introduced were the most varied and also much less radical than fully adopting equal temperament. The advantage introduced by equal temperament (i.e. the possibility to change tonality, which is technically called "modulation") became of value only within tonal harmony. Modulation allows us to exploit harmonic ratios between chords to produce new combinations and helps composers enrich melodies, give them more colours and tones, exalt particular passages, etc. (for a review on musical aspects, see the pages From sound to music, Consonance and dissonance and Psychoacoustic and musical aspects).
During the debate surrounding their adoption, equal temperament and, even before it, the less radical temperaments were accused:
- From a theoretical point of view, of shifting away from the simplicity of Pythagorean frequency ratios (in effect, we will see how the ratio that expresses the temperate semitone is actually an irrational number in equal temperament!).
- From a musician's point of view:
- of being to the detriment of harmony by tweaking the natural consonance of the intervals of the fifth, fourth and third;
- of introducing an excessive mechanicality removing the particular colouring of each tonality (Bach's "structural" genius is to be thanked for the understanding that the possibility of transposition from one tonality allows us to enrich music with elements of order, symmetry and formal transparency).
How is the temperate scale constructed?
Before proceeding with the construction of the temperate (or equal temperament) scale, we suggest you read the page How is a musical scale constructed? in order to understand the (minimum) notions for determining the amplitude of intervals and measuring them in the logarithmic scale.
- Obviously, considering the premise, a scale constructed on equal temperament is obtained by dividing the octave into twelve equal parts;
- Since the octave has a ratio of 2:1 and the frequencies are multiplied (not added), the smallest interval will be the one that, when multiplied by itself 12 times (i.e. raised to the 12th power), gives 2. This is the temperate semitone:
- ,
which is exactly equal to 100 cents (the cent scale divides the octave into 1200 parts precisely so that each tempered semitone measures 100 cents).
- Why twelve parts?
- Because twelve perfect fifths are almost equal to seven octaves: (3/2)12 ≈ 129.7 is close to 27 = 128 (the difference is the Pythagorean comma). Dividing the octave into 12 equal semitones, seven semitones give a ratio 27/12 ≈ 1.4983, which differs from the perfect fifth 3/2 by only 2 cents: the scale therefore contains almost perfect fifths and fourths. With divisions of the octave into a small number of parts other than 12 (for example 10 or 11) there is no interval so close to the perfect fifth.
The results obtained are illustrated in the following table:
| Notes | MIDI Number | Ratio | Frequency (Hz) | Cents |
|---|---|---|---|---|
| C4 | 60 | 261.6 | 0 | |
| C♯ or D♭ | 61 | 277.2 | 100 | |
| D | 62 | 293.7 | 200 | |
| D♯ or E♭ | 63 | 311.1 | 300 | |
| E | 64 | 329.6 | 400 | |
| F | 65 | 349.2 | 500 | |
| F♯ or G♭ | 66 | 370.0 | 600 | |
| G | 67 | 392.0 | 700 | |
| G♯ or A♭ | 68 | 415.3 | 800 | |
| A | 69 | 440.0 | 900 | |
| A♯ or B♭ | 70 | 466.2 | 1000 | |
| B | 71 | 493.9 | 1100 | |
| C5 | 72 | 523.3 | 1200 |
Advantages
The advantages are obviously linked to the reasons that led to the construction of equal temperament:
- the intonation of a piece of music is independent of the tonality in which it is executed; i.e. of the note that is chosen as the base of the scale. Therefore, a piece can be transposed to another tonality without having to retune the instruments;
- fixed-frequency instruments play equally well in all tonalities;
- enharmonic notes coincide (e.g. C♯ and D♭), thus simplifying the construction of musical instruments. The black keys of the piano play both C♯ and D♭.
Disadvantages
Paradoxically, the second advantage above can also be restated as a disadvantage: fixed-frequency instruments play equally badly in all tonalities. Moreover, while there are always perfectly consonant intervals in the natural scale, when equal temperament is adopted, these intervals do not exist regardless of what tonality the music is being played in. The following table presents corrections with regards to perfectly consonant intervals.
| Notes | Temperate | Natural | Pythagorean | Temperate-natural difference |
|---|---|---|---|---|
| C | 0.000 | 0.000 | 0.000 | 0.000 |
| C♯ or D♭ | 100.000 | 111.731 | 113.685 | -11.731 |
| D | 200.000 | 203.910 | 203.910 | -3.910 |
| D♯ or E♭ | 300.000 | 315.641 | 294.135 | -15.641 |
| E | 400.000 | 386.314 | 407.820 | +13.686 |
| F | 500.000 | 498.045 | 498.045 | +1.955 |
| F♯ or G♭ | 600.000 | 590.224 | 611.730 | +9.776 |
| G | 700.000 | 701.955 | 701.955 | -1.955 |
| G♯ or A♭ | 800.000 | 813.686 | 815.640 | -13.686 |
| A | 900.000 | 884.359 | 905.865 | +15.641 |
| A♯ or B♭ | 1000.000 | 1017.596 | 996.090 | -17.596 |
| B | 1100.000 | 1088.269 | 1109.775 | +11.731 |
| C | 1200.000 | 1200.000 | 1200.000 | 0.000 |
We immediately observe that:
- The corrections are minimal, especially for the intervals of the perfect fourth (+1.955) and perfect fifth (-1.955), which are the foundation of consonance.
- The corrections are important for the minor thirds that are flat (-15.641) and major thirds that are sharp (+13.686) with respect to the corresponding natural intervals. The same problem, even more marked and in the same direction, exists for the Pythagorean scale (-21.506 for the minor third and +21.506 for the major third). This matters especially for the major thirds that make up a harmonic with a relatively low order number (E7 is both the fifth harmonic of C5 and the fourth harmonic of E5) in composite sounds. As we have already explained with regards to the Pythagorean scale, this can lead to the unpleasant phenomenon of beats. In the playing of music, when performers who play non-fixed-frequency instruments (e.g. violin) are accompanied by a fixed-frequency instrument (e.g. piano or harpsichord), they introduce the necessary corrections to eliminate these beats in real time while playing.
To listen to the intervals of the temperate scale and compare them with intervals of other scales, see Comparison of scales. You will notice that, although the corrections introduced by temperament are minimal, they are perceptible, even to a non-musically trained ear; particularly during the execution of a musical passage.
Check your understanding
What is the frequency ratio of a tempered semitone? If A is at 440 Hz, at what frequency is A♯?
Answer. ≈ 1.0595, that is, 100 cents. A♯ is at 440 · 1.0595 ≈ 466.2 Hz.In equal temperament C4 = 261.6 Hz and G4 = 392.0 Hz. Do the third harmonic of C and the second of G beat? And the fifth harmonic of C with the fourth of E4 (329.6 Hz)?
Answer. The third harmonic of C is at 784.9 Hz, the second of G at 784.0 Hz: slow beats, less than 1 per second, almost imperceptible. The fifth harmonic of C is at 1308 Hz, the fourth of E at 1318.5 Hz: about 10 beats per second. This is why tempered thirds (about 14 cents sharp) sound less pure than fifths (2 cents flat).Why, on a piano tuned in equal temperament, does the same black key serve for both C♯ and D♭?
Answer. Because in equal temperament enharmonic notes coincide: C♯ and D♭ are both one tempered semitone (100 cents) above C. In the Pythagorean and natural scales they are different notes.What is gained with equal temperament and what is the price?
Answer. Every key has exactly the same intervals, so you can transpose and modulate without retuning fixed-pitch instruments. The price is that, apart from the octave, no interval is perfectly consonant: fifths and fourths are out of tune by about 2 cents, thirds by 14-16 cents.Further reading and links
- Related questions and answers:
- ↑ the final affirmation of the use of these scales dates back to the publication of Musikalische Temperatur by Andreas Werckmeister in 1691. However, in the absence of precise methods of measurement of frequencies, a variety of different temperaments were possible; at least until 1834, when Johann Heinrich Scheibler proposed a tuning method based on a set of tuning forks with precisely measured frequencies (the tuning fork had been invented by John Shore in 1711).
