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 more so, the less radical temperaments, were accused:

1. 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!).
2. 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:

${\displaystyle {\textrm {1\ semitono\ temperato}}={\sqrt[{12}]{2}}}$,

which is exactly equal to 100 cents. The results obtained are illustrated in the following table:

Equal Temperament

Notes	MIDI Number	Ratio	Frequency (Hz)	Cents
C3	60	${\displaystyle 1\;}$	261.6	0
C♯ or D♭	61	${\displaystyle {\sqrt[{12}]{2}}}$	277.2	100
D	62	${\displaystyle {\sqrt[{12}]{2^{2}}}}$	293.7	200
D♯ or E♭	63	${\displaystyle {\sqrt[{12}]{2^{3}}}}$	311.1	300
E	64	${\displaystyle {\sqrt[{12}]{2^{4}}}}$	329.6	400
F	65	${\displaystyle {\sqrt[{12}]{2^{5}}}}$	349.2	500
F♯ or G♭	66	${\displaystyle {\sqrt[{12}]{2^{6}}}}$	370.0	600
G	67	${\displaystyle {\sqrt[{12}]{2^{7}}}}$	392.0	700
G♯ or A♭	68	${\displaystyle {\sqrt[{12}]{2^{8}}}}$	415.3	800
A	69	${\displaystyle {\sqrt[{12}]{2^{9}}}}$	440.0	900
A♯ or B♭	70	${\displaystyle {\sqrt[{12}]{2^{10}}}}$	466.2	1000
B	71	${\displaystyle {\sqrt[{12}]{2^{11}}}}$	493.9	1100
C4	72	${\displaystyle 2\;}$	523.2	1200

Advantages

The advantages are obviously linked to the reasons that led to the construction of equal temperament:

1. 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;
2. fixed-frequency instruments play equally well in all tonalities;
3. 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.

Amplitude of the intervals (in cents) in the various scales and relative corrections with regards to the natural scale

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:

• There are a minimum number of corrections; 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 exists for the Pythagorean scale, albeit in the opposite direction; especially for the major thirds that make up a harmonic with a relatively low order number (E₆ is both the fifth harmonic of C ₄ and the fourth harmonic of E₄) 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 there are a minimum number of corrections introduced by temperament, they are perceptible, even to a non-musically trained ear; particularly during the execution of a musical passage.

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