Afinação do piano em conSerto por Mauro Carmo da Silva
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viu, não deu certo por isso
uma corda aqui eu não mexi
quer ver como você vai sentir a diferença?
agora bata lá outra vez
pode bate ó
essa deu certo
a outra tava baixa
aqui você sente a onda
a outra tava baixa
essa deu
e ela tava assim
por isso que não ia dar essa aqui. ó, bata mais uma vez
agora tem que erguer isso aqui ó
essa diferença aqui ó, quer ver?
vamos ter que mexer inteira
tudinho
agora vai parar
440 ó
The “Just Scale” (sometimes referred to as “harmonic tuning” or “Helmholtz’s scale”) occurs naturally as a result of the overtone series for simple systems such as vibrating strings or air columns. All the notes in the scale are related by rational numbers. Unfortunately, with Just tuning, the tuning depends on the scale you are using – the tuning for C Major is not the same as for D Major, for example. Just tuning is often used by ensembles (such as for choral or orchestra works) as the players match pitch with each other “by ear.”
The “Just Scale” (sometimes referred to as “harmonic tuning” or “Helmholtz’s scale”) occurs naturally as a result of the overtone series for simple systems such as vibrating strings or air columns. All the notes in the scale are related by rational numbers. Unfortunately, with Just tuning, the tuning depends on the scale you are using – the tuning for C Major is not the same as for D Major, for example. Just tuning is often used by ensembles (such as for choral or orchestra works) as the players match pitch with each other “by ear.”
The “equal tempered scale” was developed for keyboard instruments, such as the piano, so that they could be played equally well (or badly) in any key. It is a compromise tuning scheme. The equal tempered system uses a constant frequency multiple between the notes of the chromatic scale. Hence, playing in any key sounds equally good (or bad, depending on your point of view).
There are other temperaments which have been put forth over the years, such as the Pythagorean scale, the Mean-tone scale, and the Werckmeister scale. For more information on these you might consult “The Physics of Sound,” by R. E. Berg and D. G. Stork (Prentice Hall, NJ, 1995).
The table below shows the frequency ratios for notes tuned in the Just and Equal temperament scales. For the equal temperament scale, the frequency of each note in the chromatic scale is related to the frequency of the notes next to it by a factor of the twelfth root of 2 (1.0594630944….). For the Just scale, the notes are related to the fundamental by rational numbers and the semitones are not equally spaced. The most pleasing sounds to the ear are usually combinations of notes related by ratios of small integers, such as the fifth (3/2) or third (5/4). The Just scale is constructed based on the octave and an attempt to have as many of these “nice” intervals as possible. In contrast, one can create scales in other ways, such as a scale based on the fifth only.
Interval Ratio to Fundamental
Just Scale Ratio to Fundamental
Equal Temperament
Unison 1.0000 1.0000
Minor Second 25/24 = 1.0417 1.05946
Major Second 9/8 = 1.1250 1.12246
Minor Third 6/5 = 1.2000 1.18921
Major Third 5/4 = 1.2500 1.25992
Fourth 4/3 = 1.3333 1.33483
Diminished Fifth 45/32 = 1.4063 1.41421
Fifth 3/2 = 1.5000 1.49831
Minor Sixth 8/5 = 1.6000 1.58740
Major Sixth 5/3 = 1.6667 1.68179
Minor Seventh 9/5 = 1.8000 1.78180
Major Seventh 15/8 = 1.8750 1.88775
Octave 2.0000 2.0000
You will note that the most “pleasing” musical intervals above are those which have a frequency ratio of relatively small integers. Some authors have slightly different ratios for some of these intervals, and the Just scale actually defines more notes than we usually use. For example, the “augmented fourth” and “diminished fifth,” which are assumed to be the same in the table, are actually not the same.
The set of 12 notes above (plus all notes related by octaves) form the chromatic scale. The Pentatonic (5-note) scales are formed using a subset of five of these notes. The common western scales include seven of these notes, and Chords are formed using combinations of these notes.
As an example, the chart below shows the frequencies of the notes (in Hz) for C Major, starting on middle C (C4), for just and equal temperament. For the purposes of this chart, it is assumed that C4 = 261.63 Hz is used for both (this gives A4 = 440 Hz for the equal tempered scale).
Note
Just Scale Equal
Temperament Difference
C4 261.63 261.63 0
C4# 272.54 277.18 +4.64
D4 294.33 293.66 -0.67
E4b 313.96 311.13 -2.84
E4 327.03 329.63 +2.60
F4 348.83 349.23 +0.40
F4# 367.92 369.99 +2.07
G4 392.44 392.00 -0.44
A4b 418.60 415.30 -3.30
A4 436.05 440.00 +3.94
B4b 470.93 466.16 -4.77
B4 490.55 493.88 +3.33
C5 523.25 523.25 0
Since your ear can easily hear a difference of less than 1 Hz for sustained notes, differences of several Hz can be quite significant!