In this article I am going to speak about musical notes (A, B, C, D, E, F, G). In particular, I am going to deepen their correlation with the physical quantity which adjusts the pitch of them: the frequency, which is measured in hertz (1 hz = 1 s-1).

For those who haven't clear ideas about what is the frequency of the sound, it's useful knowing that the sound is a wave that is periodic and the frequency is equal to the number of periods that are repeated in one second (in other words, is the reciprocal the period (T): f = 1 / T)

First of all, some knowledge about the notes: there are 7 natural notes (C, D, E, F, G, A, B or, for some countries, respectively, Do (or Ut), Re, Mi, Fa, Sol, La, Si) in addition to 5 altered notes (C# or Db, D# or Eb, F# or Gb, G# or Ab, A# or Bb). These 12 notes in ascending sequence (which, in theory, could be repeated indefinitely both downwards and upwards) are:

C C# D Eb E F F# G Ab A Bb B (C C# ... ) (1)

Each note of this frequency is distant both from the previous one both from the next one by a semitone.

Explanation: Putting the symbol # (sharp or diesis) after a note means increasing it by one semitone; putting the symbol b (flat or bemolle), instead, means decreasing it by one semitone. Moreover, the only intervals of consecutive natural notes whose value is one semitone are E - F and B - C, the others are one tone, that is 2 semitones. Now it should be clear why the above-mentioned 12 notes are arranged just like in the (1) and because, for example, the C# and the Db are the same note.
An octave is the smallest distance between notes of the same name, that is the distance between the extremes of a scale of 8 -hence derives the name- natural notes (C, D, E, F, G, A, B, C); it, therefore, is equal to 12 semitones.

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Musical notes that are in different octaves but that have the same name too, have acoustically something in common and everyone can feel it (to understand this, listen the following two recordings. In the first there are random notes; in the second the notes are all an A)

From the physical point of view, two notes whose distance is an octave (or, equivalently, 12 semitones) have a ratio between the respective frequencies equal to 2. Moreover, the ratio between the frequencies of two musical notes whose distance is one semitone is constant; in other words, the respective frequencies of the notes of the sequence (1) are in a geometric progression. Once established conventionally a frequency to a certain note, it is possible to find the frequencies of all the notes.
In fact, you can get the common ratio q the geometric series; the ratio of the frequencies between two notes away 12 semitones is 2, then:

q12 = 2 → q = 21/12 (2)

Therefore, calling f0 the frequency of the reference note and n the number of semitones of distance between the note of which we want to determine the frequency f and the reference one (n is positive if the note with the unknown frequency is more acute than the reference, n is negative if it is deeper) you will have that:

f = f0 - qn = f0 - 2n / 12 (3)

or, if you want to determine to which note corresponds a certain frequency f:

n = 12 - log2(f / f0) (4)

Conventionally, as a reference note has been chosen the A medium or A4 (the number after the note indicates in which octave it is. -4- characterizes the -medium- note; the number of the octave changes in the B - C passage), which has frequency f0 = 440 hz.

Examples

Frequency of the medium C or C4. Using the (3):

f = 440 hz - 2-9/12 = 440 hz / 23/4 ≈ 261.2 hz

Notes of a sound of 1000 hz. Using the (4):

n = 12 - log2(1000 / 440) + 14.21

Thus, it's a bit ascending B5.

Observations

  • The notes that are included in the whole range of the orchestra are 88 (like the piano keys - see the picture below -. Indeed, the piano can produce all the notes) and begin from the A0 and end to the C8. Frequencies in this range go from the 27.5 Hz to (with very good approximation) 4186 hz. whole_tastiera
    In this picture the white numbers represent the octaves
  • The range of the audible frequencies by humans is from 16-20 Hz to 16,000-20,000 Hz. So, more or less, we do not hear under the C0 and above the C#10.
  • It's interesting to noticing the existence of some very good approximations that apply well to the calculation of the frequencies of the notes:
    21/6 ≈ 55/49 (the former is 1.1224620... and the latter is 1.1224489...)
    As a result, for example, if the frequency of the G4 (which would be 440 hz / 21/6 ≈ 440 Hz / (55/49) = 392 hz) was approximated exactly to 392 hz (and G3 to 196 hz, the G2 to 98 hz, etc ...) an error of less than 0.0012% would be committed;
    21/4 ≈ 44/37 (the former is 1.1892071 ... and the latter is 1.1891891 ...)
    As a result, for example, if the frequency of F' (which would be 440 hz / 21/4 ≈ 440hz / (44/37) = 370 hz) was approximated exactly to 370 hz (and the F" to 185 hz, etc ...) an error of less than 0.00151% would be committed;
    21/4 ≈ 2093/1760 (the former is 1.1892071 ... and the latter is 1.1892045 ...)
    As a result, for example, if the frequency of the C7 (which would be 440 hz - 227/12 = 440 Hz - 22 - 21/4 ≈ 1760hz - 2093/1760 = 2093 Hz) was approximated exactly to 2093 hz (and the C6 to 1046.5 hz, the C8 to 4186 hz, etc ... ) an error of less than 0.00022% would be committed.

    Moreover, notes which simultaneously played "sound good" have precise ratios of frequency that, in the case of the major chord (for example: C - E - G, that it is represented in the recording below), can be approximated (though without the excellent accuracy of the previous cases, but with an acceptable one) to simple fractions (in the following list the first note is always higher than the second):
    • G and C (7 semitones of difference) have a relationship of frequencies near 3/2 (≈ 27/12);
    • C and G (5 semitones of difference) to 4/3 (≈ 25/12);
    • E and C (4 semitones of difference) to 5/4 (≈ 24/12 = 21/3);
    • G and E (3 semitones of difference) to 6/5 (≈ 23/12 = 21/4).
    Note, also, that the following sequence (which is very catchy, listen the recording below):
    C2 C3 G3 C4 E4 G4
    represents a sequence of frequencies approximately in the ratio of 1:2:3:4:5:6. Ultimately, errors greater than 0.2% won't be committed if, for example, we consider the E4 as a note of 330 hz (3/4 of A4 which is 4 semitones more acute) or the C#5 as 555 Hz (the 3/2 of F' that, as said before, we can say that has a frequency of 370 hz).
  • With the parity of all the characteristics (material, gauge, etc ...) of longitudinal objects that put in resonance produce a note (e.g., a string, a pipe of an organ, etc ...), the length is inversely proportional to frequency. Therefore, q = 21/12 appears in some musical instruments. For example, the twelfth fret of the neck of a guitar is right in the middle of the strings. More generally, the distances of the frets of the guitar from the end of the string, that is, from the bridge, are in geometric progression with the common ratio equal to 21/12 as the bars of this histogram which represents the trend of the frequencies (or, seen on the contrary, the lengths of the above-mentioned objects) as a function of the note. Furthermore, a figure like this can be found in the strings of a piano, and it's what gives the shape of the instrument if it isn't an upright piano.

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