Most modern synths derive their tuning from an accurate crystal reference — unfortunately not all musical instruments are that easy!
Tones and Semitones
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The impact of music is determined by the relationship of successive notes, rather than absolute pitch. Pitch is subjective but frequency (f) is the measured rate at which a cyclic oscillation reaches the same point in its period. The latter is measured in cycles per second (c/s) or hertz (Hz). As frequency increases in a geometric progression (2f, 4f, 8f, etc) so pitch is perceived as increasing in an arithmetic manner (1 octave, 2 octaves, 3 octaves, etc).
For a sinusoidal waveforms the measured frequency and subjective pitch correspond — in such cases an electronic tuning meter can be used. But other waveforms contain harmonics (also known as upper partials) or noise which may cause an apparent discrepancy between frequency and pitch.
Octaves are far too large to be used as a ‘unit’ of pitch. Hence each octave must be divided up into intervals which are repeatable and musically pleasing.
Pythagoras and his Greek contemporaries of the 4th century BC divided the octave into seven intervals to form a scale. The notes which marked the boundaries of the these intervals were given the letters A, B, C, D, E and F.
 
The basic interval (the spacing between the notes) is the whole-tone or tone. Any scale which only uses these intervals is called a whole-tone scale. Unfortunately such a scale in which all steps are equal is not particularly pleasing. Hence some of the seven intervals are reduced to half a tone, also known as a semi-tone.
If the note of A is raised by a semi-tone it becomes A sharp (A#). Similarly if lowered by a semi-tone it becomes A flat (Ab). Notes which are neither sharp nor flat are called natural.
Modes
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The earliest musical mode used by the Greeks was based on the four notes of D, E, F and G. It included two semi-tone intervals which added character to the music.
Later modes. widely used in medieval music, gave the same results whatever note began the scale. This allowed instruments that were tuned differently to play together. Since there was no absolute ‘tuning reference’ all instruments were tuned by the beat-less fifths method — which allowed the musician to compare the pitch of one note with another.
In modern music the starting note for a scale is called the key. The tedious process of converting music or musical script from one key to another is called transposition.
Ionian mode represented the white notes on a piano, from C to C, and is equivalent to the modern major scale. It features semi-tone intervals between the 3rd and 4th white notes and also between the 7th and 8th notes.
 
Aeolian mode also used the white notes, this time from A to A, and is equivalent to the modern minor scale. It features semi-tone intervals between the 2nd and 3rd notes and also between the 6th and 7th notes. In the melodic minor scale both the 6th and 7th notes are sharpened. In the harmonic minor scale only the 7th note is sharpened.
 
Other Greek modes were also based on the white notes. These included Dorian (D to D), Phrygian (E to E), Lydian (F to F) and Mixolydian (G to G).
Scales
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By the end of the 17th century the modes had been replaced by modern scales. These include:-
 
Tonic Sol-fa
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This form of scale notation was introduced to aid ‘sight reading’ of music. It does not assume that sharps (fe, se, le etc) produced in an ascending scale are the same as the flats (ta, la, ba etc) produced in a descending scale. They may be the same — but with real instruments using real scales they may be slightly different!
 
Changing the key will of course shift the starting point in relation to the tonic scale — doh may be flat or sharp!
Intervals
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Apart from tones and semi-tones there are a number of other intervals used in musical terminology. Some of these are shown below, in this instance for the key of C major.
 
The 4th and 5th intervals are considered perfect because of the way they fit exactly within the octave. In a major scale the 3rd interval will be a major interval (5 inclusive half notes) whilst in a minor scale it will be a minor interval (4 inclusive half notes).
 
Any perfect or minor interval that is flattened by a semitone is diminished.
Any perfect or major interval that is sharpened by a semitone is augmented.
The Frequencies of Musical Notes
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Natural Interval Tuning
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Traditionally many instruments are tuned for natural intervals by means of a beat-less ‘fifth’ (a ratio of 2:3). This can be shown using the frequencies of 200 and 300 Hz . If the 300 Hz fundamental goes ‘out of tune’ to 300.5 Hz its 2nd harmonic will be 601 Hz. This will beat with the 3rd harmonic of 200 Hz to give a resultant of 1 Hz — or ten ‘beats’ in ten seconds.
The tuning procedure begins by setting A (the one above middle C) to the standard A of 440 Hz. You then proceed along the list using pairs of notes
down the scale in fifths
and
up in the scale in octaves
in the following sequence:-
A, D, D', G, C, C', F, F', A#, D#, D#', G#, C#, C#', F#, F#', B, E, E', A
At the end of this process A (the starting point) will come out at around
434 Hz! This considerable difference is known as the comma of Pythagoras.
Natural intervals are more widely spaced (sharper) than tempered intervals. So an instrument tuned in one key will be badly out of tune in another — by as much as a semi-tone.
Tempered Interval Tuning
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Modern keyboard music is based on the equal tempered scale, a system particularly favoured by Bach. It has the advantage of giving consistent results, whatever key is chosen. The 12 intervals are all in the ratio of 1:12 ‚àö2.
In tempered tuning only A, and its octaves, are tuned exactly. The comma of Pythagoras is then absorbed by spreading it evenly across each octave. This means that all the semi-tone intervals within the octave are slightly flat.
The differences between natural and tempered scales are particularly noticeable when stringed instruments play with a piano.
Tempered tuning uses the beats method — a variation of beat-less fifths. As before, the procedure begins by tuning A to standard A at 440 Hz. Then A' is tuned to 880 Hz in relation to A. Then D (a fifth below A) is tuned — it is first set flat and then advanced past the beat-less point until the required beat is obtained.
Adjustments should give:-
 
„ An error of 4 to 5 beats on the final E to A interval is normal.
„ Five beats of inaccuracy in 10 seconds means an error of 0.5 Hz in the
second harmonic — or just 0.25 Hz in the fundamental.
„ Some countries may use 442 Hz or even 444 Hz as standard A.
