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Octave and its Temperament (and Cents)

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Chapter - 5A
Octaves Temperaments

An octave of a note is a note that has a frequency (pitch) which is either 1/2, 1/4, 1/8, etc or twice, 4 times, 8 times, etc that of that note. To put it into an equation - for a note of frequency F Hz , an octave note is one which has frequency F1 Hz such that,

F1 = 2nF
where n is a whole number between +/- infinity excluding zero.

The "lower octaves" are the ones with a negative n, while
The "higher octaves" are the ones with a positive n.
A note with n = 0, or having the same frequency, but produced from different strings (sources) are referred to as "unison" and often written as P8.

Thus an octave is a harmonic (covered earlier) but all harmonics are not octaves.

The "higher octave" that is one higher (double the frequency) of an indicated note is sometimes written as 8va, and the octave below 8vb. A 15va is a octave with 4 times the frequency of the fundamental - 2nd higher octave. An octave is sometimes written as P8 as well.

An octave is important in music for the following reasons:
(i) It is the simplest ratio (1:4, 1:2, 2:1, 4:1,...) between two notes beyond the unison (1:1).

(ii) To the ear, octave frequencies sound the same. So any two notes that are octaves of each other are indistinguishable to the ear. The ear perceives the difference in the frequency merely as how "sharp" or "loud" or "crisp" the sound is. This is called " octave equivalency." And because it is so -

(iii) Octaves in music notation are written using the same symbols. e.g.: The middle C (written as "C4" but different in meaning from that in music sheet or chords) is 262 Hz. Its octaves C3 (131 Hz) and C5 (524 Hz) are all written, in general music notation as "C" only.

(iv) They function as landmarks in the otherwise ratio based music scaling system.

Division of the Octave and Introduction to Scales:
(There are many articles on Wikipedia that cover these topics in cast detail, and so I will restrict my discussion to the working principles only.)

The octave, in modern music, is divided into "12 half notes" so that it 7 full steps and 5 half steps to get from one named note to the same note in the octave above or below.

These 12 notes are what make up the "Diatonic Scale" when written in order from any named note in cyclical order up to the named note before it e.g. A to G# and Bb to B. - (discussed earlier).

Choosing specific notes from this diatonic scale, various other scales are produced.

The system of dividing of the octave into 12 notes is referred to as the "Twelve - Tone Equal Temperament (12-TET)" which is the standard for almost all the music we hear.
Equal temperament is a system of tuning in which every pair of adjacent notes has an identical frequency ratio. In equal temperament tunings an octave is divided into a series of equal "steps" i.e. equal frequency ratios. The reason why a geometric division is preferred to a geometric division is -
(i) The human ear distinguishes sound frequencies in a geometric way - ratios are better distinguished than the arithmetic difference.
(ii) It allows the music composed, to be played in any octave ("Transposition").

The basis of this division is that the notes produced in dividing the octave in this way sound good together - and that's what music is all about. There are other systems that divide the octave into different parts into more than 12 parts - upto 171 parts (171-TET). These are used in "Microtonal music."

Before 12-TET came into being, another ratio system was used - "Just Intonation" where simple ratios were used that sounded good. The 12-TET system sounds a lot like that system but it is a lot more mathematically sound.

So, the result of such a division is that we have 11 steps between two consecutive octaves, the 12to one being the octave itself.

The Cent System:
The 12-TET division of the octave is done geometrically. Thus we end up with a regular geometric progression of frequencies. The frequency ratios between any two consecutive notes so obtained is defined as "100 cents." This means that a cent is precisely equal to 21/1200 - the 1200th root of the number 2. Thus an octave - the ratio of a frequency with one double of is 2 = 1200 cents.

An octave interval = an interval of 1200 cents

Thus consecutive half steps are 100 cents apart and consecutive full steps are 200 cents apart.

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