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INTERVALS WITH THE TONIC

The frequency of tonic C (264 Hz) divided by that of C’ (528 Hz) is ½. The inverse ratio, the frequency of C’ divided by that of C is equal to 2/1 or 2. These intervals are called

octaves. The note that creates an octave interval with the

tonic is also called an octave. The octave note is also called the 8th natural note or the 8th degree. In this book, every

now and then, we will refer to the octave note and the octave interval as semitone 13. The natural notes of a Scale are also known as degrees, 1st, 2nd, 3rd and so on.

The frequency of note C’ is equal to the frequency of the 2nd harmonic overtone of note C. The 2nd harmonic overtone is produced by the ½ length (one of the

secondary lengths) of a vibrating string and its frequency is

2x that of note C. Notes C and C’ being octaves produce essentially the same overtones; their timbres and “colors”

are essentially the same. Notes D and D’ are essentially the same notes, as they produce essentially the same overtones; but D’ belongs to the next higher octave, scale or key. The same is true with the other octave intervals (semitone13).

The interval between the tonic note C and the 2nd

natural note D is called a 2nd interval or a Major 2nd or M2 interval.

The interval between the tonic C and the 3rd natural

note E is called a 3rd interval or a Major 3rd or M3.

The interval between the tonic C and the 4th and the

5th natural notes are respectively referred to as Perfect 4

or P4 and Perfect 5th or P5.

The interval between the tonic and the 6th natural note is called a Major 6th or M6. The interval with the 7th natural note is called a Major 7th or M7. After the octave are the 9th, 10th, 11th, 12th, 13th intervals.

The harmonic overtones produced on the secondary

lengths (½, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9 and so on) are

progressively and proportionately increasing in frequency.

Harmony with the tonic however proportionately diminishes with the increase in frequency.

The amplitudes of the waves proportionately

diminish with the secondary lengths. And because the

amplitude determines the volume of the overtones, they are progressively less heard.

Intervals with the tonic are classified into: (This

classification is arranged in descending harmony with the

Perfect Unison Octave Dominant Subdominant Major Mediant Submediant Supertonic Leading Note Accidentals Augmented Diminished Minor PERFECT INTERVALS

Unison and Octave

The most pleasing and harmonious intervals are those from a (1) unison, the superposition of waves whose frequencies are the same, and from (2) octaves, the superposition of waves of the tonic and of waves whose frequencies are its halves or doubles.

The simultaneous playing of notes whose frequencies are an octave apart is practically a unison. Notes that are an octave apart produce practically the same overtones. The prime sign (‘) after the letter note is used to signify a note which is one octave higher, like C’ is one octave higher than C. The notes C, D, E, F, G, A and B will sound essentially the same as the notes C’, D’, E’, F’ G’, A’ and B’ although they are one octave apart.

The overtone created on the ½ length, the 2nd

harmonic overtone creates the octave interval with the tonic.

As mentioned earlier, the term octave also refers to the 8th natural note, like C’ is the octave of C, and the other

way around. Octave is also conveniently used to mean a

group of 8 natural notes. The group of the notes namely C,

C#, D, D#, E, F, F#, G, G#, A, A#, B and C’ has 13 notes. Although it has 13 notes, this group of notes is still called an

octave, because it has only 8 designated natural notes.

Dominant

The overtone that creates the next most pleasing interval with the tonic is the 3rd harmonic overtone. The 1/3

secondary length produces this. Its waves dominate the

them as to loudness and as to influence in determining the final waveform of the tonic. The 1/2 partial of this overtone is assigned to the dominant and 5th natural note of the major

diatonic scale of its tonic.

The frequencies of the notes within an octave are equal to the partials of the frequencies of the harmonic overtones of the tonic.

The range of frequencies of the notes in an octave starts with the frequency of the tonic and ends with the frequency of the 8th natural note (the octave note, semitone 13, the 2nd harmonic overtone or the overtone produced by the ½ secondary length). In our reference scale the Key of C, the frequencies of the notes start with 264 Hz (tonic C) and ends with 528 Hz (note C’, the 8th natural note, the octave note of C, semitone 13 or frequency of the 2nd harmonic overtone of tonic C).

The frequency of the 3rd harmonic overtone of note

C is 264 x 3 = 792 Hz. This is beyond the 264 to 528 Hz range. If we divide 792 by 2 we get the frequency 396 Hz. This frequency is assigned to note G, the designated 5th

natural and dominant note of this reference scale, the Key

of C. As we shall see farther, the frequencies of the other notes in this scale, natural and accidental, are portions/parts

of or partials of the frequencies of the harmonic overtones produced by the secondary lengths.

The interval that the dominant note G creates with its tonic (note C) is called a 5th interval because the assigned

note to it is the 5th natural note of the C Scale. This interval

is also known as Perfect 5th or P5. The interval is 8

semitones. The frequency of note G is 264 x 3/2 = 396

Hertz.

Subdominant

Another perfect interval with the tonic is the Perfect

4th (P4). The interval it creates with the tonic is less

harmonious than that created by P5. The interval is 6

semitones. Its frequency is a partial of the 4th harmonic

overtone. The 4th harmonic overtone is produced by the ¼ length of a vibrating string. !/3 of this frequency is assigned to the 4th natural note of the diatonic scale of its tonic.

The 4th harmonic overtone of tonic C is 264 x 4 = 1056 Hz. 1/3 of 1056 is 352 Hz. This frequency is assigned to note F, the designated 4th natural note of the Key of C, In

the Key of C, note F is a natural note below the dominant note G. Note F is less dominant to and less harmonious with tonic C. Note F is consequently called the subdominant of

the Key of C

MAJOR INTERVALS

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