The overtone that creates the next most pleasing interval with the tonic is the 5th harmonic overtone. The 1/5
length of a vibrating string produces it. The frequency of the 5th harmonic overtone of note C is 264 x 5 = 1320. The ¼ partial of this frequency is 330 Hz. This frequency falls right in the middle of the frequency of the tonic (264 Hz) and that of the 5th note (396 Hz). This note is consequently called the mediant of the Key of C. This note is designated as the
3rd natural note of the C major diatonic scale. The interval it
creates with the tonic is called a Major 3rd (M3) or 5
semitones. The mediant of the C Scale is note E. Its
frequency is 264 x 5/4 = 330 Hertz.
Submediant
A closely related note to the mediant is the
submediant, because its frequency is also a partial of the
5th harmonic overtone. The submediant is less
harmonious to the tonic than the mediant is, thus the
secondary length of a vibrating string. The frequency of
the submediant is 1/3 of the 5th harmonic overtone. The
submediant of the C Scale is note A. The interval it
creates with the tonic is called a Major 6th (M6) or semitone 10. The frequency of note A is 264 x 5/3 = 440 Hertz.
Let us temporarily stray away from the present topic and go to a subject deemed important and interesting at this point. This information will allow us to use the easily available piano music sheets in playing the same melody, in the same Key, in our selected music instrument, even if the assigned key/pitch of that instrument is not the same as that of the piano. Musical instruments have different assigned/recognized pitches or keys.
In an international music convention, the frequency 440 Hz has been set as the standard frequency for note A of the middle octave on the piano keyboard. But even if this has been set as the international standard, still 264 Hz, the frequency of note C of the piano keyboard, is used as the reference frequency in identifying the pitch of other music instruments.
Almost all Western music instruments can produce the popular Do-Re-Mi sound together with its accidental
notes. The frequency of the sound Do on an Alto Saxophone differs from the frequency of the sound Do on a standard trumpet.
The frequency of the sound Do of an Alto Saxophone is 313.5 Hz. This is the same frequency as that of note Eb on the piano. The Alto Saxophone is accordingly called an Eb instrument. The frequency of the sound Do on a standard trumpet is 231 Hz, same as the frequency of the note Bb of the piano. The standard trumpet is accordingly called a Bb instrument.
In producing the Do-Re-Mi- sound, the fingering (which holes are close or open) is the same for all kinds of saxophones (bass, tenor, alto, soprano), clarinets, flutes and piccolos. This means that if one knows how to play the do-re-mi sound on one of the mentioned music instrument, he/she can play the do-re-mi sound with the same fingering on all the other instruments. They will however differ as to Key or pitch. The key/pitch is determined by frequency of the Do- sound of the instrument, its assigned tonic. If the frequency of the tonic is the same as that of the C of the piano, the pitch/key of that instrument is C. If it is equal to that of note D of the piano its pitch/key is D.
The tonic determines the pitch or key of a Key or Scale. We have the 12 Keys or Scales, C, C# (Db), D, D# (Eb), E, F, F# (Gb), G, G# (Ab), A, A# (Bb) and B depending on the tonic. The other Keys or Scales have the same letter names but they belong to lower or higher octaves. They are signified by the use of small letters or prime signs.
Going back to intervals:
The next most pleasing interval with the tonic is that produced by the 7th harmonic overtone. It is produced by
the 1/7 length a vibrating string. The ¼ partial of this frequency is not assigned to any natural note in its scale. It is assigned to an “accidental note”. Its frequency is a semitone lower than the 7th natural note (semitone 12) and
thus is known as the minor 7th (m7), semitone 11 or note
Bb. Because its frequency is also a semitone higher than the 6th natural note (semitone 10) it is known as Augmented
6th (A6), semitone 11 or note A#.
Note A# and note Bb although called by different names have the same frequencies. They are referred to as enharmonic equivalents. The same is true with C# and Db, D# and Eb, F# and Gb and lastly G# and Ab.
Historically C#, D#, F#, G# and A# had lower frequencies than their flat equivalents until it was agreed during an international music convention to adjust, modify or temper the frequencies of the notes of the music scale and
assign them equal frequencies and call them enharmonic equivalents. Present day music scales are tempered. Semitone intervals are not exactly 16/17 or 17/16 in the same way that whole tone intervals are not exactly 16/18 or 18/16. They are however essentially and practically 16/17 and 16/18 intervals.
A natural note that is lowered or made “flat” by a semitone is signified by the flat sign b after the letter symbol of the natural note. A natural note made higher or made “sharp” by a semitone is signified by the sharp sign # after the letter symbol of the natural note.
The frequency of this accidental note is 264 x 7/4 = 462. Please read further on Accidentals.
Interestingly, note A# or Bb, even if its only an
accidental note has a higher ranking in the hierarchy of
harmony than a Major 7th (M7) semitone 12 or note B, a
natural note. Historically, it took many years of exposure to
the M7 before listeners came to appreciate the sound of this interval. M7 is a characteristic interval/sound of Jazz music.
Supertonic
The next most pleasing interval is that with the 9th
harmonic overtone. This is produced by the 1/9 secondary length of a vibrating string. The 1/8 partial of this frequency
corresponds to the natural note, one note superior (above) the tonic; it is thus called the supertonic. The supertonic of the C Scale is note D. It creates a Major 2nd interval, M2
(semitone 3) with the tonic. The frequency of note D is 264 x 9/8 = 297 Hertz.
Leading Note
Among the natural notes of a major diatonic scale, the least pleasing harmony/interval is produced by the 15th
harmonic overtone. It is produced by the 1/15 secondary length of a vibrating string. The 1/8 partial of this frequency
corresponds to the 7th natural note and creates a major 7th interval (M7) with the tonic (semitone 12). This is note B. It creates the greatest “tension” with the tonic, and being so, must invariably resolve or lead to the tonic. This note is thus called the leading note. The frequency of note B is 264 x 15/8 = 495 Hertz.
ACCIDENTALS
Augmented Intervals
Perfect and major intervals are augmented when they are lengthened or increased by a semitone. Raising the higher note a semitone does this. An augmented note is signified by the sharp sign # after the letter note. A perfect 5th, semitone 8 (C to G,) is augmented by raising note G a semitone, Thus a C to G# is an Augmented 5th, A5, +5 or
semitone 9. Another example is, a Major 6th, semitone 10