HARMONIC SERIES MUSICAL SYSTEM The acoustics of the natural trumpets (without keys-valves) e.g. in Bb (and overtone flutes) are such that the column of air is producing at least 16 overtones or harmonics in 3 octaves of frequencies 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 times the fundamental frequency a.
This is by itself a natural scale which constraints the one that plays it to sound always within that scale. This scale has a natural harmony in the sense that if harmonic x is such that divides as order the harmonic y, then the y is already an overtone of x. The degree of consonance of two frequencies x y depends on how many common overtones they have. The more common overtones the more harmonic the musical interval. Probably one psychological disadvantage of the natural harmonic series as the musical scale is that as we proceed to higher numbers although it is an arithmetic progression of equally spaced frequencies the human perception of the frequencies after the logarithm transformation perceives them as of decreasing relative distances thus decelerating increase of pitch.
OVERTONES-UNDERTONES AND HARMONICS-SUBHARMONICS
When we utilize the undertones or subharmonics the effect of minor sad chord apprears . In other words if a is a fundamental frequency the undertones are the 1/2a ,1/3 a, 1/4a , 1/5a etc
In a string of length l giving frequency a the undertones will be produced by multiplying the length of the string from l, to 2l , 3l 4l 5l etc.
Similarly a fretboard of n equal length l of frets will produce the n undertones of mini-string of length l (but not oft he whole string of n frets)
WHAT IS VERY INTERESTING IS THAT THE INITIAL MAJOR CHORD IN OVERTONES HAS A CORRESPONDING MINOR CHORD OF UNDERTONES!
If we start with C the initial overtones chord is the C major, but the initial undertones chord is the F minor! This can be computed by the frequency of the note e.g. C4 261.63 and a plying the subharmonics 1/2 C3 , 1/3 F2 , 1/4 C2 1/5 Ab2 (see e.g. http://pages.mtu.edu/~suits/notefreqs.html )
This is also significant in understanding the sad emotion correlated with the minor chord as it is by contraction and lowering of a fundamental frequency compared to expansion and raising of fundamental frequency by overtones which gives the major chord.
An example of an instrument Mbira (kaliba) based on the right on evertones and he left on undertones is the arithmetic Array Mbira by Bill Wesley
https://www.youtube.com/watch?v=B_owNkjewGU
An example of an instrument Mbira (kaliba) based on the right on evertones and he left on undertones is the arithmetic Array Mbira by Bill Wesley
https://www.youtube.com/watch?v=B_owNkjewGU
ABOUT THE STANDARD DOMINANT 7TH RESOLUTION FROM ANXIETY TO SERENITY
Interval explanation by notes in one semitone distance
This is called the standard Dominant resolution e.g. G7->C , D7->G,
C7->F etc The key to explain of this anxiety-serenity resolutions is that
the dominant 7th chord e.g. G7 is the superposition of the Gmajor+Bdiminished,
and the diminished chord is by itself a bit sad with anxiety, which is
correlated with the Cmajor chord which has serenity because it contains at
least one notes of one semitone distance of notes of the C major. These are the
notes B, with one semitone distance from the C of Cmajor. The same is the
explanation why the Cdim7 below
resolves to G, because it has the notes D# and F# with one semitone
distance from the notes D, G of Gmajor. Sometimes even Cm will be considered resolving to G, because of the D# note
at one semitne distance from the D note of G, and the major chord sounds with
more serenity than the minor chord!
And the same is the explanation why in the
Andalusian Cadenza Dm, C, Bb
A. , the Bb resolves to A, because it has 3 notes one semitone distance apart
from corresponding notes of A.
Similarly as the E7 resolves to A so the Em7b5 resolves to A too because it
has the 3 notes D, F , G# one semitone apart from the C#, E , A of A chord. And
similarly the Edim7 resolves to A as
it has the notes D, F one semitone apart from the C#, E of A
For the same reason if D7 resolves to G, then also Daug resolves to G because
Daug has the notes F#, C# at one semitone distance from the notes G, D of
G major chord and Daug sounds in anxiety.
We support here the philosophy that the emotional content of the harmony of the chord progressions is different and is not included in the emotional content of the sequence of notes of the melody. In other words the harmony is not summarizing only the emotions of the melody but is carrying and suggesting a sequence of emotions by itself. The emotions of the harmony and the emotions of the melody must of course be compatible.
We support here the philosophy that the emotional content of the harmony of the chord progressions is different and is not included in the emotional content of the sequence of notes of the melody. In other words the harmony is not summarizing only the emotions of the melody but is carrying and suggesting a sequence of emotions by itself. The emotions of the harmony and the emotions of the melody must of course be compatible.
OVERTONES-UNDERTONES AND HARMONICS-SUBHARMONICS
When we utilize the undertones or subharmonics the effect of minor sad chord apprears . In other words if a is a fundamental frequency the undertones are the 1/2a ,1/3 a, 1/4a , 1/5a etc
In a string of length l giving frequency a the undertones will be produced by multiplying the length of the string from l, to 2l , 3l 4l 5l etc.
Similarly a fretboard of n equal length l of frets will produce the n undertones of mini-string of length l (but not oft he whole string of n frets)
WHAT IS VERY INTERESTING IS THAT THE INITIAL MAJOR CHORD IN OVERTONES HAS A CORRESPONDING MINOR CHORD OF UNDERTONES!
This is also significant in understanding the sad emotion correlated with the minor chord as it is by contraction and lowering of a fundamental frequency compared to expansion and raising of fundamental frequency by overtones which gives the major chord.
