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**

MARKING IN THE OUD FINGERBOARD FRETS FOR 1) BACH EQUAL TEMPERED 12-TONES SCALE (BLUE LINES) B) 16 OVERTONES OR HARMONICS NODES (RED LINES) 3) 16 UNDERTONES OR SUHARMONICS LEVELS (GREEN LINES)

WHAT IS REMARKABLE IS THAT THE ARE A LOT OF COINCIDENCES OF UNDERTONES LEVELS OVERTONES NODES AND BACH EQUAL TEMPERED SCALES, MAINLY PER PAIRS.

See also

http://barthopkin.com/tangular-arc/