236. 3 TYPES OF PERCEIVED MUSICAL PITCH SCALES SYSTEMS: 1) NATURAL HARMONICS IN LOGARITHM DECELERATING PROGRESSION. 2) BACH SYSTEM IN ARITHMETIC UNIFORM PROGRESSION AND 3) INVERSE HARMONICS ACCELERATING.

 3 TYPES OF PERCEIVED MUSICAL PITCH SCALES  SYSTEMS: 1) NATURAL HARMONICS IN LOGARITHM  DECELERATING PROGRESSION. 2) BACH SYSTEM IN ARITHMETIC UNIFORM  PROGRESSION AND 3)  INVERSE HARMONICS  ACCELERATING.




0)It is known that when a frequency of series is produced f0 f1 f2 f3 f4 f5 f6 ....fn the human body for reasons of economy perceives then after transforming them by the logarithm function. Thus a geometric or exponential progression of frequencies (like bach-scale) will be perceived as equal distances musical pitches.

1) HARMONIC SERIES (DECELERATING , LESS HAPPY ) 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.

2) BACH SYSTEM OF UNIFORM (NEUTRALLY HAPPY )PITCH PROGRESSION The Bach equally tempered musical scale removes this by utilizing a geometric progression of frequencies which after taking the logarithm of the human perception of the musical pitch it results into an arithmetic progression as if of equal relative distances of musical pitch. 

3) INVERSE HARMONIC SERIES ACCELERATING (HAPPIER) MUSICAL SYSTEM

The harmonic series exists also in the vibrations of a string e.g. of the guitar. Still, there are no frets for most of them to play them. If we take the inverse of the harmonic series (overtones) of the string as relative distances but still of increasing frequencies we should start from small differences frequencies and go to larger interval frequencies. If we divide the string in equal distances, as the overtones do , 1, 2, 3, 4, 5, 6, etc and take the first lower node (not last higher) of each harmonic then in frequencies we get the INVERSE HARMONIC (OVERTONES) SERIES (n+1/ n ), n/(n-1), ... 2/1
The inverse harmonic series creates also musical pitches with inner affinity due to overtones. E.g. two successive such frequencies over a fundamental frequency a , that are x1=((n+1)/n)a and
x2=(n/(n-1))a have successive overtones nx1=(n+1)a and (n-1)x2=na  that are already successive overtones of the fundamental frequency a, thus with the musical affinity of overtones. Of course, each one of the frequencies of the inverse harmonics series e.g. the x1=x1=((n+1)/n)a, has an nth-overtone that all overtones above it are also overtones of the original fundamental frequency.

E.g. if we utilize the first 12 harmonics or overtones then these nodes will have frequencies 

1   12/11   11/10    10/9   9/8  8/7  7/6  6/5  4/3  3/2  2 

which as notes will be 

C C#1 C#2 C#3 D  D#1 D#2  E  F  G  C' 

where #1 #2 #3 are fractions of the one sharp interval. 

This 10 notes scale is of accelerating relative intervals it contains the standard notes C D E F G (but not A, B) and as an accelerating scale to higher pitches, it is a happier scale compared to equal distances pitches, or decelerating intervals,  pitches as the direct harmonics (overtones) scale. 

In a fretless guitar, we can easily set frets for them as the above numbers of frequencies multipliers by inverse 1/x function become rations of distances of the frets from the lower bridge of the guitar.

For a guitar of 66 cm scale length thus of an octave on the fretboard of 33 cm, these frets as distances from the upper bridge would be

2.75 cm,  3cm, 3.6 cm 4.125cm , 4.75 cm 5.5 cm, 6.6 cm 8.25 cm 11  cm 33 cm


But this can be done also in  Oud, that is not fretted, by marking with a pen the distances, 0, 1/12, 1/11, 1/10 ....  2/3 , 1/2 of the scale-length, from the left lower bridge on the fingerboard and playing the strings in the marking lines.

If the strings in the oud or guitar are tuned by 5ths (like mandolin or violin ) I think is the best option.

UNDERTONES AND SUBHARMONICS METHOD

In this method to produce accelerating pitches, is to utilize the undertones or subharmonics. 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.


Almost uniform distribution of nodes


As alternative if we want a more even distribution of pitches we just take all the nodes of the harmonics (overtones) up to some order. Thus in general the pitches will be of the type not only x1=((n+1)/n)a for an overtone but all xi=(ki/n)a where ki=n+1 , n , n-1 ....1 . An again for any two such frequencies x1=(k/n)a, x2=(l/m), the nx1 and mx2 are both overtones of the original fundamental frequency a, thus with an harmonic affinity.

So in order to take nodes of the harmonics within the first octave from a to 2a, and say derive 12 such frequencies , then the next nodes must be taken:

Let as assume a scale length of 66 cm thus a first octave fret at 33cm

From the 3rd harmonic the only node from 0 to 33 cm is the 2/3*66cm=44cm which as distance from the upper bridge is is the 1/3*66=22 cm

From the 4th harmonic the only nodes from 0 to 33 cm and as distances from the upper bridge are the 1/4*66cm=16.5cm

From the 5th harmonic the only nodes from 0 to 33 cm and as distances from the upper bridge are the 1/5*66cm=13.2cm and 2/5*66cm=26.4

From the 6th harmonic the only new nodes from 0 to 33 cm and as distances from the upper bridge are the 1/6*66cm=11cm

From the 7th harmonic the only new nodes from 0 to 33 cm and as distances from the upper bridge are the 1/7*66cm=9.42cm  , 2/7*66cm=18.85 and 3/7*66cm=28.28cm

From the 8th harmonic the only new nodes from 0 to 33 cm and as distances from the upper bridge are the 1/8*66cm=8.25cm  , 3/8*66cm=24.75

We already have 12 frets that as distances from the upper bridge are

0 , 8.25cm  , 9.25 cm, 11cm, 13.2 cm,  16.5 cm, 18.85 cm, 22 cm, 24.75 cm, 26.4 cm, 28.28cm,  33cm

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.

The fretless oud  with the above tuning by 5ths, allows for beatiful solo improvisations outside any scale! The sliding of the fingers is stopped by the feelings at the right point to produce the  right pitch which sounds most satisfactory, based on the previous notes. It is a blind solo improvisation. Because of the tuning by 5ths, the two imediately lower strings relative to the string that the solo takes place  can be used for a drone-5th interval which is repeated, while the solo takes place. A practice common to other  instrumenst tuned by 5ths like the Sazi,  the Boulgari, tambouras etc.