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.
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.
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
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.
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 which is about the 7nth fret of the Bach fretboard.
From the 4th harmonic the only nodes from 0 to 33 cm and as distances from the upper bridge are the 1/4*66 cm=16.5cm which is about the 5th fret of the Bach fretboard.
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.
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 which is about the 7nth fret of the Bach fretboard.
From the 4th harmonic the only nodes from 0 to 33 cm and as distances from the upper bridge are the 1/4*66 cm=16.5cm which is about the 5th fret of the Bach fretboard.
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 which is about the 4th fret of the Bach fretboard. and 2/5*66cm=26.4 which is about the 9th fret of the Bach fretboard.
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 which is about the 8th fret of the Bach fretboard.
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
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 which is about the 8th fret of the Bach fretboard.
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
These levels at the fretboard need not be frets necessary. Just marks on it to find the nodes of the first 8 harmonics and play under-tones of them by just touching the strings at that point.
From the above we may notice that for the first 8 harmonics the usual bach pitches (frets) that also nodes of them are the 4th 5th 7nth , 8th and 9nth.
1) Thus we may say that the first 8 harmonics (these are also the harmonics that a trumpet utilizes) there is the next 6-note scale in semitones intervals that are also nodes of them 4-1-2-1-1-3 which is known in indian music as raga Sarvati. Starting from C it would be C-E-F-G-G#-A-C.
2) On the other hand if we stop at the 7nth harmonic the next pentatonic is such a scale too
4-1-2-2-3 In notes C-E-F-G-A-C
3) If we stop at the 9nth harmonic the next 7-notes scale is such a scale too
2-2-1-2-1-1-3 in notes C-D-E-F-G-G#-A-C
4-1-2-2-3 In notes C-E-F-G-A-C
4) Finally if we stop at the 12 harmonic then the next 9-tones scale of the bach-fretboard is also on nodes of the above harmonics
2-2-1-1-1-1-1-1-2 in notes C-D-E-F-F#-G-G#-A-Bb-C
From the above we may notice that for the first 8 harmonics the usual bach pitches (frets) that also nodes of them are the 4th 5th 7nth , 8th and 9nth.
1) Thus we may say that the first 8 harmonics (these are also the harmonics that a trumpet utilizes) there is the next 6-note scale in semitones intervals that are also nodes of them 4-1-2-1-1-3 which is known in indian music as raga Sarvati. Starting from C it would be C-E-F-G-G#-A-C.
2) On the other hand if we stop at the 7nth harmonic the next pentatonic is such a scale too
4-1-2-2-3 In notes C-E-F-G-A-C
3) If we stop at the 9nth harmonic the next 7-notes scale is such a scale too
2-2-1-2-1-1-3 in notes C-D-E-F-G-G#-A-C
4-1-2-2-3 In notes C-E-F-G-A-C
4) Finally if we stop at the 12 harmonic then the next 9-tones scale of the bach-fretboard is also on nodes of the above harmonics
2-2-1-1-1-1-1-1-2 in notes C-D-E-F-F#-G-G#-A-Bb-C
