243. WHISTLES , QUENAS AND PANFLUTES WITH A MEMBRANE REED ON THE FIPPLE FOR LOW IMPEDANCE

WHISTLES AND QUENAS WITH A MEMBRANE  REED ON THE FIPPLE 

It is possible to have a reed on the soundhole of the fipple of a whistle! For this the sound hole must be larger, and then cover the extra part with a plastic or other material membrane reed. It could also be a cane reed. Or it could be a cooper foil  or aluminum foils,  of total thickness 0.08 mm .  Or it can be a double sheet of rubber surface from a balloon.  The vibration of the air which is vertical to the surface of the membrane will set in vibration the membrane-reed also which will influence the color and quality of the sound.  This idea exist also in the dizi chinese flutes, but they put the membrane not on the soundhole but on an extra hole just after the mouthpiece. It adds to the sound of the flute the sound of the membrane which vibrates as the air flows parallel to it. I used a membrane-reed from a (white) balloon, and actually I put it in double sheet as it was the balloon before inflating. It has to be well stretched and fixed on the fipple by tape. In the photo we see an example on a wooden-plastic Fipple of a Low D4 whistle with thin wall aluminum body of inner diameter 25 mm . The open sound hole that is left, has ratio of transverse to longitudinal sides 2:1.  The area that the membrane-reed covers is about double the uncovered one (Uncovered : 10mm*5mm Covered about 10mm*10mm ) Before the balloon reed, the fipple had large sound hole with very breathy sound and difficult 2nd octave. After the balloon reed the sound hole become 10mm*5mm and the low octave acquired very clear (not breathy) and soft sweet sound, while the 2nd octave acquired again very clear (not breathy) and loud melo sound,and somehow it is  more pleasant and easy to play in the higher octave than in the lower as is the case with the concert Bohm flute too.  This suggest happier melodies that star in the 2nd octave move in the low 1st octave and finally end again in the 2nd octave as in the upper registry whistles (see post 199 THE UPPER REGISTRY WHISTLES (28< Length / bore ID ratio  <48) FOR HAPPIER MELODIES ) .
The membrane on the fipple acts as intermediate impedance from that of the air vibration and the material of the tube at the sound-edge vibration. In total it lowers the impedance of initiating a vibration , especially on the 2nd octave.

 In winds there two types of impedance a) the vibration impedance responsible of propagating the vibration from the air column of the mouthpiece to the air column of the tube,and then outside the tube b) The air-flow impedance which is the resistance of the flow of the air. Large bore tubes have reverse effects on the two types of impedance: They increase the vibration impedance because a larger mass of air must vibrate , but they decrease the air-flow impedance because lower flow speeds are necessary to drive the blowing outside the tube compared to thin bore tubes. The air-flow impedance is by far of larger effect on the felt "resistance in playing and producing sound compared to the vibration impedance. The membrane on the fipple does not affect much the air-flow impedance but lessents the vibration impedance.

The overall result was a substantial improvement of the sound and easier playing response of the whistle. Somehow it has also the color of the sound of a quena.  Suddenly this cheap Low D4 whistle become the best sounding and playing Low D4 whistle among many other expensive ones that I have! The material of the reed influences the color of the sound. I did not try it but I assume that if I would put a cane reed from say an alto sax cane reed, the sound would be more woody. Obviously this can apply also to soprano C5 and D5 whistles and might allow rather wide bore such whistles (more than 14 mm)  to have easier , clear and loud 2nd octave, even if the Length/bore ratio is less than 20. This in particular gives very interesting applications on the native American  flutes , that traditional have Length/bore ratio<20 and do not have 2nd octave. With such membranes on the soundhole of the native American flutes they acquire easily 2nd octave  , but in order to be uniform in blowing it requires also a more appropriate hole sizing where the lower pitch holes are larger and the higher pitch holes smaller and not vice versa.

In the next 2 photos we see 6  whistles from PVC . An G4  an A4  and  a C4 (2nd  photo) with 20 mm inner diameter of the  bore and 25 mm external diameter and 3 a Bb3 , a C5 and and a D5 with 16.5 mm inner diameter of the bore and 20 mm external diameter . These whistles were initially crafted as tunable quenas , and they still have their quena mouthpieces. Their inner diameter of the bore gives Length/bore ratio less than 20 , thus normally they should not be able easily to have a 2nd octave with ordinary fipples. But with these membrane-reed fipples , they retain both the 1st octave full rich quena-like sound due to wide bore and big holes (for jazz partial holing) and also have a fine clear and loud 2nd octave because of the vibration of the membrane at higher pitches, which partially substitutes the adjustment in angle and air speed that we can do at the 2nd octave with the mouth and the quena rim-blowing mouthpieces. They are certainly easier to play and have more smooth and melo full  sound compared to them as quenas. 

In the next photo we see the originally crafted as tunable Bb4 quena , with its quena mouthpiece, but also with the membrane-fipple mouthpiece which makes it a whistle and an alto saxophone mouthpiece, that makes it also clariphon (chalumeaux) at Bb3 (one octave lower) and because of laws of acoustics capable of 1st only octave compared to quena and whistle that are capable of at least 2 octaves. In order  to have also the clariphon (chalumeaux) to play at least a 2nd octave too, either we should use a 8+1 holes system instead of 6+1 here,  or keep the 6+1 hole system and use (as in yamaha venova) a initial branche (chimney) as in post  244 that would make the close-open acoustics to open-open acoustics.

Obviously this idea of membrane that reduces the impedance and makes initiation of the sound easier but also vibration  self-maintenance shorter CAN APPLY TO QUENAS NOTCH AND TO PAN-FLUTE BLOWING HOLES!

242. HOW TO CONVERT A ROMANIAN CAVAL FLUTE TO PLAY THE HARMONIC MINOR DOUBLE HARMONIC MINOR , NEAPOLITAN MINOR ,AND NATURAL MINOR. THE CHROMATIC MINORS FLUTE

We remind the reader that in the online notes here we call a sequence of 7 interval that sum-up to 12 semitones a mode and all cyclic permutations of it as the scale that the mode belongs. Thus any cyclic permutation of the 2-1-1-3-1-1-3  is considered again as the parachromatic Byzantine scale but at a different mode of it.

See also the post about the parachromatic Byzantine flute (3rd chromatic Byzantine minor scale) post 235. 

Other very well known 7-notes Byzantine scales but different from the parachromatic or 3rd Byzantine chromatic minor , also in  the chromatic family are the harmonic minor or 1st Byzantine chromatic minor (https://en.wikipedia.org/wiki/Minor_scale#Harmonic_minor_scale ) and double harmonic minor or 2nd Byzantine chromatic minor (https://en.wikipedia.org/wiki/Double_harmonic_scale).

OF SCPECIAL INTEREST IS THAT INVERSE OF THE HARMONIC MINOR 2-2-1-2-1-3-1  IS DERIVED FROM THE FIRST 13 OVERTONES OR HARMONIC SERIES OF A STRING OR NATURAL TRUMPET IF  THE 7NTH HARMONIC (E.G.  C-D-E-F-G-G#-B-C )  if we perceive the 7nth overtone as B rather than Bb (in reality it is somewhere in the middle)  AND SO IT  is the inverse 7-notes scale of the first 13 overtones  ON THE OTHER HAND THE MELODIC MINOR OR THE 7-NOTE SCALE 2-2-1-2-1-2-2 E.G. C-D-E-F-G-G#-Bb-C WHICH IS DERIVED FROM THE FIRST 13 OVERTONES HARMONIC SERIES IN A NATURAL TRUMPET IF WE PERCEIVE THE 7NTH HARMONIC AS Bb rather than B.   BOTH SCALES  ARE   CONSTRUCTED FROM THE 4-CHORD 2-2-1 AND THE 5-CHORD. 


To convert a romanian 5-holes caval flute (see e.g. https://www.youtube.com/watch?v=p9j8DlakBuQ) to a 7 holes flute which can play both the harmonic minor and double harmonic minor we simply ad 4th hole to the upper 3 holes in the romanian caval between the upper 3 and lower two which is one semitone away from the last lower 3rd upper hole and also a thumb hole which is 3-semitones higher than the highest front hole and one semitone lower than the next 2nd harmonic root. Of course instead of a 4th upper front hole we may just make the lower of the upper 3 front holes of the caval larger and apply partial holing.

 Alternatively instead of a single large thump hole we may add a double thump hole (in vertical order not in horizontal order as in recorders double holes). Thus in a romanian caval flute rooted in A3 starting from the 5 holes that give  the notes A3-B3-C4-D#4-E4-F4 (in semitone intervals 2-1-3-1-1 which is a 6-notes scale sub-scale of the 7 notes double harmonic minor ) we will result in to the 7 holes  A3-B3-C4-D-D#4-E4-F4-G#4-A4 , (in semitone intervals 2-1-2-1-1-1-3-1 which is an 8 notes scale) and which will allow playing the 1st mode of the harmonic minor (1st Byzantine chromatic minor ) 2-1-2-2-1-3-1 or in notes A3-B3-C4-D4-E4-F4-G#4-A4, known to me as 1st mode of A3 harmonic minor and also the 1st mode of the double harmonic minor (2nd Byzantin chromatic minor scale) 2-1-3-1-1-3-1 or in notes A3-B3-C4-D#4-E4-F4-G#4-A4 known to me as 1st mode of A3 double harmonic minor. If we have double thump hole (or large size thump hole to apply partial holing) we have also the note G4 (in total A3-B3-C4-D#4-E4-F4-G4-G#4-A4 ) which allows also to play the natural (A4) minor.
In addition as the thump hole can play also the G, instead of G# ( e.g. A3-B3-C4-D#4-E4-F4-G4-A4) in semitones 2-1-3-1-1-2-2  then we get the 7-notes Neapolitan scale , instead of the double harmonic minor scale  2-1-3-1-1-3-1
See the photos below

The previous modification of the Romanian caval (which plays a 6-notes scale) so as to play both 7-notes scales , the harmonic minor and double harmonic minor was the minimal modification with 2 only additional holes.

Here are the photos of 2 soprano at B4 and D5  such modifications of a Caval flute with double thumb hole

WE SUMMARIZE:

We take now the example of D4-Caval Flute.

1) The Romanian Caval flute is a 6-notes scale 2-1-3-1-1-4 E.g. at D4 it is D4-E4-F4-G#4-A4-A#4 

This scale is one note less than a full 7-notes scale which is widely known among different cultures as double harmonic minor (see e.g. https://en.wikipedia.org/wiki/Double_harmonic_scale) When we say scale of notes , to differentiate it from a mode of a scale , we meana lso all 7 cyclic permutations of it as the same scale (all different 7 modes as modes of t he same scale)

So to make it a full double harmonic minor 2-1-3-1-1-3-1 or D4 it is D4-E4-F4-G#4-A4-A#4-C#5 you just have to open at hump hole  3-semitones higher than the highest front hole. And better be a large hole for partial holing for reasons that I will explain below. Or it can be a double hole but close together , by one semitone away so that with the thump we can cover one only or both. 

Now you already have the chords D minor , G minor , F minor , F major ,  A major , C# major  C#minor  C major With partial holing on the thump to give C5 we can play also the chord A minor

In addition we may play  the D-melodic minor  scale D4-E4-F4-G4-A4-B4-C#5-D5 with interval structure 2-1-2-2-2-2-1 (see e.g. https://www.basicmusictheory.com/d-melodic-minor-scale)

2) Many folk versions of Romanian Caval in Romania (as you can see in videos in youtube) add also a 4th hole to the upper group of 3 holes of the original caval and below the lower again one semitone apart, which gives the note G4. With this 8-notes scale D4-E4-F4-G4-G#4-A4-A#4-C#5, you can already play both the D4 double harmonic minor and also the D4 harmonic minor 2-1-2-1-1-3-1 or D4-E4-F4-G4-A4-A#4-C#5. (see e.g.https://en.wikipedia.org/wiki/Minor_scale#Harmonic_minor_scale  ) Plus you can play the natural D4 minor (with partial holing on the thump) that is D4-E4-F4-G4-A4-A#4-C5 with interval structure 2-1-2-2-1-2-2
And now the chords that you can play are the previous plus the  C minor .  

4) Sometimes some people add still another hole which is  the D#4 note. If you do so then  in addition you have the chord C minor so that all the minor triad of chords Dm-Gm-Cm is possible.
5) If on the otherhand we a the thump hole to a 5-hole caval  so that it is only one tone above the highest front hole, then we get the 7-notes Neapolitan scale (e.g. A B C D#, E, F, G, A) , instead of the double harmonic minor scale  (e.g. A B C D#, E, F, G#, A) .

VERSION OF THE FLUTE  IN  DOUBLE HARMONIC MINOR  WHICH IS CLOSED OR INVARIANT TO THE 3RD OVERTONE 

Nevertheless, the 3rd harmonic and on the same holes  in such a flute (and on the original 6-notes Romanian  Caval too) will not give nots inside the double harmonic minor as it does say in  a major mode flute . The simple harmonic minor with interval structure 2-1-2-1-1-1-3-1 is already non-symmetric in a shifting higher by an interval of 5th as does the 3rd overtone. Nevertheless the double harmonic minor has a mode the 5th mode which is symmetric to shift higher by an interval of 5th and in interval structure it is 1-3-1-2-1-3-1. Therefore if we open holes for this 5th mode of the double harmonic minor we will get again a flute in double harmonic minor which has the additional advantage that the 3rd overtones on the same holes gives again notes of the same scale.

E.g. if the flute starts again from g, the notes that we should open holes are the 

1st hole open gives g#4

2nd hole open gives b4

3rd hole open gives c5

4th hole open gives  d5

5th hole open gives  d#5

6th thumb hole gives f#5

This scale is the a mode of the c5 double harmonic minor . By partial holing on the 2nd hole and on the 6th hole we may play also the c5 simple harmonic minor and natural c5 minor .

A VERSION THAT INCLUDES THE BYZANTINE PARACHROMATIC SCALE 

If we add a 8th hole which is one semitone lower than the lowest hole of the current modified caval flute, then we get a flute  that includes the Romanian Caval, and the Byzantine parachromatic flute and can play a) The double harmonic minor b) the harmonic minor c) the byzantine parachromatic scale (see post 235)

A 8 HOLES VERSION WHICH IS UNIFORM IN THE SEMITONES AND CAN PLAY ALL THE ABOVE SCALES

Finally if we add still another holes in between the 1st front 2 and the upper front 4 , making thus in total 7 front holes in a row all one semitone away from its next we get a 9 notes scale 

2-1-1-1-1-1-1-3-1 that can play all the above scales a) double harmonic minor b) simple harmonic minor  c) parachromatic and much more while being uniform in semitones at the front 7 holes. 

We may compare it with some soprano (and not only) cavals that are about t he same except that the tump hole is only a semitone away from the highest front gibing the 9-notes scale 

2-1-1-1-1-1-1-1-3 . This arrangement though cannot play the double harmonic minor, although it seems more convenient at the fingers and in particular at the thump hole.

241. THE LOOSE CONCEPT OF DIATONIC SCALE TONALITY IN EARLY CLASSICAL JAZZ (NEW ORLEANS) : AT LEAST 50% OF THE TIME ONLY IN THE SCALE

As with almost all folk happy music, the early classical jazz of New Orleans had harmony in the songs was based initially on the simplicity of a triad of major chords and in particular of 1-4-5 (the 12-bars blues deviated later to 5-2-1, substituting 4 with its minor relative 2).

Nevertheless even early classical jazz is known for its reach chromatic harmony, and this seems as a contradiction but it is  not!

What happens here is what happens with a free accompanying of an improvisational melody with a chord. The chord fits as long as at least half of the the time that notes of the melody sound , are from notes that belong to the chord at various octaves.

Similarly an early classical jazz song is supposed to be on  diatonic scale (or mode of it) as long  as at least half of the the time that notes of the melody sound , are from notes that belong to the diatonic scale  at various octaves. 

Thus the triad 1-4-5 will work at least half of the time, with other chords either from this diatonic scale or from the chromatic 12-notes scale appearing if necessary as a closer follow-up of the melody! 

Here is an example of such early classical jazz songs played by the marvelous band Tuba Skinny:

https://www.youtube.com/watch?v=pZ7mg9Kl-RU

ACCOMPANYING WITH  SINGLE POWER CHORD:
And there is more on this!
The harmony of a jazz improvisation song can be simplified nor to three but to a single chord! 
E.g. If the improvisation song is in Bb, the single accompanying chord will be the power-5 chord on the root Bb3-F4-Bb4. Then two completely different melodies that are at least 50% in the scale and that follow the same rhythm, will very well be accepted as improvisational parallel countermelodies  within the base simple harmony.  (see also posts  71 ,126, 128, 135, 138, 141 )

240. THE EMOTIONAL DUALITY SAD/HAPPY OF MINOR/MAJOR CHORD AS COMING FROM THE DUALITY OF OVERTONES-HARMONICS AND UNDERTONES-SUBHARMONICS

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 

https://en.wikipedia.org/wiki/Undertone_series

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/

239. SCALES IN 12 NOTES EQUAL TEMPERAMENT THAT ARE ALSO ALMOST NODES OF HARMONICS

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 

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.

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

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

238. A SYSTEM OF FRETBOARD FOR STRING INSTRUMENTS BASED ON THE NODES OF HARMONICS (OVERTONES)

A SYSTEM OF FRETBOARD FOR STRING INSTRUMENTS BASED ON THE NODES OF HARMONICS (OVERTONES)




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.


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

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. 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. 

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

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

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




MARKING IN THE OUD FINGERBOARD FRETS FOR
1) BACH EQUAL TEMPERED 12-TONES SCALE (BLUE LINES)
2) 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.

237. AN ACCELERATING 2 OCTAVES MUSICAL SCALE 1-2-2-3-4-5-7

This scale at its first octave is the inverse of maximal harmonic pentatonic scale and at the 2nd octave the inverse of a power -5 chord.

As semitones interval structure it is

1-2-2-3-4-5-7

e.g. B C D E G A B D B  or

E F G A C E A E

The 1st octave part 1-2-2- 3-4 is the inverse of the maximal harmonic pentatonic scale 4-3-2-2-1

(see post 117 e.g. C E G A B C  ) while the 2nd octave 5-7 B-D B is the inverse of a power chord.

Its psychological effect is excitement and joy because the intervals are of increasing size .It has 4 chords 2 major one minor in the 1st octave and a power chord in the 2nd octave.
E.g. C E G,    E G B ,   G B D.