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.

235. THE PARACHROMATIC BYZANTINE (BLUES WITH LEADING TONE) OVERTONE WHISTLE. HEXATONIC BLUES AS CHROMATIC 7-NOTES DIATONIC TONALITY

This whistle is like an overtone whistle with high Length/(inner_diameter_of_Bore) ratio (here it is 55.5cm/1cm=55.5 ) and it is based on the Byzantine (para)chromatic 7-notes scale in semitones 

2-1-1-3-1-1-3  (see post 227  and also https://books.google.gr/books?id=tvxMDwAAQBAJ&pg=PA281&lpg=PA281&dq=Byzantine+parachromatic+scale&source=bl&ots=CM8asY99sh&sig=ACfU3U2I6jjEtOsdTJhQYb1U3ygHl_yK7g&hl=el&sa=X&ved=2ahUKEwjB-9W1rfHiAhWq-ioKHfHMBDIQ6AEwC3oECAkQAQ#v=onepage&q=Byzantine%20parachromatic%20scale&f=false )

Here some modes and invere modes under various names of this scale. 

1 1 3 1 1 3 2	Chromatic Mixolydian
1 3 1 1 3 2 1	Chromatic Lydian, Raga Lalit, Bhankar
3 1 1 3 2 1 1	Chromatic Phrygian
1 1 3 2 1 1 3	Chromatic Dorian, Mela Kanakangi, Raga Kanakambari
1 3 2 1 1 3 1	Chromatic Hypolydian, Purvi That, Mela Kamavardhani, Raga Shri, Pantuvarali, Basant, Kasiramakriya, Suddharamakriya, Puriya Dhanashri, Dhipaka, Pireotikos: Greece
3 2 1 1 3 1 1	Chromatic Hypophrygian, Blues scale III
2 1 1 3 1 1 3	Chromatic Hypodorian, Relative Blues scale, Raga Dvigandharabushini
2 3 1 1 3 1 1	Chromatic Mixolydian inverse
1 1 2 3 1 1 3	Chromatic Phrygian inverse
1 1 3 1 1 2 3	Chromatic Hypophrygian inverse
3 1 1 3 1 1 2	Chromatic Hypodorian inverse

1 3 1 1 2 3 1

Raga Lalita, Persian, Chromatic Hypolydian inverse, Raga Suddha Pancama

Its characteristic tetrachord is the 1-1-3 (instead of the 1-3-1 in the harmonic minor and harmonic double minor).

Here also  as  Parachromatic, new genus 5 + 5 + 20 parts 7 in The mathematical theory of tone systems 

https://books.google.gr/books?id=o2K1DwAAQBAJ&pg=PA274&lpg=PA274&dq=Parachromatic,+new+genus+5+%2B+5+%2B+20+parts+7&source=bl&ots=bFLsv8tBf4&sig=ACfU3U3FRBsIjZd2OmwnlcZ8-tbr_FMPHQ&hl=en&sa=X&ved=2ahUKEwjkjrP_653zAhWNNOwKHSKJDhMQ6AF6BAgCEAM#v=onepage&q=Parachromatic%2C%20new%20genus%205%20%2B%205%20%2B%2020%20parts%207&f=false

The flute is of lowest note D4 and it is end-blowing (fipple) it is made from elder wood.

The Byzantine (para)chromatic 7-notes scale here is

D5, E5, F5, Gb5 ,A5, Bb5, B5 , D6

or as interval structure in semitones

2-1-1-3-1-1-3 . or of it assumed as the 5th mode of the  next version Gb3   parachropmatic= 5#-6#-7-1-2#-3-4-5# Thus like the double harmonic minor 1-2#-3-4-5#-6-7-1' but with an extra sharp at 6. It has also the chord 5#m. The Byznatine parachromatic flute in the photo is the 5th mode of it

It can be considered chromatic tonality of the B major diatonic scale in improvisation e.g. with 12-bars blues. 

The double harmonic minor would be on this flute as

D5, E5b, F5, Gb5 ,A5, Bb5, B5 , D6 or in semitones 1-2-1-3-1-1-3 in other words with flat on the 2nd note E4. 

While if we gave flat on the 5th note D5, E5, F5, Gb5 ,Ab5, Bb5, B5 , D6   it becomes in semitones

2-1-1-2-2-1-3  called  also Mela Namanarayani, Raga Narmada, Pratapa, Harsh Major-Minor (see post  227)

The Byzantine parachromatic flute is also the 2nd mode of the 7-notes Blues scale with  leading note. Here the first note of the Blues scale with leading note which would be the B4

The 7nth mode of the first version of the parachromatic in intervals is the 

3-2-1-1-3-1-1, is realized in a diatonic scale as

1-3b-4-5b-5-7b-7-1 

or 

1-2#-4-5b-5-7b-7-1 

or 

1-2#-3#-4#-5-6#-7-1  

The 5th mode of it is as follows

1-2#-3-4-5#-6#-7-1'  or insemotones 3-1-1-3-2-1-1 Thus like the double harmonic minor 1-2#-3-4-5#-6-7-1' but with an extra sharp at 6. It has also the chord 5#m. The Byznatine parachromatic flute inthe photo is the 5th mode of it. 

Compared to the diatonic scale 1-2-3-4-5-6-7 , the 1st mode it has the altered or chromatic tonality chords 7M, 2#m and 1m.

It is also called BLUES WITH LEADING TONE SCALE.

And by skipping the 7nth note we get the 6-tones blues scale 

1-3b-4-5b-5-7b-1 (1-2#-3#-4#-5-6#-1) or another mode (1-2#-4-5#-6#-7-1) ) and in semitones

 3-2-1-1-3-2.

An alternative Blues hexatonic would be 2-3-1-1-3-2. (e.g. 1-2-4-5b-5-7b-1  and it is known as Bebop hexatonic scale.  As  7-notes 1-2-4-5b-5-7b-7-1  in semitones 2-3-1-1-3-1-1 which is the inverse of  the Byzantine parachromatic ) and it can be as chromatic tonality over the same diatonic as in the standard blues hexatonic. 

An interseting 8-notes scale derived from it might be called parachromatic is the 

1-2#-3-4-4#-5-6#-7-1  or in semitones 3-1-1-1-1-3-1-1

7M, 2#m , 1M and  1m.

In other words if in the parachromatic 7-notes scale we start from the 7nth and we skip the 6th note we get the 6-notes blues scale. 

This scale is derived by combining the chromatic family pentachord 2-1-1-3 (which is also known as  Samba) with the chromatic family tetrachord 1-1-3 known also as parachromatic tonal tetrachord.

The Byzantine parachromatic scale is called also inverse Persian or  Purvi Theta scale.

If we want to tune a portable Celtic harp with levers for sharps it could be e.g. the next mode of it. 

c-d#-e-f-g#-a#-b-c

or  the 7nth mode of its (blues scale with leading note).

c-d#-e#-f#-g-a#-b-c

From the 4 "blue" notes of diatonic scale (e.g. at natural a minor a, b, c , d , e , f , g , a the 4 blue notes would be in this order g#, d# , f#, c# ). They are by definition the neighboring by one semitone notes to the interval of 5th c-g thus g# f# around g and c# around c and the same with the best next interval of major 3rd c-e thus d# . There cannot be other as first 4 notes other than the above 4 blue notes.   The Byzantine parachromatic minor utilizes the 3rd blue note f# if it would be a modification of the a natural minor In other words it would use f# instead of f and as a parachromatic minor it would be a b c db e f gb=f# a  or in semitone intervals 2-1-1-3-1-1-3 .  The 3rd blue note f# is utilized also in 6 notes  blues scales and country jazz music 2-1-1-3-2-3 or in notes a, b, c , c# , e , f# , a . The harmonic minor uses the 1st only Blue note that is g# instead of G in a natural minor and the double harmonic minor both 1st and 2nd blue notes the g# instead off g and D# instead of d. The Byzantine parachromatic minor scale is slightly less chromatic compared to the double harmonic minor and it was used in cosmic celebrations while the harmonic and double harmonic minor in Religious hymns. The scale that would use only the d# blue note therefore the a-b-c-d#-e-f-g-a,  in semitone intervals is the 2-1-3-1-1-2-2 is nothing else than the Mela Sulini, Raga Sailadesakshi, Raga Trishuli, Houzam: (Greece) or Neapolitan scale. The scale that would use both the c# and d# blue notes would be the a-b-c#-d#-e-f-g-a,  in semitone intervals is the 2-2-2-1-1-2-2 which is the double melodic minor or Arabian scale also known as Kokilapriya, Raga Kokilaravam, Heptatonia tertia. The 7-notes scale as variation of the a natural minor that would use both the f# and d# blue notes would be the  a-b-c-d#-e-f#-g-a,  in semitone intervals is the 2-1-3-1-2-1-2 which is nothing else than the Bayat-e-Esfahan, Sultani Yakah, Zhalibny Minor, Armoniko minore: (Greece) or the 4th mode of the harmonic minor. The 7-notes scale that would use only the f# blue note is the  a-b-c-d-e-f#-g-a,  in semitone intervals is the 2-1-2-2-2-1-2 which is nothing else than the dorian mode of the G major scale. 

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.

Other very well known 7-notes Byzantine scales but different from the parachromatic , also in  the chromatic family are the harmonic minor (https://en.wikipedia.org/wiki/Minor_scale#Harmonic_minor_scale ) and double harmonic minor (https://en.wikipedia.org/wiki/Double_harmonic_scale). 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. Thus in a romanian caval flute rooted in C4 starting from the 5 holes that give  the notes C4-D4-Eb4-F#4-G4-Ab4 (in semitone intervals 2-1-3-1-1) we will result in to the 7 holes  C4-D4-Eb4-F4-F#4-G4-Ab4-B4-C5 , (in semitone intervals 2-1-2-1-1-1-3-1) which will allow playing the harmonic minor (1st Byzantine chromatic minor ) 2-1-2-2-1-3-1 or in notes C4-D4-Eb4-F4-G4-Ab4-B4-C5 , known to me as 1st mode of C4 harmonic minor and also the double harmonic minor (2nd Byzantin chromatic minor scale) 2-1-3-1-1-3-1 or in notes C4-D4-Eb4-F#4-G4-Ab4-B4-C5 known to me as 1st mode of C4 double harmonic minor. See the photo below

The whistle looks like a romanian caval flute too except the latter has not the lowest hole of the parachromatic byzanatine whistle (see e.g. https://www.youtube.com/watch?v=p9j8DlakBuQ) .  Thus a Romanian caval flute that we add a lowest hole one semitone lower than the lowest of the Caval becomes a Byzantine parachromatic flute.

The whistle looks also like the holes pattern of a Ney flute , and because of its high L/B ratio plays easier in  the 2nd octave than in the 1st octave. A Turkish Ney has also two groups of 3 holes with distances between each group of one semitone, but the distance between the two groups of 3 holes is almost a tone, while here it is a 3-semitones. If the scale of the Ney flute would be converted to the equal tempered 12-notes scale it would be the 8-notes scale 2-1-1-2-1-1-3-1 e.g. a-b-c-c#-d#-e-f-g#-a , which is obviously different from the parachromatic scale but also different from the neapolitan  or inverse of it and different from the harmonic minor or inverse of it. The Ney 8-notes scale resembles a combination of the melodic minor and the harmonic minor. A Ney is also rim-blowing wind while here it is with a fipple as it is usual with overtone winds. There is not thumb hole for the chromatic Byzantine whistle as it not necessary as overtone flute. But in Ney too that does exist a thumb hole it is mainly used in the first harmonic octave.

The Turkish caval seems  much closer to the Byzantine parachromatic flute than the Romanian caval.

Some how this Byzantine parachromatic scale whistle  resembles also a suling flute.

The tube is made from elder wood.

Here are two photos of it 

Finally if we add a 7nth hole between the 3rd and 4th of the current flute and one semitone lower than the 4th , and also a thumb hole which is 3 semitones higher than the highest front hole of the current flute we get a  7+1 holes 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.

I have ordered also in Seydel harmonica manifacturing company though their online customization in their site ,an harmonica tuned on the parachromatic Byzantine scale on the root D4.

234. THE MAGIC AEROPHONE TRUMPET FOR IMPROVISATIONAL MEDITATION. AEROPHONE DIDGERIDOOS(CHROMATIC/OVERTONES FUJARAS) WITH TRUMPET ACOUSTICS.

THE MAGIC AEROPHONE-FIPPLE TRUMPET. AEROPHONE DIDGERIDOOS(CHROMATIC/OVERTONES FUJARAS)  WITH TRUMPET ACOUSTICS.
LONG TUBE AEROPHONES (CHROMATIC/OVERTONES WINDS)  WITH FIPPLE ,SIMILAR TO FUJARAS/DIDGERIDOOS, TRUMPETS AND BRASS WINDS (ratio 80< Length/Bore and about equal 125 as in trumpets ) ?

It is known that many  long tube winds have been invented with cane reeds (like  coils either double reeds  e.g. the Rackett  https://www.youtube.com/watch?v=QwPgiVCVxEA  https://www.youtube.com/watch?v=d9_ma4ITRSc or single reed like Mr curly of Lindsay Polak https://www.youtube.com/watch?v=Iu60MwpMiow  https://www.youtube.com/watch?v=XNO97-mPP8U&t=14s) or lips-reeds in trumpets and other brasswinds. A trumpet that the sound is produced by the lips is a  lips-reed wind with closed-open acoustics that normally have only odd number harmonics (overtones) . Nevertheless because of the gradual cone at the end, as with the saxophone  eventually because of a complicated phenomenon of acoustics  both odd and even number of overtones. Still strictly speaking an aerophone is not producing the sound from a reed, but by a fipple or rim blowing. A modern trumpet at Bb was tube length of 1.482 m and is producing 8 overtones, the first is not possible to play so from 2nd to 8th are C4 G4 C5 E5 G5 A5 C6. These 7 notes are essentially part of an arpeggio of the C major chord with 7nth within 2 octaves. The keys-valves allow for 6 more series of such harmonics each one a semitone lower. 

So what if the same acoustics are derived as aerophone winds with fipples? Fujara and didgeridoos are simplistic  cases of them with close air acoustics at the side of the mouth but they can be rendered to open-open acoustics with fipples. Such winds have not been elaborated in the same way with valves etc as the brass lips-reeds winds.



The fujaras that are known if they have large bore inner diameter and thus length/bore ratio less than 80 will not belong to the acoustic and musical effect described here (sliding whistle over a dense scale of higher harmonics)
Examples of Fujaras
https://www.youtube.com/watch?v=y8Wzb3tLPCs

In the next we not only use the length to bore ratio between 80 and 125, 80< Length/Bore and about equal 125, but also to tune it to a particular note e.g. Bb we use as length the wavelength of the note two octaves lower. E.g. the wavelength of Bb3 is 148 cm (see http://pages.mtu.edu/~suits/notefreqs.html) but the root note will be by the 4th harmonic that is two octaves higher Bb5. Thus these open-open acoustics as far as Length/bore ratio and harmonics are closer to the french-horn rather than trumpet.

I made one such aerophone trumpet from a 97 cm long PVC pipe with inner diameter 12mm (thus Length/bore ratio= 970/12=80.83) and with a fipple from a Sweet tone Clarke whistle C5. I opened 4 holes above the middle of the tube (corresponding to the 2nd harmonic) that as in the trumpet keys give an increase of the tone by 1 semitone by 2 by 3 and by 4 semitones (one more key than trumpets) .The action is mainly on the 6th and 5th octave. The base not is F5.

As 2nd version o the holes we open a hole at the 2/3 of the length (corrected higher by above 80% of the more diameter) which corresponds the 3rd harmonic and two holes  that give sequentially 2 semitones above it. And we also we mark the middle of the tube (that would correspond to  thump hole giving the roo one octave higher, thus 2nd harmonic) but we do not open a hole there, we only open again 3 holes that give sequentially 1 tone  1  semitone and 1 semitone above it. In this setting of 5 holes (3+2) we incorporate almost the natural minor, the double harmonic minor and the harmonic minor in the main octave (which contains the 1-3-1 semitones pattern and has 5# and 2# in place of 5 and 2. The harmonic minor has only 5# in place of 5 and 2 while the natural minor has 5 and 2 ) which is chromatic feeling that complements the strongly harmonic feeling of the arpeggio of the major chord of each hole as higher harmonics). Strictly speaking the holes scale is the parachromatic Byzantine which is in semitones 1-2-3-2-1-1-3 (in steps of  F major it is 5, 5# 6, 1, 2, 2# 3 5).When we open all the lower 3 holes a major 3rd interval appears , while of we open only the upper two holes a minor 3rd interval appears. Alternating these two we get nice harmony.  In the modern trumpet the keys or valves add seven semitones within an interval of 5th thus  creating  in total 7 different series of overtones a semitone each away from the previous starting from the first one defined by the tube. In a guitar this would correspond to have 7 strings each one having pitch one semitone higher than the previous. Each string defines by its own a full sequence of harmonics. This is somehow what we could do here also except as it is not easy to reach the holes at the end of the tube we could open 4 above the  5th note and including 5th , 5 holes .  Thus as a scale it would be the chromatic (5, 5#, 6, 6#  7 )

Then I made a 2nd one which feels even better (because the ratio is higher 104 and the bore larger) without holes  . I made this 2nd   aerophone didgeridoo from a 172 cm long PVC pipe with inner diameter 16.5 mm (thus Length/bore ratio= 172/1,65=104.24) and with a fipple from a generation whistle at Bb . I did not opened holes as the variation of pitch was relatively easy and desne in the harmonics.The action is mainly on the 6th  and 5th octave. The base note is G5.



These winds should not be confused with the Mosenos that are similarly long and with a fipple but they still play at the 1st and 2 harmonics mainly with small Length/bore ratio (e.g. 20) compared to 80.

The most important characteristic of these aerophone trumpet, winds are

1) They need not be bass , but on octaves from 7nth to 4th etc
2) The sound is not really one note but rather like a chord with many hidden harmonics
3) They must have very high Length/bore ratio above 80 and preferably around 125
4) They behave in the pitch of the sound like sliding whistles with a continuum of pitches but not entirely on all frequencies but on a dense grid of harmonics (certainly more than 12 chromatic notes in the octave) 
5) They are very satisfying in playing like natural human whistling but unlike it or unlike the violin without frets, the harmonics give a natural scale of "frets"  that the pitch falls in it which is very harmonic to listen . It is so because the relative change o the pitch is determined always by harmonics.
6) The harmonics are very easy to obtain not by overblowing (as in standard overtone whistles)  but by simple blowing even under-blowing. 
7) Holes are not necessary for hem. Only if the Length/bore ratio is much less than 125, e.g. 80 then 4 choles (reachable by the hands) at equal pitch distances of one semitone are adequate to help small chromatic changes of 1, 2 3 or 4 semitones (in trumpets the 3 keys help for 1,2 and 3 semitones). The reason that 4 semitones is added is because  a  sequence of alternating minor-major thirds gives chords on every 3 notes.
8) Such winds are so much so as to read an existing  written melody from musical notes and play but a melody inside you to improvise and meditate .

When the L/B ratio is above 100 there is not even need for holes because the change of pitch by the change of blowing power is very sensitive and effective. But if one insists in making it easily chromatic then one hole close at the end of the tube at 0.9438 of the total length higher (plus about one inner diameter) will  give a hole of one semitone higher pitch. In the modern trumpet the keys or valves add seven semitones within an interval of 5th thus  creating  in total 7 different series of overtones a semitone each way from the previous starting from the first one defined by the tube.  Here in the picture below are two such aerophones with trumpets acoustics  one in G5 176.02 cm and one in Bb5 (like the lips-reed trumpet)  of 148.2 cm The length is more or less the theoretically expected by the wavelength at the 3rd octave  (G3 , Bb3)  e.g. here http://pages.mtu.edu/~suits/notefreqs.html The actually sound seems also as if having hidden root frequencies at the 2nd octave (G2 , Bb2 ) which makes it beautiful and harmonious. There is also a soft resistance in blowing which makes it satisfying.
The inner diameter of the bore is about 12mm and the tube I used is irrigation pipe. The pictures below for a first experimental assessment. As fipple I used that of generation whistles at Bb4. The feeling of the harmonics fixing your "whistling"at the right exact pitch is very satisfying, as well as the fiddling at any desired pitch like a violin with very dense frets on its  fingerboard, defined by the harmony of the overtones.  The sound is like that of waters, or the sound that often pigeons  make when they are flirting. Still it is harmonic like chord and whistling. The sound is not loud but that is exactly what sometimes needs in order not to disturb the neighbors, it is sweet and very appropriate for meditation. I prefer these aerophone trumpets for improvisational meditation compared the Shakuhachi flute and Nay flute that also serve for meditation

More experiments that are not irrelevant by Nikolas Brass

https://www.youtube.com/watch?v=R35N43h8M5U&t=684s

https://www.youtube.com/watch?v=D8YJXMCSbLg

https://www.youtube.com/watch?v=oKnK5h_y3pg

https://www.youtube.com/watch?v=qzOLO5YHL9Q

and    Xavi Lozano

https://www.youtube.com/watch?v=6ZW2UWu2IWU&t=145s

https://www.youtube.com/watch?v=-tzk94TK4SY&t=532s

https://www.youtube.com/watch?v=XNO97-mPP8U&t=14s

233 . THE MAGIC OVERTONE WHISTLE FOR IMPROVISATIONAL MEDITATIVE CHEERING UP. : DIATONIC AND CHROMATIC OVERTONE WHISTLES WITH HALF OCTAVE ONLY HOLES (3 DIATONIC 5 CHROMATIC)

(see also post 199) 
THE ENCHANTED OVERTONE WHISTLE  THE  DIATONIC OVERTONE WHISTLES (48< Length / bore ID ratio<80  ) FOR CHORD ARPEGGIOS AT EACH HOLE.

These whistles are relatively unknown and i am not aware of any traditional flute in this category. The closest traditional flutes are the overtone flutes but they do not have usually but 1 or 2 only  holes to play a diatonic or other scale. The  flutes I suggest here have the peculiar characteristic that they are so long and with thin tupe (L/B ratio above 48) that when blowing or even under-blowing they cannot lay the 1str harmonic registry (octave). The only play with the 2nd or higher harmonics. They have more than 8 harmonics that sound easily with blowing softer or stronger. These flutes are a combination of the traditional overtone flutes and the standard diatonic flutes like irish whistles. As far as it is  concerned the acoustics relevant to the L/B ratio (which is between 48 and 80  ) , they are in between the trumpet acoustics with ratio around 125 and classical flutes and whistles with ratio between 16 to 48.  They play with less than half action  "horizontally" with their holes as diatonic scale (e.g. 2 octaves) and more than half action  "vertically" at each hole more than 8 overtones (3 ,4 octaves) that are essentially arpeggios of major chords (up to the 7nth harmonic). Up to the 10 harmonic, the overtones of each hole belong to the scale (with the exception of the 7nth harmonic which makes the arpeggio , that of a major chord with  7nth). The playing is best when it is improvisational. It is nice to play a melodic theme and be able to play also over each note of it the arpeggio of  power-chord or major chord. The overall range of such flute is considerably larger than the standard whistles. The fipple sound hole is preferably smaller than the usual to help the higher overtones. Such winds are so much so as to read an existing  written melody from musical notes and play but a melody inside you to improvise and meditate .

The fujaras that are known if they have large bore inner diameter and thus length/bore ratio less than 80 will also belong to the acoustic and musical effect described here if made by smaller size bore and thus also shorter length (thus maybe not bass flutes). The fujaras use to have only 3 holes at the diatonic 4-chord in semitones 2-2-1. E.g. if the root is C3, then the holes give the notes D3-E3-F3 but at the 3rd harmonic G4 they will give the notes  A , B , C  thus eventually all the diatonic scale. Similarly if we open 5 holes chromatically in semitones 1-1-1-1 as in the style of valves of trumpet,  for the first harmonic it would be C#3-D3-D#3-E3-F3, the 3rd harmonic will give on the same holes G#, A , Bb, B, C , thus eventually all the 12-note chromatic scale
Examples of Fujaras
https://www.youtube.com/watch?v=y8Wzb3tLPCs

See also

https://www.youtube.com/watch?v=qzOLO5YHL9Q


If we want less than 7 notes "horizontally" for such an overtone flute, then the next best choice of the 7-notes diatonic scale is  the  is the maximal harmonic pentatonic scale (see post 117) I, iii, V, vi, vii 

The general rule to make an overtone whistle say in X base tone (e.g. C5, G4, D4, C4 etc) is to utilize the same bore inner diameter size with the ordinary whistle but about double the length. This is expected in general to give the Length/Bore ratio between 48 and 80. The resulting whistle is an overtone whistle that the basic tone (one octave lower than the corresponding whistle tone) cannot be played in the instrument , but the next octave (2nd Harmonic) is playable which is the original whistles root note.  Furthermore if we want a diatonic overtone whistle we utilize only 3 holes (the  lower pitch 3 holes if it would be an ordinary whistle) If the length is very long (for alto or low whistles) we "coil" the tube with turns and rounds as in t he brass wind instruments.  If we want a chromatic such overtone whistle we simple fill-up the gaps of the 3-holes of the diatonic overtones whistle as above with all their sharps or flats getting so 5 holes in the same area each one semitone away from the next , which is about the lower 1/3 of the tupe. In this way with the 2nd and 3rd overtones we have all the 7-notes diatonic scale or all the 12-notes chromatic scale. 

In the next picture I made from 16 mm bore  (inner diameter 12 mm) PVC and sweet tone Clarke whistles mouthpieces (fipples of D5 and C5 ) two such "magical" diatonic overtone whistles in D5 and C5 (although the length is as if of Low D4 and C4). The have respectively L/B ratios the D5 57/1.2=47.5 and the C5 64cm/1.2cm=53.33
They have 6+1 holes, but if we seal with a tape all except the highest 3, as in Fujara, we still have with the 1st and 3rd harmonic all the 7 notes of the diatonic scale.

In the next C5 and F5 overtone diatonic whistles only 3 holes are opened and suffice for a full 7-notes diatonic scale even only with the 2nd and 3rd harmonic.

Here is a bass in Bb3 diatonic from PVC of 32mm external diameter ,  of length about 148.2 cm (the same length with a Bb4 trumpet) , again with only 3 holes. In semitone intervals 2-2-1 or in notes Bb3 C4  D4 Eb4 . That is all it is necessary for a 7notes scale because it already starts with the 2n harmonic (due to the double than normal length)   and the 3rd harmonic gives in the same octave and on the same 3 holes  the the notes  F4-G4-A4-Bb4 thus all the Bb3 7-notes major scale. As it is a bass one, it falls in the category of Fujara. The high pitch ones I call overtone whistles. By changing the last part of the coil-tubes with another of appropriate length and holing we may have more roots on the same upper body.

In the next  photo we see a  C5 full chromatic overtone  whistle  that utilizes the 2nd and 3rd harmonic and with the 6 holes to give the chromatic half octave [C5]   C5# D5 D#5 E5 F5 F#5 with the 2nd harmonic and with the 3rd harmonic to give on the same holes the [G5]  G#5 A5 A#5 B5 C5 C#5 gives thus a full chromatic 12 notes octave. The 1s tharmonic  is not possible to play as it is an overtone whistle with Length/bore ratio L/B= 57  It is made again by PVC tube of external diameter 16 mm and internal 12 mm.  The idea is the same with the trumpet and the valves that produce 5 notes each a semitone away from the previous. Actually the whistle would be as well in Bb4 till F5 instead of C5 to G5  to be in accordance with the Bb scale of the trumpet.Or it could be in Eb5 to Bb5 .
We see 6 holes in the front (there is not thump hole) These 5 notes that divide a 5th (also a 4th) give the concept of ancient Greek 4-chords and 5-chords as basic building block for an octave scale. Compared to the full chromatic Ney in C5 in post 247, it has many more overtones than just the 2nd and 3rd because of the considerably larger Length/bore ratio. The chromatic Ney in post 247 is just an upper registry whistle while this here is a chromatic  overtone whistle. It has also an ordinary fipple (from a sweet tone clarke C5 whistle) and not a fipple with membrane as the Ney in post 257. 

232. ATTRACTOR AND REPULSOR NOTES IN A CHORD OF THE DIATONIC SCALE.. TRINITY VERSUS DUALITY IN THE DIATONIC SCALE

Let us assume that we are improvising a melody strictly inside the diatonic scale.

A general  trinity of the music in the diatonic scale is the triad of factors 

1) Chromatic dimension (intervals of 2nd/7nth, 1 or 2 semitones etc)

2) Melodic dimension (intervals of 3rd/6th , 3 or 4 semitones etc)

3) Harmonic dimension (intervals of 4th/5th or 8th , 5 or 7 semitones etc)

E.g. we ascend harmonically ,we descend melodically and we change triad of chords from minor or diminished to majors or vice versa chromatically  (from 7-3-6  to 1-4-5 chromatically at the chord relations 7-1 and 3-4 )

Let us also assume that when moving from not x1 to note x2 with a waving melodic vector (see post 231) we apply the expanding and contracting waving from the 7 independent melodic themes patterns as in post 231, that as picture are



and


Then we should start from a minimum possible amplitude and intervals (repulsor note ) and end in such one  (attractor note) . But in the diatonic scale the minimum interval is a semitone. There are two only poles and positions in the diatonic scale with semitones: The root pair b-c and the pair e-f
Therefor a note which is in one of the two semitones of the diatonic scale will be either the attractor or the repulsor of the melodic motion.

It happens that each of the chords of the diatonic scale has at least one attractor or repaslor which we will cal singular point of the chord. Here is the enumeration of them.

C major c-e-g has singular notes the e and c

D minor d-f-a has singular note the f

E minor e-g-b has singular note the e and b

F major f-a-c has singular notes the f and c

G major g-b-d has singular note the b

A minor a-c-e has singular notes the c and e

B diminished b-d-f has singular notes the b and f.

The above analysis and information helps someone to know where to start waving and were to end waving in melodic themes patters as above. 

It is well known that the two poles (duality) of the diatonic scale is the root and dominant. That are 5th apart. Utilizing twoocatvesa lso the duality we describe has the two poles a 5th apart.

Only two chords have only one pole: The Dm and the G. The major chords C and F have both poles and are consecutive in the cycle of 4ths. The same with the minor/diminished  chords Bd- Em and Am.

If we want to complement the described here chromatic duality of b-c and e-f in the diatonic scale , it is here where the dominant pair g-a enters. So the trinity are the next 3 pairs of notes and areas of the scale


b-c   / e-f /  g-a 

Of course this trinity in the diatonic scale is not much different than grasping the scale from the triad of major chords 1-4-5 (I-IV-V) or triad of minor chords 3-6-2 (iii-vi-ii).

Another general trinity of the music in the diatonic scale is the triad of factors 

Chromatic dimension (intervals of 2nd/7nth, 1 or 2 semitones etc)
Melodic dimension (intervals of 3rd/6th , 3 or 4 semitones etc)

Harmonic dimension (intervals of 4th/5th or 8th , 5 or 7 semitones etc)

Example of a nice melody which utilizes waving ending in such attractor and repulsor notes is the Irish St Anne's Reel

https://www.youtube.com/watch?v=lwllcWC_FL4

231. THE 3 CLASSES OF 7 VARIATIONAL INDEPENDENT MELODIC PATTERNS FROM THE COUPLING OF THE POSITIVE AND NEGATIVE EMOTIONS. THE 3 SIMPLICIAL DOLPHIN WORDS

THE 3 CLASSES AND 7  INDEPENDENT PATTERNS OF MELODIC THEMES FROM THE COUPLING OF POSITIVE AND NEGATIVE EMOTIONS.


WE MAY COMPARE THE NEXT PERCEPTIONS AND TECHNIQUES WITH THE 3 BASIC SOUNDS IN SOLOING 1) THE CHROMATIC (MAINLY INTERVALS OF 2NDS)  2) THE MELODIC (MAINLY INTERVALS OF 3RDS) 3) THE HARMONIC (MAINLY INTERVALS OF 4THS/5THS/8THS )

All the 3-classes of dynamic coupling of positive and negative emotions have been applied also in the psychology of social behaviour, and even in the psychology of behaviour of couples in the marriage 

(see "The mathematics of marriage" by Gottman, Murray, Swanson, Tyson, and Swanson, MIT press 2002. See also for the Voltera's equations "Models in Ecology" by J. Maynard Smith, Cambridge University press )

All of the above 3 classes of non-linear coupling have been solved and their solutions have been classified. 

The  3 classes of melodic patterns from the coupling of positive and negative emotions are

1) The ascending (or descending ) melodic pattern (3 patterns)

2) The stationary waving melodic pattern.  (3 patterns)

3) The isocratic note melodic pattern  (1 pattern)

The 1st class the ascending (or descending ) melodic pattern are subdivided by three  patterns

a) The straight stepwise ascending (or descending) melodic theme

b) The waving ascending (or descending) melodic theme



c)The abrupt jump (spike) interval ascending (or descending) by an interval larger than an octave without intermediate steps.






2) The The stationary waving melodic patterns are divided to 3 cases

a) The stable amplitude stationary waving pattern






b) The diminishing (contracting) amplitude stationary waving pattern









b) The expanding   amplitude stationary waving pattern

Obviously these patterns (and "Dolphin words" as defined e.g. in post 114 ) are variational independent, in the sense that no translation or inversion can produce one from another. On the other hand pitch translations within the scale or pitch inversion within the scale will not change the type and order-topological shape of the melodic theme pattern ( a straight ascension will remain such or a stationary waving will remain such etc)

On the other hand if we pass from one such pattern of the above  6 cases to another such pattern we have called this transition as mutation of the melodic theme.

Therefore the above 3 classes (and 6 cases of types which when including their inverses become 11 ) make  base of melodic themes for a melody or a melodic seed for a melody of a song.

Their nature in melodies is similar to the type of chords in chord progressions (major , minor, diminished, augmented,, seventh etc)


We may compose after these patterns melodies utilizing rules for the rhythm, scales, underlying harmony and musical morphology e.g. as in posts 229,221, 214, 203

THE MELODIC IMPROVISATION AS A SEQUENCE OF "ROTATIONS". EACH "ROTATION" HAS THE NEXT 3-LEVEL  RHYTHMIC OR MORPHOLOGICAL STRUCTURE

THE 4 PARTS A1 A2 A3 BI STRUCTURE 

This structure is rhythmic morphological and comes as a pattern from the 15-syllable poetry.

2 lines in a 15-syllables poetry are divided to 4 parts A1 A2 A3 B1, each of the Ai having 4 syllables and the B1 3 syllables. Similarly here in the melody the A1 A2 A3 , B1 are of 2 measures.  Each of the A1 A2 A3 B1 contain at least one melodic theme. If we put the restriction that both the melodic theme and its variations are consisting from intervals that are 2nds in less that 1/3 (=33%) of the cases, then we get an "Irish" but also "south american", melodic/harmonic sound in the melody. The melodic theme of A1 is translated or inverted to that of the A2 and that of the A2 to that of A2. Finally the melodic theme of A3 is mutated to that of B1. The simplicial sub-melody has one note in each measure therefore two notes in each of the A1 A2 A3 B. These two notes signify a MELODIC MOVEMENT (VECTOR) which is the basis of an emotion.  If we superimpose the melody at the first two levels, the detailed improvisational notes of the melodic theme (1st level) and the melodic move or vector of the simplicial submelody (2nd level) we get a new musical entity and concept that we may call WAVING MELODIC VECTOR. The wavings are as in the above 7 melodic themes patterns. Another classification of them is that they are of 3 classes a) Chromatic (of length an interval of 2nd) b) Melodic (of length an interval of 3rd) c) Harmonic (of length an interval of 5th). The choice in the improvisation of one of these 3 classes is similar to the choice in a chord progression if the chord-transition will me chromatic , melodic or harmonic.
Very often the wavings are of diminishing amplitude as we reach the melodic center-end note of it. This is very convenient to have in mind when we improvise because we think that we will move from a persisting note a (melodic center) to a persisting note b (melodic center ) in a scale but with improvisational waving way that we may change each time at willAll the melodic movements of the A1 A2 A3 B make a DOLPHIN WORD (as defined also in post 101, 114 ). Each part A (thus melodic move too) has an underlying chord. The chords of two parts like A1 A2 may be the same or different chords. The melodic move can be also a melodic triad (alternating major minor vector interval of 3rd as in post 208) . The Dolphin word of the "rotation" may be a closed polygon (e.g, triangle) of waving melodic vectors , so that the length of the melodic vector indicates also the length-interval of the waving melodic vector, which may also be the distance as musical interval of the middle notes of the underlying chords, of the melodic move. Each side of the polygon has its own waving melodic theme pattern as the above 7 .
This melodic polygon allows for simple and very concise, simple and visually beautiful  methods of writing the improvisation with not less information than the necessary but no more than a minimum too, so that each time we see the written paper and play the melody a different melody will emerge (mainly at the first layer/level of it) while still it will be "the same" song. 

When playing again such an improvisation the simplicial sub melody is repeated (thus the "rotation" A1 A2 A3 B or Dolphin word of it), but the other notes inside the measures that are embellishments may change. All together the "rotations" or Dolphin words make the total melodic phrases of the song. Therefore we have here at least a 3-level structures of the melody a) notes b) melodic moves c) "rotations" or Dolphin words,and in time it is 1, 2^3=8 (2 measures)  and 2^5=32 (A1A2A3B parts).

If we put the restriction that these patterns of  melodic theme and their variations are consisting from intervals that are 2nds in less that 1/3 (=33%) of the cases, then we get an "Irish" but also "south american", melodic/harmonic sound in the melody.

See also post 282

Examples of such progressions of simplicial melodic themes (or Dolphin words) are the next  (each vector-arrow is an oriented interval that fits to a single or more  underlying chord(s)).

Or

Or

Or

AS THE COMBINATION SIMPLICIAL MELODIC MOVES (ORIENTED INTERVALS SEE POST 282) CREATE PATTERNS THAT ARE CALLED "DOLPHIN WORDS" , WE MAYS AS WELL CLASSIFY THE "SIMPLICIAL DOLPHIN WORDS" . The simplest such patterns are of course the 3: 1) THE CYCLE 2) THE ASCENDING SEQUENCE 3) THE DESCENDING SEQUENCE.

230. CONTINUOUS NOTES OF REED-WINDS AND BOWED STRINGS VERSUS PLUCKED STRINGS NOTES. COUNTING THE RHYTHM WITH THE LEFT-RIGHT MOVES OF THE BOW

One may think that the musical notes of bowed string instruments like violin cello etc or of reed-winds like clarinet , saxophone , etc are of the same  description musical notes as the notes by plucking the strings of musical instruments like guitar, mandolin etc. This by far it is not so! If the mousical note by plucking the string of an instrument like guitar has 3 dimensions : 

1) Pitch, 

2) duration 

3) volume, 

the corresponding note of a bowed string instrument like violin or cello etc has 5 dimensions 

1) Pitch, 

2) Bows-speed/pressure  timbre (color)  and  reed-pressure or reed-length-in-the-mouth degree of winds.
3) Bow moving left-right direction changes and rhythmic pattern of it

4) duration 

5) volume.

By playing a single musical note of the latter type of fixed pitch and volume we may even conduct the rhythm on its continuous duration by changing the speed of the bow, thus by changing the Bows-speed timbre (color) which in spectral decomposition of the sound means changing the weight center of the harmonics of the sound higher or lower. Therefore somehow this Bows-speed timbre (color) is also one-dimensional quantity.  Similarly how much we put the reed in the mouth and how much we press it changes in a quantitative way the timbre of the note.

Here is ancient 2-voices music from the kazakhs violin which is called kobyz, which makes it more obvious

https://www.youtube.com/watch?v=P-qwTp1aVrs&t=1647s

 COUNTING THE RHYTHM WITH THE LEFT-RIGHT MOVES OF THE BOW

THE RHYTHM IN IMPROVISATIONS REFLECTED ON THE BOW LEFT-RIGHT DIRECTION CHANGES REGARDLESS OF THE MELODY NOTES CHANGES:

In the bowed instruments when improvising we have a very simple and efficient way to reflect the rhythm and this is by making the bow left-right direction changes exactly at the rhythm of a measure in other words as the up or down measurements of the rhythm.
And we do so either if we play a single long lasting note (long note that nevertheless reflects the rhythm too) or when we play many notes, each note a single or an  integral number of bow right-left moves. (so we avoid many notes with the same  direction of the bow move and if necessary we jump to the next division of 2 in realizing the rhythm). In this way we shape the left-right moves of the  bow with a simple pattern that of the rhythm rather than random left-right moves which might feel more complicated in improvising. In this way a part of the melody with a more desne number of notes might still be played with the same density of  left-right moves of the bow. If necessary we may jump to the next division of 2 in realizing the rhythm, but always counting the rhythm.
Based on the above we may say that the bow has as modes of speeds of change of directions the powers of 2 of the written melody (integrals, halves, quarters, eighths, sixteenths, etc)