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Thursday, July 4, 2019

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 )

Tuesday, June 25, 2019

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 
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/

Friday, June 21, 2019

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



Thursday, June 20, 2019

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.






Tuesday, June 18, 2019

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.


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.

Monday, June 17, 2019

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 


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


1 1 3 1 1 3 2Chromatic Mixolydian
1 3 1 1 3 2 1Chromatic Lydian, Raga Lalit, Bhankar
3 1 1 3 2 1 1Chromatic Phrygian
1 1 3 2 1 1 3Chromatic Dorian, Mela Kanakangi, Raga Kanakambari
1 3 2 1 1 3 1Chromatic Hypolydian, Purvi That, Mela Kamavardhani, Raga Shri, Pantuvarali, Basant, Kasiramakriya, Suddharamakriya, Puriya Dhanashri, Dhipaka, Pireotikos: Greece
3 2 1 1 3 1 1Chromatic Hypophrygian, Blues scale III
2 1 1 3 1 1 3Chromatic Hypodorian, Relative Blues scale, Raga Dvigandharabushini
2 3 1 1 3 1 1Chromatic Mixolydian inverse
1 1 2 3 1 1 3Chromatic Phrygian inverse
1 1 3 1 1 2 3Chromatic Hypophrygian inverse
3 1 1 3 1 1 2Chromatic 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=p9j8DlakBuQto 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.