10. The mathematics of rhythms: The dimension , the order and the height of rhythms. Simplistic rhythm harmony. Their writing and the appropriate software to play them.

The basic idea here is to conceive the rhythm also as a very low frequency  sound, of so low frequency that the ear does not conceive as musical sound but as sequence of separate beats. Nevertheless the frequency of beats is not a different mathematical measure from the frequency of the pitch of notes. This will allow know concepts about the pitch to apply to rhythms, like the concept of chord for notes now it will become "chord" of beets, and as with chords we have the concept of 3-notes or 4-notes chords so with rhythms we will have the concept of dimension of the rhythm, 3-dimensional and 4-dimensional (but also 2-dimensional similar to intervals) rhythms. While the number of beats of each of its dimension within a cycle will define the order of the dimension of the rhythm, and the maximum of the the order of the rhythm as a whole. In addition the maximum number of subdivisions by powers of 2 in the speed of the rhythm  (similar to the octaves in melodies) defines the height or deepness of the rhythm. But these three numbers are not enough to describe mathematically the rhythm . We need an ordered sequence of zeros 0 and 1's of the length of the order for each dimension so as to have the full information. In no way the classical musical notation with two numbers only  like 3/4 or 3/8 etc can ever define fully the rhythm as drummers and elaborate musician require. 

This video gives the idea of dimension and order of rhythms but also all the necessary combinatorial information required to define it mathematically
https://www.youtube.com/watch?v=2UphAzryVpY&feature=youtu.be





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

http://cgm.cs.mcgill.ca/~godfried/publications/Nanning-Proceedings-Paper.pdf


For rhythm analysis
http://www.rhythmrascal.com/

Here is another interesting video bout rhythm perception and practice by utilizing ordinary musical writing of time. So although the rhythm is not only the 2 numbers at the beginning of the pentagram, we can still convert the 0;s and 1's to ordinary musical writing. Here is how:

https://www.youtube.com/watch?v=Y5_27Gc28ls

https://www.youtube.com/watch?v=Ri0w4gLeeZE

https://www.youtube.com/watch?v=RuvA4b_2pk0

https://www.youtube.com/watch?v=6dA1aG54QHg

And here a video about polyrhythm

https://www.youtube.com/watch?v=c1pejTgLuhA

THE THREE ORGANIZATION LAYERS OF A RHYTHM.

1) The first scale or organization layer is of course the duration of the whole note in the melody notation which determines how fast is realized the writing of the notes.

2) The second organization layer is the rhythm of the measure , with its factors of dimension, order, and height.

3) The third organization layer is the rhythm is the pattern of repetitions of the melodic themes or melodic phrases, which in the current blog and book has been called the poetical measure of the melody as it corresponds to the poetic measure of the parallel lyrics (when they exist) e.g. 7-syllables, 111-syllables, 15-syllables 17-syllables etc.



There are 2 important remarks to make about rhythm while playing music and songs

1) Although  in a musical piece or a  song, it is supposed that when one is playing a musical instrument or is singing , one keeps a constant beat (usually measured with a mechanical or electronic ,metronome) the truth is that the best interpretation of the music is when the beet is slightly irregular, and it is accelerating or decelerating according to the  emotional content of the piece. It is like the human heart which is accelerating or decelerating according to stress or relax , joy or sadness. This is natural when one is laying alone. When he is playing with others usually the beat must be stable, unless all of the players agree to accelerate or decelerate it simultaneously.

2) The strumming pattern when playing a string instrument like a  guitar, need not me  the same through out the song. many jazz guitarists will assure you that although they try to keep the beet constant they change the strumming pattern to fit the chard changes of the song. 

Both of the above remarks show that although the rhythm and to keep the rhythm is very important in a musical piece or a song, also slightly changing the beet and making the rhythmic pattern more rich and irregular is also part of the freedom and better interpretation by the musician. 


TWO NUMBER RHYTHMS (M,N)

The rhythms is rarely determined by only two numbers. But there is a category of simple rhythms that are defined by only tow numbers. We call then two-numbers rhythms.

The way to define them is as follows. If m, n are the two numbers (e.g. m=3,n=4) then we take their least common multiple (m,n) here (3,4)=12, and we define an underlying but silent  constant beat of 12 hits (000000000000). Now every 3 of them a beat is sounding ,and also every 4 of them again a hit is sounding. The total results is (100110101100). 

If the numbers e.g, where m=2, n=3, then (2,3)=6, and the next rhythm would be defined

(101110). If the numbers where m=2,n=5,then (2,5)=10 and  the next rhythm would be defined (1010111010). Etc.

DOUBLING RHYTHMS

Such rhythms are common in Latin music and Latin jazz. The main idea, is that while playing, suddenly we double the number of beats within the same tempo, or quadruple them. In general multiply with power of 2. Then we return to the slower beating etc.

Here is an example of a tango rhythm

(11011010)

Another tango rhythm (Cuban Habanera) is the next

(10011010)

or simply (10101011)

A simple waltz rhythm would be

(100100100)



X, X+1 RHYTHMS
Such rhythms, start with a pattern x, then they repeat it as x, but they add also a one more beat 1. E.g. the Zembekiko of Greek Folk music: x=(11011010) x+1=(110110101) , in total
(11011010110110101).

ESCALATING AND DE-ESCALATING RHYTHMS (x, x+1, x-1, x-1) or (x, x+1,x+2,x)

These rhythms start with a sequence of n beats, then they continue with a sequence of n+1 beats, and then they fall back to two or one sequence of n-1 beats. E.g. this simple Samba rhythm E.g. n=3 , 304020020 or in detail (1110 111101100110) or simpler such rhythm which is not called samba n=2, 203010010 or in more detail (110111010010).

Or if x=11, then x+1=111, x+2=1111 and the samba rhythm here becomes the 16 beats (1101110111101100)=(203040200) .Notice that after the last 11, we have a pause of double duration than the usual. 
The previous simple samba rhythm can be considered also as two 7-beats and 9 beats rhythms as follows 7-beats (1101110) and 9-beats (111101100).
Other such rhythms may be 12-beats (101101110110)

The samba rhythm can be considered also as an attempt to fit a sequence of 3-beats and of 4 beats, within 16 beats. In other words this pattern (1100110011001100) with a pattern like (x00x00x00) , which be superposition result to the 16-beats (1110 111101100110).

ESCALATING AND DE-ESCALATING DOUBLING RHYTHMS (2x, 4x, x ,x) 
or (2x, 4x, 2x ,x)
A more peculiar such type of rhythms is when the escalation is by doubling. So they start with a pattern of 2n beats, they continue with 4n beats , and they fall back to n beats. E.g.

12-beat such rhythm (110111101010) or 16-beats such rhythm (1100111100100100)
Or  (11001111001100100)

A Gypsy jazz rhythm:
(11101010)

CETAN MUSIC AND IRISH MUSIC RHYTHMS

Some rhythms that I have found in Cretan and irish folk music are the next

(1010100010001000) (condillies)

and 

(1110)  (pidichtos Cretan dance)

also 

(1010)

Scottish reel groove (e.g. https://www.youtube.com/watch?v=EBC1pQawfy4 )

(10010010)

FLAMENCO RHYTHMS

SOLEA
(100100101010)

It is composed by two halfs (compass) the first is the (100100) and the 2nd is (101010)
SOLEA POR BOOLERIAS
(100100011010)

ALSO

(010010001101)

https://www.youtube.com/watch?v=yliXnfftN50




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