General
– Regularly ordered Patterns of Duration and Strength of Notes
– Percussion as Cymbal, Taiko, Bongo etc. have different Timbres
– We can use Poetry to Introduce a “Rhythmic Language” and Form Sentences similar as in Music
– It´s about how many Notes you put in a Particular Measure (or how many Words in a Sentence)
– African Music uses more of a 3:4 to Form 12 Beat Groups (more Complex as the Duration is Longer and more options are available as the Ratio is 0.75 instead)
Example
– How many Ways are there to put Short Sibilances & Long Sibilances together to Form a Pattern?
– The Resulting Numbers are called “Fibonacci Numbers”
– Each One is the Sum of the Previous 2
– Count “1+2=3”, “2+3=5”, “3+5=8”, “5+8=13”, “8+13=21”, “13+21=34”
1) In a 1 Beat Pattern = 1 Way, either Short or Long
2) In a 2 Beat Pattern = 2 Ways , i.e. Short + Short or Long
3) In a 3 Beat Pattern = 3 Ways, i.e. Short + Long, Long + Short, or Short + Short + Short
4) In a 4 Beat Pattern = 5 Ways
5) In a 5 Beat Pattern = 8 Ways
6) In a 6 Beat Pattern = 13 Ways
7) In a 7 Beat Pattern = 21 Ways
8) In a 8 Beat Pattern = 34 Ways….etc.
Notation
- We use Notation in a Time Signature
- e.g. if we Do something in 3/4 Time (3 Quarter Notes per Bar), Each Measure has 3 Beats = Top Number (# of Beats per Measure) & Each Beat is 1 Quarter Long = the Bottom Number (Length of each Beat)
– The Top can be any Number while the Bottom Number has to be a Power of 2 (means Bottom stays Dividable by 2)
– Use “p” and “q” to Look for common Subdivisions as within Mathematical Fractions,
– i.e. “p” (Top Number) against “q” (Bottom Number) LCM “p:q”
– e.g. in a 3:5 Signature you need 15 Divisions so they meet each other again (at 1 and then at 15)
– Whole Note = 4 Beats
– Half Note = 2 Beats
– Quarter Note = 1 Beat
– Eight Note = 1/2 Beat
– 16th Note = 1/4 Beat
Rests
– Whole Rest = 4 Beats
– Half Rest = 2 Beats
– Quarter Rest = 1 Beat
– Eight Rest = 1/2 Beat
– 16th Rest = 1/4 Beat
– If we Want a Note that is Not a Fraction with the Power of 2 in it, we can use other Notation as Triplets or Quintuplets (also a Beat, but No longer divided into 2, 4th, 8ths or 16ths etc.)
– a Dot after a Note is to use Half of the Length of the Note, means if we had a 2 Beat Line we would add the Half of that 2 Beat Line = 1 Beat additionally to the 2 Beats = 3 Beats (same works with Rests)
Single Rhythm
– A Single Part playing different Rhythms
– Standard Time Signatures are 3/4, 2/4, 2/2, 6/8
– 6/8 is particular Interesting cause 6 = 2 x 3 (which is a Part of two different Primes), you can think of 6 as 2 Groups of 3 or you can Think of it as 3 Groups of 2
– 2 Groups of Triplets
– 3 Groups of 2
– 1 1/2 is the Ratio of the Length of the 2 Groups known as “Hemiola”
– you may try Out other Mathematical Stuff based on that Hemiola (Brubeck’s “Time Out” has a lot of Interesting Patterns)
Polyrhythm
– Two Parts playing different Rhythms in the same Span of Time
– Time Signatures as 4/2, 3/4, 3/2, 3/5, 4/7, 5/11
– use to create a Sense of Instability before resolve Rhythm (similar to Dissonance<->Consonance)
– Yves was famous for using Church / Traditional Melodies and he sometimes used 2 Conductors for the Orchestra to Manage different Rhythms
– People use Words as Lead to Accent a particular Rhythm
Simple Example 4/2
4 (4 Quarter Notes in Part 1) against 2 (2 Half Notes in Part 2)
Medium Example 3/4
3 (3 Quarter Notes Per Bar Part 1) against 4 (1 Beat per Bar Part 2)
Advance Example 3/2
– Hemiola Formula used Simultaneously instead Sequentially
3 (3 Quarter Notes Per Bar Part 1) against 2 (2 Beats per Bar Part 2)
Original Piano with the “Agitative” Feeling
Complicated Example Chopin
– Remember, p against q LCM = p:q, means in a 3:5 Signature you need 15 Divisions (p and q look for common Subdivisions as within mathematical Fractions) or 28 Divisions in 4/7 (basically means you Split Bottom Time Signature Number into 28 tiny Fractions or in Live a Grid Division of 28 Lines per Bar)
Combining Pitch & Polyrhythm in a 3/5 Signature Example
– Melodies together Repeat every 15 Notes (again you take the Least Common Multiple of 3 and 5 you get 15)
– And since there is 5 Notes in each of these Bars it Repeats every 3 Measures
– You get a Sense of Returning, imagine two Gears where one is Smaller that the other and Turns a lot more, sometimes they are in Sync and then they Drift away for 15 Beats until they Match Up again
Math and Music Coda
– in Math, when we keep on adding only Positive Numbers until Infinity, we do Not End in Positive Numbers but it Converges to 1 instead Growing (Geometric Series in Math)
We Proof that through Music:
1) Start Off with a 4/4 Whole Note of 4 Beats per Measure = 1 Whole Note Equals 1 Measure = 1
2) Now replace that Whole Note with 2 Half Notes which is 2 Beats each, mathematically we still look to 1/2 and another 1/2 Note but it Equals 1 Measure
3) Now replace the 2nd Half Note with 2 Quarter Notes which are 1 Bar each, mathematically we have a 1/2 plus 1/4 plus 1/4 and that Equals 1
and so on…
- So each Time we keep taking the Last Note and replacing it by 2 Notes, each of which is Half as Long as the One we replaced, keeps the Measure always the same – 1 Measure Long
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