# Music as a Pattern Process is a way to design patterns. Let\'s zoom out and think about pattern generally: Working theory: ![[PUBLISH/media/music-as-process-media/image8.png]] ## Binary Patterns With any two elements A B there are a finite number of patterns of a given length, each with an inversion. We can ignore rotations (AAB = ABA = BAA). (What's any element? Anything, of any size - a note, a phrase, a section of a piece.) Length Pattern Inversion 2 AB BA 3 AAB BBA 4 AABB BBAA ABAA BABB **Work out a few of the 5-long patterns in As and Bs, and then choose one to animate a two-note texture.** ## Nesting / Mapping Patterns often nest inside of patterns. For example, drummers practice the pattern 'paradiddle' which alternates between one pattern and its inversion. On the level of individual notes, you have an 8 item cycle. But, zooming out slightly, we can also think of it as a simple AB: ABBA BAAB = *A B* Our letters can represent any thing at any scale. So far we've been mapping our As and Bs to individuals pitches, but you can map them to anything you want. Here are some examples: Create a nested pattern, invent your own mapping, and write it out. ## Ternary Patterns Patterns with three elements: Length Patterns: 3 ABC ACB 4 ABCA ABCB ABAC ACAB ... 5 ABCBA ABCAB ... 2. Work out a few of the 6-long patterns in ABCs, and then choose one to walk through a 4 chord progression. Here's an example with a 5-long pattern ABCAB, walking through this chord progression: 1\) Notice how I keep the voice-leading consistent --- 'A' begins as the F and remains the lowest note of each chord. B is always the middle, C is always the top. 2\) Notice how the pattern loops every 5 but our meter (4/4) makes groups of 4. The patterns spill over the bar lines. We hear the \'cycle-resets\' as a steady pulse going against the meter. ![[image11 2.png]] ## Rhythmic Dyads **Pulse** is a series of events of the same length. **Rhythm** arises from events of different lengths. The simplest, smallest rhythm is two events in a three-pulse cycle: \|: x x . :\| A A B Short-long --- a heartbeat. Notice it is still a binary pattern, only our B is now a rest. Because it has two events, we'll call it a dyad. Here it is in 8th note subdivisions and 8th note triplets: `missing image` 1. Write out the heartbeat pattern in sixteenth note subdivisions, and then quintuplet sixteenth notes. Each time, repeat the pattern until it comes out even. How many times does it take to come out even? Handy formula: **length of cycle (3) x length of other cycle (5) = total number needed to come out even (15)** (Credit to Miles Okazaki\'s *Fundamentals of Guitar* for some of this.) ## Rhythmic Triads Three-event patterns in cycles of different length. (Limiting the maximum length of any event to 3). \|\|: x x x . :\|\| 3 events in a four-pulse cycle A A A B these are all still binary patterns \|\|: x . x . x :\|\| 3 in 5 2+2+1 \|\|: x x . . . :\|\| 3 in 5 1+4 \|\|: x x . x . . :\|\| 3 in 6 1+2+3 \|\|: x x . . x . :\|\| 3 in 6 1+3+2 \|\|: x x . . x . . :\|\| 3 in 7 1+3+3 \|\|: x . x . x . . :\|\| 3 in 7 2+2+3 \|\|: x x . . x . . :\|\| 3 in 7 1+3+3 4. Translate one of these patterns into notation, choose a base duration (quarter notes, eight notes, triplets, etc) AND METER, and work out the rhythm a few cycles until it comes out even. ## Maximally Even Rhythms Consider this cycle: \[ x . . . x . . . x . . . x . . . \] We have 16 steps with 4 strikes, equally distributed. By our definition it's not really a rhythm, it's a pulse. Rhythms are uneven time-patterns, and there's a particular type of un-even-ness that humans seem particularly attracted to: rhythms that are **imperfectly but maximally even**. Close-but-not-quite. Start by choosing a number of strikes and steps that don't divide evenly. Spread out the strikes as evenly as you can. \[ x x . x . . . . \] - too clumped at the front \[ x . . x . . . x \] - clumps when it repeats \[ x . . x . . x . \] - just right These are also called Euclidean Rhythms. There's an algorithm he described 2200 years ago that finds ideal solutions. We don't necessarily need ideal solutions, we're going to do them just by feel. Many of the traditional musics from around the world are built on these kinds of rhythms. Persian *Khafif-e-ramal* (13th c.) \[ x . x . . \] Bulgarian *Ruchenitsa*: 4 strikes in 7 steps: \[ x . x . x . x \] ![[ruchenitsa.mp3]] Tango: 5 in 8 \[ x x . x x . x . \] Arab *Agsag-Samai:* 5 in 9 \[ x . x . x . x . x \] Ewe Bell Pattern: 7 in 12 \[ x . x x . x . x x . x . \] Clave: 5 in 16 \[ x . . x . . x . . . x . x . . . \] For more on this topic, see papers by Eric Bjorklund and Godfried Toussaint. 5. Chose a number-of-steps and number-of-strikes that don't divide evenly. Put an x in the first position, and then find a way to spread the rest out that seems even-ish. Translate to notation. ## Pentatonic and Diatonic Rhythms Here are two special Maximally Even Rhythms: Consider the pentatonic scale in numbers of half steps: 0 1 2 3 4 5 6 7 8 9 t e the twelve chromatic notes C D E G A the pentatonic scale x . x . x . . x . x . . 2 + 2 + 3 + 2 + 3 This is the 'Son clave,' the fundamental rhythm of Afro-Cuban music. It also appears in a treatise from the 13th century Baghdad (Kitāb al-Adwār, which calls it the al-thaqil al-awwal, and is known in rock and roll music as "the Bo Diddley beat."  Now let's do the same with the diatonic scale: 0 1 2 3 4 5 6 7 8 9 t e the twelve chromatic notes C D E F G A B the diatonic scale x . x . x x . x . x . x 2 + 2 + 3 + 2 + 3 You end up with the 'standard bell pattern' or *bembé*, the most commonly used key pattern in sub-Saharan Africa.  ![[standard bell.mp3]] Ex. *Bénin: Rythmes et chants pour les vodun (1990)* Isn't it surprising that the two most common tonal sounds in the world are structural analogs to two of the worlds most important rhythms! We like maximal even-ness. Part of the magic of these two rhythms is that they can be felt in either duple or triple time. Rhythms built around them can play a game of emphasizing the 'two-ness' or the 'three-ness' in different ways. To see how this works: **Write Son clave in 3/4 and the standard bell pattern in 3/2. (All notes and rests stay 8th notes, only the beaming and bar lines change).** ## The 6/8-3/4 Game Aka *Hemiola*. Here, Bach does it with phrasing (and harmonic rhythm) --- the eighth notes are in groups of 3s and then groups of 2s for the cadence.  Here's an example from *Short Ride in a Fast Machine* by John Adams. Here 3 half notes in the horns are played against two half-note-long clarinet loops: Ravel writes the game into his time signature in his *String Quartet in F*. You can see how the inner voices divide the bar into two groups of 3 (6/8) while the first violin and cello divide it into three groups of two (3/4):![[ravel 2.mp3]]  **Write a melody in 6/8 that sometimes divides the bar into two groups of three and sometimes into three groups of two.** **Write two lines against each other, one moving in groups of 2s and one moving in groups of 3s.** ## Polypulse Now let's generalize this idea. This is the two against three we've been looking at: x . x . x . x . x . x . \[2\] x . . x . . x . . x . . \[3\] Any pair of numbers can be set against each other this way: x . . . . x . . . . x . . . . \[5\] x . . x . . x . . x . . x . . \[3\] We've divided the common underlying pulse (every x or . ) into two new pulse streams pulsing at different rates. How many pulses does it take for 5s and 3s to line back up? 5 x 3 = 15. In notation things can get a bit gnarly. It's simple enough when your meter reflects the common pulse, as in the first two examples below. But when one of the polypulses is the meter, tuplets are usually needed (look at the top voice of \#1... now imagine the group of 5 squeezed into one quarter note, and the bottom group of 3 with it... that's \#4). The trick is to say: **"5:3 --- every fifth triplet --- every third quintuplet"** as in the examples below: **Choose a different two numbers that don't divide evenly. Work out all four ways of notating it, just like the example.** ## Polyrhythm Same as polypulse, except instead of pulses (x's are evenly spaced) --- x . . . . x . . . . x . . . . x . . . . x \[5\] x . . x . . x . . x . . x . . x . . x . . \[3\] --- we use rhythmic patterns (x's are not evenly spaced). I'm introducing 'o' as a rest to distinguish it from the periods of our underlying common pulse. Here\'s our heartbeat: x x o A A B x . . . . x . . . . o . . . . x . . . . x . . . . o \[5\] x . . x . . o . . x . . x . . o . . x . . x . . o . \[3\] Here are the same four notation examples as in the previous lesson, now with rhythms (note the bottom two examples are incomplete (you have to continue the pattern, not loop the bar.)  **Write a rhythm. Repeat it along your two pulses from the last exercise.** ## Phone Number Notation Since we\'re thinking about number games in music, it\'s a good time to introduce this tool. I learned this from the group **Sō Percussion**. They often write music as a series of numbers, starting with something like a phone number, or a sequence of the numbers of letters in the words of a phrase. These numbers can be rendered into a rhythm in the main pulse, or in subdivisions of the main pulse. For example, for the phone number 325-1268  These can be layered in polyrhythm: ![[image20 2.png]] Or they can determine a pattern of changing subdivisions: ![[image20 3.png]] In **Jason Treuting's** *Go* from *Amid the Noise*, he uses a number sequence to determine phrase lengths. Once you learn the complete phrase of music (bar \#6)...  Then all you need to play the piece is a short score like this. (Letters are for another player.) One benefit of this approach: **complex rhythms that are easy to memorize.** I think their whole show Invisible Cities was based on phone numbers and addresses from their childhoods. The core material for a 75+ minute show was already memorized. Another benefit of this approach: you\'re thinking in numbers, so number-based transformations, games, and processes are easy to invent and deploy. Write your phone number as a rhythm using one of these strategies. ## [[Combination and Permutation]] Representing musical information -- rhythms, notes, anything -- as a string of numbers is a powerful tool for working out patterns. In the 20th century, many composers sought to establish a \'rational\' basis for modern music, which usually involved making lots of charts of numbers before making a piece. But it\'s a very old tool. For example, *change ringing* is a tradition of bell ringing originating in 17th century Anglican church. The goal of *change ringing* is to ring the church bells in strictly controlled, non-repeating permutations. These are executed by a group of bell ringers, one per bell, standing in a circle, from memory. There are tens of thousands of compositions; an old standard is the Plain Bob Minor which weaves bells 1 and 2 in linear paths through the changes:  Plain Bob Minor mapped onto the floor:  Compare to Tom Johnson's floor map score for *Nine Bells*.  Compare to *[merukhand](https://www.scribd.com/document/357378952/MERUKHANDA#logout)* [[Combination and Permutation|permutation]] practice in Indian music pedagogy. Change-ringing and *merukhand* share some values with much of the thinking in avant-garde composition in the 20th century West. - **completeness** (exploring every possibility) - **non-repetition** (nothing unnecessary) - **series** (order matters) - **process** (iterative transformation) ## Lou Harrison Lou Harrison has an interesting approach to permutational music design:   ## Pattern Inspiration ### Crazy Quilts Think about the ratio of pattern to non-pattern, the oscillating rhythm of predictability and unpredictability.   ### Wallpaper Groups There are only 17 possible 2D tiling patterns exhibting different lines of symmetry, rotation, and inversion. How many ways can you make your pieces fit together?  ### Formal Grammars **Axiom: B** **Rewriting rule: A\>AB B\>A** Generation: 0 B 1 A 2 AB 3 ABA 4 ABAAB 5 ABAABABA 6 ABAABABAABAAB 7 ABAABABAABAABABAABABA These strings can be read as a list of instructions. For example: A: step up, make a note B: leap down a third, make a note Gives you:  Invent your own characters, rewriting rules, and interpretation instructions. Rewriting rules can involve chance. Lindenmeyer or L-systems are a type of formal grammar used for modeling botanical structures. The strings are used as drawing instructions for line segments at angles.  ## Number Sequence Research For the numerically interested, the [On-Line Encyclopedia of Integer Sequences](https://oeis.org) is an incredible research resource. You can search for any series of integers and it will tell you all the interesting places they appear in mathematics.