OpenLLM-France/wikimedia huggingface dataset:
# Intuitive Trigonometry/Radians and Arc Length
## Radians and Arc Length
### Finding the formula for arc length

First, let us examine the formula for arc length. Imagine some arc, and then extend each of the ends. Eventually, you will end up with a circle (see Figure 1). We know that the circumference of a circle is $2\pi r$, which we can imagine as the circumference of a full circle. Every arc length must then be equal to or a fraction of $2\pi r$.
Next, imagine a circle, with an angle in it that opens onto a section of the circle (see Figure 2). As the angle varies, clearly the arc length varies. In fact, if the angle is $360^{\circ }$ it is easy to see that the arc length must be $2\pi r$. Using this, we can see that to find the length of any arc, all we must do is multiply the circumference of a circle (a "full" arc), $2\pi r$ by the angle measure out of $360^{\circ }$, leaving us with the formula $\ell _{\text{arc}}=2\pi r\left({\frac {\theta }{360^{\circ }}}\right)$.

### Definition of a radian

To derive the conversion rate between radians and degrees, we must know what exactly a radian is. A radian is defined as
    A unit of angle, equal to an angle at the center of a circle whose arc is equal in length to the radius.

### Deriving the conversion rate between radians and degrees

You can derive the relation $2\pi {\text{ rad}}=360^{\circ }$ using the formula for arc length that we just derived. Take the formula for arc length, or $\ell _{arc}=2\pi r\left({\frac {\theta }{360^{\circ }}}\right)$. Assume a unit circle; the radius is therefore one. Knowing that the definition of radian is the measure of an angle that subtends an arc of a length equal to the radius of the circle, we know that $1=2\pi \left({\frac {1{\text{ rad}}}{360^{\circ }}}\right)$. We can further simplify this to $1={\frac {2\pi {\text{ rad}}}{360^{\circ }}}$. Now we can multiply both sides by $360^{\circ }$ to get $360^{\circ }=2\pi {\text{ rad}}$.

### Example conversion between radians and degrees

Suppose we are given the degree measure $30^{\circ }$ and asked to convert it into radians. We know that $2\pi {\text{ rad}}=360^{\circ }$, so we can set up a proportion: ${\frac {x{\text{ rad}}}{30^{\circ }}}={\frac {2\pi {\text{ rad}}}{360^{\circ }}}$. We can then cross multiply, getting $360^{\circ }\times x=2\pi {\text{ rad}}\times 30^{\circ }$. Dividing each side by $360^{\circ }$, notice first that the degree signs cancel, so we are left with $x={\frac {60\pi {\text{ rad}}}{360}}$ which simplifies to $x={\frac {\pi }{6}}{\text{ rad}}$. The problem is solved (notice it is customary to leave it in fraction form as it is more exact).

### Problems

All solutions in Appendix 1.
* Find the length of an arc whose related angle is $10^{\circ }$ and whose related circle has a radius of $2$.
* Convert ${\frac {\pi }{2}}{\text{ rad}}$ into degrees.
* Convert $45^{\circ }$ into radians.
* Find the measure of the angle related to an arc of length $5$ on a circle with a radius of length $4$.

### References

* ↑ Google definition
# Introspector/Toolchain

Toolchain :
A set of tools that interoperate, exchange data. A Suite of tools. Normally this means a compiler toolchain.
http://www.linux-mips.org/toolchain.html "" A toolchain is a complete collection of compiler and binutils programs and can be run either as a cross-compiler, or native on the target (if performance allows). ""
# Intuitive Trigonometry/Introduction
## Introduction

The purpose of this book is to provide a way of trigonometry that is simple, intuitive, and where it is clear where the ideas come from. In most trigonometry classes, the material is simply thrown at the student, with no explanation as to why anything is. Here, it is attempted to build the subject from the ground up. At the same time, it tries to avoid anything overly rigorous. There is no need for advanced, perfect, research-quality proofs at every step. However, there is an appendix with some details on how to write a good proof. The student should have an understanding of basic algebra and geometry, however, as much is explained in the book as possible. The last chapter contains an application of calculus to trigonometry; this section is not necessary for learning the subject, but does require knowledge of calculus.
The book starts from the very basics, explaining the conversion between radians and degrees, then progressing through the trigonometric functions, the unit circle, trigonometric identities and solving trigonometric equations, graphing trigonometric functions, and finally, the application of calculus to trigonometry. Problems are included at the end of each chapter, to check one's understanding of the material; solutions to the problems are provided in Appendix 1.
Hopefully this book will more easily guide you through trigonometry, and act as both a self-learning resource and a good accompaniment to whatever classes you may be taking.
# Music Theory/Real Book Errata
### The Real Book, Volume II, Second Edition

The Real Book, Volume II, Second Edition. Published by Hal Leonard Corporation, Milwaukee WI. ISBN 0-634-06021-X
Songs that were in the original Real Book Volume 2 that are absent in the Hal Leonard version of the Real Book Volume 2:
"728"
All God's Chillun Got Rhythm
April
Backstage Sally
Ba-Lue Bolivar Ba-Lues-Are
Beethoven's Blue 3rd
Better Git It In Your Soul
Beyond All Limits
Bills' Hit Tune
Blue A La Mode
Blue And Sentimental
Blue Moon
Blues Connotation
Blues For Philly Joe
Bohemia After Dark
Good Bait
Boogie Stop Shuffle
Brazilian Beat
Bright Moments
Brown Skin Girl
Bud's Bubble
But Not For Me
Byrdlike
Can't We Be Friends
Careful
Careless Love
Catch Me
C'est What
Chase
Confessin' (That I Love You)
Cookin'
Cork 'n Bib
Cute
Day In Vienna
Decision
Deep Purple
Diane
Domingo
E.K.'s Blues
Embraceable You
Enchance
Evening In Concert
First Moves
Flim Flam
Flintstone's Theme
Fly By Night
Fools Rush In
For Minors Only
Freddie Froo
Free
Free Cell Block "F", Tis Nazi U.S.A.
Friday the 13th
Funk Dumplin'
Games People Play
Gentle Rain
Gentle Wind and Falling Tear
Ghost Of A Chance
Golden Notebooks
Groove Merchant
Groovy Samba
Harlequin
Hello
Here's That Sunny Day
Horacescope
Horizon
Hummin'
Humpty Dumpty
I Cover the Waterfront
I Get A Kick Out Of You
Ice Cream Konitz
If You Could See Me Now
I'm Getting Sentimental Over You
Incentive
Is It Really True?
It Had To Be You
It's Only a Paper Moon
I've Found a New Baby
Jaco
Je Ne Sais Pas
Jorgie's
Joy To the World
Jump For Me
Just You, Just Me
Kentucky Oysters
La Nevada Blues
Lady's Blues
Last Page
Laura
Lennie-Bird
Lester Leaps In
Liberated Brother
Lies
Light as a Feather
Love For Sale
Lunar Tune
Lydian April
Man I Love
May-Reh
Mellow Mood
Modesty Blues
Monk's Sphere
Moontrane
Morning
Moving Out
Mr. Broadway, Theme From
Mr. Clean
Mr. Jones
Mr. Magic
Mr. Sims
Mrs. Miniver
Muezzin'
Mystic Touch
Nancy Joe
New Thing
No Me Esqueca
Oh, Lady Be Good!
On A Clear Day
One for Daddy-o
On the Stairs
On the Trail
Once In A While
Opus De Funk
Our Love Is Here To Stay
Palo-Alto
Patterns
Penny Arcade
Perdido Line
Progression
Preacher
Quicksilver
Ramblin'
Recado Bossa Nova
Remark You Made
Round Trip
Samba Du Bois
Scene Is Clean
September In The Rain
Serenity
Seventeen West
Shiny Stockings Short
Riff
Short Stop
Sister Sadie
Sleeping BeeSliver's Serenade
Slow Hot Wind
Someone To Watch Over Me
Song for My Lady
Sound Lee
Spring can Really Hang You Up the Most
St. Thomas
Star Eyes
Summer Knows
Summer Time
Sunshine Express
Sweet Georgia Brown
Tadd's Delight
Taking A Chance On Love
Tautology
Their Tears
Things Ain't What They Used To Be
Thriving On a Riff
Tickle-Toe
Time After Time
Toad's Place
Two Not One
Vonetta
Walk Tall
Wall Street
Watersign
Weekend Blues
Wrong Is Right
Yana Amina
You Better Leave It Alone
You Stepped Out Of A Dream
# Music Theory/Necessary Tools

One of the wonderful aspects of composing music is that you can use as few or as many tools as you want and need.
The minimum you'll need is probably:
1. Some way of storing the music you write:
2. Some way of listening to the music you write:
3. Some inspiration to help you write the music you write.
If you go with the simplest setup, some notes you sing to yourself written on lined paper (which works wonderfully if you have a good ear for pitch), it'll cost you the price of a pencil, a few sheets of paper, and a couple erasers.
However, if you'd like to invest thousands of dollars in a complex MIDI-enabled computer, keyboard, drum-set equipped, high-fidelity sound environment, well, you can do that too. Just don't expect to finish setting that up for a few weeks.
The rule of thumb is: the more complicated you want your composition environment to be, the longer it'll take you to actually start writing music.
# Music Theory/New Age

New Age music is a style of music that corresponds with the New Age movement. Like the spiritual movement with its amalgam of religious, spiritual and cultural practices, New Age music is characterized by a blend of musical elements appropriated from different cultures.
Harmony is typically simple and often modal, and there is a strong emphasis on ephemerality. New Age music often features "pads" and "ethnic" vocals of indeterminate Middle-Eastern/African/South Asian origin.
# Music Theory/Plainsong

Plainsong is a form of music that originated in Christian churches early in the church's history. Christians wanted to praise God in a special way, and so chose to sing the entire liturgy when they held services. At first, with archaic and variant practices, such early plainsong was linked to specific communities and based on folk melody. However, as specific forms of liturgies were cemented, particularly the Liturgy of St. John Chrysostom, the Liturgy of St. Basil, and the Liturgy of Pope Gregory I (the Mass), plainsong grew to be a formalised and integral part of christian worship.

## Notable Features

Plainsong is characterised by a number of key features:

### Sonority and Instrumentation

* It is usually unaccompanied. The phrase "a capella" literally means "from the chapel/church", and implies an unaccompanied texture. In a modern context, sometimes instruments such as the organ can provide basic harmonic support, but this is minimal and very rare.
* It has a reasonable range. As it is designed to be sung by anyone and everyone in a church congregation, the range of notes for a particular piece of plainsong rarely exceeds one octave, and it is not very virtuosic.

### Melody

* It is melody-driven. At its root nearly all plainsong is a single monophonic line.
* It is scalic. Where there are leaps, they are at most a perfect 5th. Most plainsong is dependent on scalic motions.
* It can be very ornate. Melismas on particular words are very common.

### Harmony

* As plainsong is typically unaccompanied, harmony does not apply. Where harmony is used by certain traditions, it is block homophony, and quite simple.
* Pedal notes and drones are sometimes used to underpin the melody.

### Tonality

* Plainsong used all modes available, with 8 identified for primary use, and 4 others used archaically.

### Texture

* Monophonic. A single solo melodic line is ubiquitous.
* Homophonic. In certain traditions, harmony (or pedals and drones) are used as a basic accompaniment. The melody drives this.

### Structure

* Strophic. The earliest plainsong hymns (as an example, Tantum Ergo by Thomas Aquinas) are based on repeated verses.
* Through-composed. Most plainsong is simply designed to match the words, so melodies are not repeated.

### Tempo, Rhythm, and Metre

* Tempo is at natural speech rhythms. In a modern context, plainsong is often sung quite slowly, but it can be sung at a very fast pace.
* Metres were non-existent when plainsong was written. This was in favour of natural speech rhythms.
* Rhythms are simple, mainly to do with lengthening certain syllables with longer notes for emphasis.

## Composers

It is often difficult to pin down exactly who wrote each piece of plainsong. Certain authors of texts, such as St. Thomas Aquinas, are well recognised. Ironically, even though plainsong is often all painted under the brush "Gregorian chant" it is difficult to categorically say whether Pope Gregory I wrote any plainsong at all. Other composers are know. Hildegard of Bingen is known for her developments based on plainsong. Many composers have been known to use plainsong in their liturgical works.
# Music Theory/Modulation

Modulation is the process of changing key center. Modulation occurs in a number of ways, most of them involving some way of making the current key ambiguous and reaching a cadence in a new key. The key chosen is usually one closely related to the original key, those that share the key signature or have only one sharp or flat different in the key signature (for example, in C major, A minor, G major, and F major are all closely related keys). Modulation to other keys is not unknown, however, particularly in the Romantic era.
There are several means used to modulate. One is simply to introduce a new key, though this does not always sound very convincing and many consider it quite crude. In classical music, modulation is usually prepared and carried out through the use of pivot chords, which exist in both the starting key and the intended destination. Especially during the Romantic era, the possibilities of chromatic pivot chords, such as the diminished seventh, were heavily exploited.

## Pivot chord

A pivot chord is a chord that is common to both the key you are moving from and the key you are moving into. The pivot chord begins as a particular chord in the original key (eg. iv) and is then used as a different chord in the new key (eg. ii). This creates a smooth transition to the new key.
Eg. To modulate from C major to G major, the chord a- is common to both keys, and can be used as a pivot chord as follows:
C -> a -> D7 -> G
(I -> vi/ii -> V7 -> I)
Another method of modulation is to treat a major chord in a given key as the dominant chord in a new key by adding a minor 7th above. Commonly, the tonic chord is used like this to modulate to the sub-dominant:
Eg. in C major, modulating to F major
C -> C7 -> F
(I -> I7/V7 -> I)
Or to modulate to Bb, one may follow this progression:
C -> F -> F7 -> Bb
(I -> IV -> IV7/V7 -> I)

## Uses of modulation

Modulation is integral to sonata form, where a modulation to a (usually) related key in the exposition sets up an important tonal contrast in the development. Modulation is also used in other forms like the fugue and rondo to create contrast within unity.
# Music Theory/Playing by Ear

Playing by ear is both inherently difficult and inherently easy due to two things:
* The music is played as heard — therefore, rhythmic values and expression are easier to pick up and replicate in one's own playing.
* The music is, more often than not, unknown as a piece to play to whomever is playing it, and as a consequence he or she, depending on the attention to which they feel inclined to play the piece, will have to work out key, chord changes, melody, harmony and technicalities like fingering on their own.
Some people have a gift known as perfect pitch, that is, they are able to tell from listening to a specific note or chord what it is. This may aid them immensely when they try to transcribe a piece of music or play it by ear.
The best way to learn to play by ear is to first develop your ear, that is, play easy songs without notation. Play along to the song, so you know you are in the right key and always use an instrument that is in tune (if applicable). If you are blessed with a particularly good memory, you will be able to memorize songs and chord changes without too much difficulty. This gift does not have a name, but is comparable to perfect pitch in its use and applications. Playing songs in one's head in the right key is not difficult after some ear development, so play by ear when you don't have notation.
Not all people have the gift of perfect pitch, but one skill that you can develop is the ability to recognize relative pitch. That is, being able to tell the interval between a note and a reference note in a melody, being able to play a specific interval between a melodic note and bass note (typically 1, 3, or 5 intervals below the melodic note), and being able to play the notes of a specific chord based on the a given bass note (e.g., 1, 3, and 5 intervals above the bass note). This ability not only enables you to play a song by ear, but also play a song in any key. For example, you hear the first three notes of the melody of a certain song. Although you would not be able to exactly pinpoint what the actual notes are, you would still be able to duplicate what you heard (but perhaps not in the original key) by identifying the intervals between those three notes.
Learning this skill means learning to play, and more importantly, hear the different intervals between notes: unison, minor 2nd, major 2nd, minor 3rd, major 3rd, perfect 4th, and so on.
# Music Theory/Reggae

Reggae is a style of music which originated in the Caribbean. Bob Marley is one of the most famous reggae musicians along with Peter Tosh.
Reggae usually has a 2/4 or 4/4 with accented 2nd notes time signature, and a low tempo.

## Theory

Reggae can only be played in a 4:4 or 2:4 measure, meaning it has to have 4 beats per bar.

## Instruments
### Drums

Most of the time there are eighth notes on the high hat. A loud snare or rim shot can almost always be heard on the third beat. The bass drum varies: sometimes it's just on the first beat, the first and the third, every beat, or even just the third.

### Guitar

The standard rhythm guitar in reggae uses quick downstrokes on the offbeat (The 'And' between the beats). 1 and 2 and 3 and 4 and. This is often referred to in reggae as the "skank" and is what gives reggae its characteristic sound. Often you will hear the rhythm being played on the 'and' beats as one quaver beat or two semi quaver beats

### Bass

A reggae bass line embodies the "backbone" to the rest of the overall "riddim". Its important to stay simple while occasionally leaving space to coax the rhythm forward. Three things to aim for are denseness, tension, and momentum. Start simple focusing on the root or the 3rd. Inversions are used frequently while using the 5th sparingly adds a certain sense of satisfaction when you finally hear it. Coming in on the strong beats 1 and 3 are helpful while making good use of the 16th note and the triplet is where your groove can be found. Find a motif, use the call and response technique, and play the octave to end a phrase or finalize a musical statement. All the high frequencies (above 1000 Hz) should be filtered out. Bass is often considered the "lead" instrument in reggae because it is responsible (along with the drums) for establishing a song's groove.

### Vocals

The vocals in reggae are often religious or telling some sort of story but they don't have to be. For example, there are songs about love, bank robberies, and seemingly nothing but a feel good vibe . As it is music "Of the people, for the people, by the people", Reggae lyric often speaks to the masses about raising awareness and vibe, and freeing all people.

### Keyboards

Keyboards are mainly responsible for the "bubble", which aids the "skank" to add to the momentum of a given "riddim". Most common reggae keyboards are the Hammond Organ, Rhodes and Clav. They share the rhythm of the guitar usually played in a staccato pattern over the "riddim". The following video demonstrates an organ "bubble/shuffle" - https://www.youtube.com/watch?v=JV0ANC2r6SA

### Horn Section

Horns in Reggae are the cherry on top of the "Riddum". Usually utilised to show the difference between "Versions", or simply to deliver after lyric.
# PBASIC Programming

This book is about PBASIC programming using the Parallax BasicStamp family of microcontrollers.

## Table of Contents

* Introduction
* History
* Hardware and Development Kits
* PBASIC Editor

### Computer Arithmetic

* Data Types
* Variables and Constants
* Basic Arithmetic

### Branching and Control Flow

* Labels and GOTO
* Branches
* Loops
* Subroutines

### Port I/O

* Input and Output
* Serial Communications
* PULSIN and PULSOUT
* RCTIME

### Advanced Arithmetic

* Trigonometry Functions
* Other Functions

## Resources

* Wikiversity:Introduction to Robotics
* Embedded Systems/PIC Programming lists PBASIC and other programming languages for the PICmicro
# Oz Programming/Syntax

{Browse 'Hello World'}
# Oz Programming/Data Types/Value

Value is the root of all data types in Oz. The Oz language is dynamically-typed in the sense that when a variable is created, its data-type as well as its value are unknown. Only when the variable is bound to an Oz value, does its type become determined.

## Subtypes of data type Value
### Number

* Int
** FDInt
*** Char
* Float

### Record

* Tuple
** Literal
*** Atom
*** Name
**** Bool
**** Unit

### Chunk

* Array
* Dictionary
* BitArray
* Class
* Object
* Lock
* Port
# PBASIC Programming/Branches
## Comparison Operators

A computer processor, such as the BasicStamp is capable of comparing two numbers and testing their relationship. The BasicStamp can compare two numbers and determine if they are equal, not equal, greater than, or less than. We can use special comparison operators to ask the BasicStamp to compare numbers.

### Equal

    To determine if two numbers are equal, we use the "=" operator.

### Not Equal

    To determine if two numbers are not equal, we use the "<>" operator.

### Greater Than

    To determine if one number is greater than the other, we use the ">" operator.

### Greater Than or Equal

    To determine if one number is greater than or equal to another number, we use the ">=" operator.

### Less Than

    We can use the "<" operator to determine if one number is less than the other.

### Less Than or Equal

    We can use the "<=" operator to determine if one number is less than or equal to another number.

## IF / THEN Branches

Once we compare two numbers, we can ask the BasicStamp to take action depending on that comparison. Using the IF / THEN structure, we can perform a jump if the condition is true, and we will not jump if the condition is false. The format of the instruction is:
Where [Condition is a comparison of two values (two variables or a variable and a constant, for instance). If the comparison is true, then the control flow jumps to the [Label].

## IF / THEN / Else Structures

Sometimes we want to have a second option. for instance, we want to do one thing if the comparison is true, and we want to do a different thing if the comparison is false. Here is an example, using a single IF / THEN branch, and GOTOs:
Only one set of code in this case will be executed, depending on whether the condition is true or false. There are other ways to arrange this, of course.

## Simple Loops

We can use GOTO to create a simple infinite loop. The program will repeat the code in the middle of the loop forver:
We can use IF / THEN branches to create more advanced loops as well. However, most of the advanced loops that we can create can be made in a better way. We will discuss more advanced loops in the next chapter.