Chapter_2_Pitch_Classes_and_Intervals.jl

### A Pluto.jl notebook ###
# v0.19.27

using Markdown
using InteractiveUtils

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include("post_tonal_tools_min.jl")

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md"""
# Chapter 2: Pitch Classes and Intervals
"""

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md"""
## Pitch Class

A pitch class (```PC```)[^1] is any name associated with the sensation of a quasi-discrete sounding pitch, for example:

	C, do, etc.

## Pitch Space

A pitch space (```PS```) designates where a ```PC``` is situated in a pitch space region, e.g.:

	C4

```C4``` known as middle ```C```, MIDI pitch ```60```, or ```261.625565```Hz, consists of a ```PC``` name and a unique number with an assigned pitch space region (```PCR```). The ```PCR``` is named so, because it is a specific collection of ```PCs```, e.g.: ```C4``` is located in region 4, but so are ```C#4```, ```D4```, and ```D#4```. ```F5``` would then be located in region 5, alongside its other siblings like ```C5```, etc.

## Pitch Class Integer Notation

If we want to measure distances between pitch classes (```PCs```) it would be beneficial to assign integers to pitch ```PCs```. In the equal temperament system, or ```12-ET``` we can generate the following table of ```PC``` to integer correlations: 


| PC name | Singable[^2] | Integer |
|:---|:---:|---:|
| C | do | 0 |
| C# | di | 1 |
| D | re | 2 |
| D# | ri | 3 |
| E | mi | 4 |
| F | fa | 5 |
| F# | fi | 6 |
| G | so | 7 |
| Ab | le | 8 |
| A | la | 9 |
| Bb | te | 10 (T)[^3] |
| B | ti | 11 (L)[^4] |
| C | do | 0 |


Can other tuning systems exist, and thereby create other musical universes? Yes, each tuning system comes with its own musical universe, one should really consider them multiverses. The ensuing table shows what a 24-ET system would look like:


| PC name | Singable[^5] | Integer |
|:---|:---:|---:|
| C | c | 0 |
| C1q# | cas | 1 |
| C# | cis | 2 |
| C3q# | cos | 3 |
| D | D | 4 |
| D1q# | das | 5 |
| D# | dis | 6 |
| D3q# | dos | 7 |
| E | E | 8 |
| E1q# | eis | 9 |
| F | F | 10 |
| F1q# | fas | 11 |
| F# | fis | 12 |
| F3q# | fos | 13 |
| G | G | 14 |
| G1q# | gas | 15 |
| G# | gis | 16 |
| G3q# | gos | 17 |
| A | a | 18 |
| A1q# | aes | 19 |
| A# | is | 20 |
| A3q# | aus | 21 |
| B | B | 22 |
| B1q# | bas | 23 |
| C | C | 0 |


We will limit our discussion to the ```12-ET``` system's universe, for now, muahahaha...

[^1]: ```PC``` singular; ```PCs``` plural.
[^2]: Really Italian note names, but singable due to their monosyllbic nature.
[^3]: In the duodecimal system of mathematics we would use A here, but in music A is already given to a note as a ```PC``` name, so ```T``` for ten has been chosen. 
[^4]: In the duodecimal system of mathematics we would use ```B``` here, but in music ```B``` already has been given to a note as a PC name, so ```L``` for ```lev``` (short for ```eleven```, yet singable) has been chosen. 
[^5]: German note names can also be sung, since they are monosyllabic as well. In this case I added the ```-as``` suffix for one-quarter-sharp, and the ```-os``` suffix for three-quarter-sharp, whenever possible, with the exceptions for PC A where I have added substitutions. Any other syllables or numbers could be borrowed or made up as long as they remain monosyllabic.

"""

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md"""
@TODO
"""

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md"""
Using a minimized file (for now) to get the post tonal tools into the Pluto.jl workspace.
"""

# ╔═╡ 4fb9ddbd-9004-4e60-86c0-76542fe59025
melody = [64, 62, 60, 62, 52, 76, 64]

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fr_Aug6 = [0, 2, 6, 8]

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md"""
## Enharmonic Equivalence
"""

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md"""
Now that we have asigned integers to pitches, we can assign ```PC``` names to their corresponding integers in a matrix. The ```n_equiv``` matrix's first column consists of the integer ```PC``` names, the second column lists the integer names as they could be sung numerically, the third column shows the English ```PC``` names favoring sharps, the fourth column shows the English ```PC``` names favoring flats, etc. The enharmonic equivalence matrix can hypthetically be expanded indefinitely.
"""

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"""
	n_equiv

Enharmonic pitch class array.
"""
n_equiv = [0 "Oh" "C" "C" "B#" "Do" "Do" "C" "C";
    1 "One" "C#" "Db" "Bx" "Di" "Ra" "Cis" "Des";
    2 "Two" "D" "D" "Cx" "Re" "Re" "D" "D";
    3 "Three" "D#" "Eb" "D#" "Ri" "Me" "Dis" "Es";
    4 "Four" "E" "Fb" "Dx" "Mi" "Mi" "E" "E";
    5 "Five" "F" "F" "E#" "Fa" "Fa" "F" "F";
    6 "Six" "F#" "Gb" "Ex" "Fi" "Se" "Fis" "Ges";
    7 "Sev" "G" "G" "Fx" "Sol" "Sol" "G" "G";
    8 "Eight" "G#" "Ab" "G#" "Si" "Le" "Gis" "As";
    9 "Nine" "A" "A" "Gx" "La" "La" "A" "A";
    10 "Ten" "A#" "Bb" "A#" "Li" "Te" "Ais" "B";
    11 "Lev" "B" "Cb" "Ax" "Ti" "Ti" "H" "H";]

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n_equiv

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"""
	enharmonic(pc)

The *enharmonic(pc)* function finds the enharmonic pitch class from any traditioanl pitch class name (English only, for now) and converts it to an integer pitch class.
"""
function enharmonic(pc=0)
	pitch = findall(x -> x==pc, n_equiv)
	return n_equiv[pitch[1][1]]
end

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md"""
Accessing different PC names, resulting always as integer PC name 1:
"""

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enharmonic("Ra")

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enharmonic("Di")

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enharmonic("Cis")

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enharmonic("Bx")

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md"""
## Octave Equivalence
"""

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"""
	o_equiv(mel_seq)

Reduces MIDI pitches to pitch class integer notation.
"""
function o_equiv(pcc=[64,62,60,62,52,76,64])
	return [i % 12 for i ∈ pcc]
end

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o_equiv(mel_seq)

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md"""
## Intervals
"""

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md"""
In case one is so inclined of wanting to use 'regular' music terms for intervals, one can create an array of interval names. The first index of the array carries the interval in enharmonic interval integer notation, followed by a column of English interval names, followed a column of German interval names. When additional music terms are desired an additional column can be created within the array. 
"""

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"""
	interval_names

An array of interval names.
"""
interval_names = [
	0 "unison" "reine Prime" "unisson";
	1 "minor second" "kleine Sekunde" "seconde mineure";
	2 "major second" "große Sekunde" "seconde majeure";
	3 "minor third" "kleine Terz" "tierce mineure";
	4 "major third" "große Terz" "tierce majeure";
	5 "perfect fourth" "reine Quarte" "quarte";
	6 "tritone" "Tritonus" "triton";
	7 "perfect fifth" "reine Quinte" "quinte";
	8 "minor sixth" "kleine Sexte" "sixte mineure";
	9 "major sixth" "große Sekunde" "sixte majeure";
	10 "minor seventh" "kleine Septime" "septième mineure";
	11 "major seventh" "große Septime" "septième majeure";
	12 "perfect octave" "reine Oktave" "octave juste";
	13 "minor ninth" "kleine None" "neuvième mineure";
	14 "major ninth" "große None" "neuvième majeure";
	15 "minor tenth" "kleine Dezime" "dixième mineure";
	16 "major tenth" "große Dezime" "dixième majeure";
]

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function get_interval_int(interval="perfect unison")
	interval_name = findall(x -> x==interval, interval_names)
	return interval_names[interval_name[1][1]]
end

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get_interval_int("seconde mineure")

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"""
### OPI
    opi(pcc)

Create Ordered Pitch Intervals (OPI) from a pitch class collection (pcc).
"""
function opi(pcc)
	return [pcc[i+1] - pcc[i] for i ∈ eachindex(pcc) if i < length(pcc)]
end

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opi(fr_Aug6)

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opi(mel_seq)

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md"""
### AIS
    ais(pcc)

Create an adjacent interval series (AIS), like the opi function, from a pitch class collection (pcc).

The AIS is identical to the OPI.
"""

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ais(fr_Aug6)

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ais(melody)

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md"""
### UPI
    upi(pcc)

Create Unordered Pitch Intervals (UPI) from a pitch class collection (pcc).
"""

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upi(fr_Aug6)

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upi(mel_seq)

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md"""
### OPCI
    opci(pcc)

Create Ordered Pitch Class Intervals (OPCI) from a pitch class collection (pcc).
"""

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opci(fr_Aug6)

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opci(mel_seq)

# ╔═╡ 09294b3a-2390-4226-9bf8-dc9b4d9b98ac
md"""
### UPCI
    upci(pcc)

Create Unordered Pitch Class Intervals (UPCI) from a pitch class collection (pcc).
"""

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upci(fr_Aug6)

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upci(mel_seq)

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md"""
## Interval Class Vector (ICV)
"""

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md"""
### IC
    ic(pcc)

Create Interval Classes (IC) from a pitch class collection (pcc).

The IC is identical to the UPCI.
"""

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ic(fr_Aug6)

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ic(mel_seq)

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#md"""
## Spacing and Register
#"""

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