# Preliminaries

There are 12 notes in the musical system:
 C * D * E F * G * A * B C 0` 1` 2` 3` 4` 5` 6` 7` 8` 9` 10` 11` 0`

# Scales

### Major

 0 2 4 5 7 9 11 0

### Minor

 0 2 3 5 7 8 10 0

### Chords

Today I will give a reference table for standard chords including jazz chords. To be able to construct these quickly we will explain what the notation means:

Some of the notation uses numbers, like X^9, these represent scale degrees of a standard major scale:

 Scale Degrees 1 2 3 4 5 6 7 8 Intervals 0 2 4 5 7 9 11 0

continuing to go further yields

 Scale Degrees 8 9 10 11 12 13 14 15 Intervals 0 2 4 5 7 9 11 0

Also note that for the symbol X9, we can see it uses the intervals 0 2 4 7 10, so it's a combination of X7 plus the 9th (remember 9th is the interval 2 based on the tables above). Also note that for X13 we have the intervals 0 2 4 5 7 9 10 so it's combining X9 with the 13th scale degree. In practice we don't need to play all those notes at once to get the idea of X13 across, in general we can remove intervals which represent simple harmonics (being a harmonic means that it is very similar to the root tone), so it can be simplified to 0 2 5 9 10 (there are other options).

So when we have a chord with a number after it, it specifies all notes up to that N should be added (like in the X13 example above), but if we wrote Xadd11 for example we would just add the 11th scale degree and not everything up until the 11th scale degree so it would be 0 4 7 5.

Note when a chord is notated as alt (for altered) we take the chord and additionally add any subset of the set {1, 3, 6, 8} to the chord (those are intervals)

Here is the table of all of the chords:

Chord Symbol Semitones Above
X 0 4 7
X5 0 7
X2 0 2 7
X+ 0 4 8
Xo 0 3 6
Xh (half diminshed must be a 7th chord, by def) 0 3 6 10
Xsus 0 5 7
X^ (def'd to be 7) 0 4 7 11
X- 0 3 7
X^7 0 4 7 11
X-7 0 3 7 10
X7 0 4 7 10
X7sus 0 5 7 10
Xh7 0 3 6 10
Xo7 0 3 6 9
X^9 0 2 4 7 11
X^13 0 2 4 7 9 11
X6 0 4 7 9
X69 0 2 4 7 9
X^7#11 0 4 6 7 11
X^9#11 0 2 4 6 7 11
X^7#5 0 4 8 11
X-6 0 3 7 9
X-69 0 2 3 7 9
X-^7 0 3 7 11
X-^9 0 2 3 7 11
X-9 0 2 3 7 10
X-11 0 2 3 5 7 10
X-7b5 0 3 6 10
Xh9 0 2 3 6 10
X-b6 0 3 7 8
X-#5 0 3 8
X9 0 2 4 7 10
X7b9 0 1 4 7 10
X7#9 0 3 4 7 10
X7#11 0 4 6 7 10
X7b5 0 4 6 10
X7#5 0 4 8 10
X9#11 0 2 4 6 7 10
X9b5 0 2 4 6 10
X9#5 0 2 4 8 10
X7b13 0 4 7 8 10
X7#9#5 0 3 4 8 10
X7#9b5 0 3 4 6
X7#9#11 0 3 4 6 7 10
X7b9#11 0 1 4 6 7 10
X7b9b5 0 1 4 6 10
X7b9#5 0 1 4 8 10
X7b9#9 0 1 3 4 7 10
X7b9b13 0 1 4 7 8 10
X7alt 0 1 3 4 8 10 6
X13 0 2 4 5 7 9 10
X13#11 0 2 4 6 7 9 10
X13b9 0 1 4 5 7 9 10
X13#9 0 3 4 5 7 9 10
X7b9sus 0 1 5 7 10
X7susadd3 0 4 5 7 10
X9sus 0 2 5 7 10
X13sus 0 2 5 7 9 10
X7b13sus 0 5 7 8 10
X11 0 2 4 5 7 10

### Tritone

Given any 7 11 2 5 you can replace it by a 1 5 8 11. The reasoning behind this substitution is that it maintains the same chord quality with two matching notes

# Tone Collections

In general, in music we can find a certain pattern across all of the main scales we can use, the invariant I have personally taken away from it is that in a scale is a tone collection such that there are no more than two consecutive semitones with seven notes. Based on this we can construct all tone collections, they have been grouped by similarity

• 0, 1, 3, 4, 6, 7, 10 | 0, 1, 3, 4, 6, 7, 9
• 0, 1, 3, 4, 6, 8, 10 | 0, 1, 3, 4, 6, 8, 9 | 0, 1, 3, 4, 6, 9, 10
• 0, 1, 3, 4, 7, 8, 10 | 0, 1, 3, 4, 7, 9, 10
• 0, 1, 3, 5, 6, 8, 10 | 0, 1, 3, 5, 6, 8, 9 | 0, 1, 3, 5, 6, 9, 10
• 0, 1, 3, 5, 7, 8, 10 | 0, 1, 3, 5, 7, 9, 10 | 0, 1, 3, 6, 7, 9, 10
• 0, 1, 4, 5, 7, 8, 10 | 0, 1, 4, 5, 7, 9, 10 | 0, 1, 4, 6, 7, 9, 10
• 0, 2, 3, 5, 6, 8, 10 | 0, 2, 3, 5, 6, 8, 11 | 0, 2, 3, 5, 6, 8, 9
• 0, 2, 3, 5, 6, 9, 10 | 0, 2, 3, 5, 6, 9, 11
• 0, 2, 3, 5, 7, 8, 10 | 0, 2, 3, 5, 7, 8, 11
• 0, 2, 3, 5, 7, 9, 10 | 0, 2, 3, 5, 7, 9, 11
• 0, 2, 3, 5, 8, 9, 11
• 0, 2, 3, 6, 7, 9, 10 | 0, 2, 3, 6, 7, 9, 11
• 0, 2, 3, 6, 8, 9, 11
• 0, 2, 4, 5, 7, 8, 10 | 0, 2, 4, 5, 7, 8, 11
• 0, 2, 4, 5, 7, 9, 10 | 0, 2, 4, 5, 7, 9, 11
• 0, 2, 4, 5, 8, 9, 11
• 0, 2, 4, 6, 7, 9, 10 | 0, 2, 4, 6, 7, 9, 11
• 0, 2, 4, 6, 8, 9, 11 | 0, 2, 5, 6, 8, 9, 11
• 0, 3, 4, 6, 7, 9, 10 | 0, 3, 4, 6, 7, 9, 11
• 0, 3, 4, 6, 8, 9, 11 | 0, 3, 5, 6, 8, 9, 11