Clock notation

I am a guitar player, who has learned playing exclusively with tablatures. Now I am a bit tired of playing like a robot without understanding why the songs I like are written as they are. I can feel that there is an underlying structure and logic common to many songs. I have tried to read and understand some material on music theory and harmony, but I generally find it non-logical, confusing, and difficult to apply to guitar playing.
In this blog I will try to explain harmony by using numbers and geometric figures, which in my opinion is more natural, logical, and easy to remember. I hope this will be helpful for guitarists or keyboard players without a background in music theory.

Rock around the clock

You probably already know the major scale, C D E F G A B C, or Do Re Mi Fa Sol La Si Do. The distance or interval between two notes is expressed in tones and semitones.  2 semitones = 1 tone. You may also know that there are two semitones (one tone) between most notes, such as C (Do) and D (Re), but just one semitone between E and F and between B and C:

• C 2 D 2 E 1 F 2 G 2 A 2 B 1 C

This notation maps nicely to the white keys of the piano keyboard:

• When there is a black key between two white keys, the keys are separated by 2 semitones. 
• When there is no black key (B - C and E - F), the keys are separated by 1 semitone. 
• A black key is 1 semitone above the preceding white key. The note's name is suffixed with a sharp #.

Therefore, unlike the guitar, the piano keyboard is not uniform: two adjacent white keys can be separated by 1 or 2 semitones, depending on their position. This is why some people build alternative keyboards: http://www.altkeyboards.com

Now, why on earth do we use consecutive letters, which look like a regular interval, when the intervals between letters are not all the same ? This makes calculating intervals between two notes quite cumbersome. For example the interval between C and F (3 letters "above" C) is 2.5 tones, but the interval between F and B (3 letters "above" F) is 3 tones...
Playing music with traditional notation looks like doing arithmetic with roman numerals !

Fortunately, there is a simpler notation, called clock or integer modulo 12 notation:

Each note is named with a number, from 0 to 11. With this notation, calculating intervals between two notes is as simple as doing additions and subtractions:

• The interval between 0 (C) and 5 (F) is 5-0 = 5 semitones (2.5 tones). 
• The interval between 5 (F) and 11 (B) is 11-5 = 6 semitones (3 tones).
• The interval between 11 (B) and 2 (D) is 2 + 12 - 11 = 3 semitones (1.5 tones). 

Whenever an operation's result is above 12 or below 0, just subtract / add 12 to stay in the 0 to 11 range.
The 0 (C) major scale becomes:
0 2 2 2 4 1 5 2 7 2 9 2 11 1 0

Now, try to memorize 2 2 1 2 2 2 1, the sequence of intervals of the major scale. Start with any key note, add these numbers, and you obtain the notes of the major scale in any key.
For example, the 5 (F) major scale is:
5 2 7 2 9 1 10 2 0 2 2 2 4 1 5
F   G  A   A#  C   D   E   F

Interval names

Now we know how to calculate the interval between two notes, using the number of semitones. However most musicians prefer to use these awkward names:

Semitones	Name	Position of the note in the C major scale
0	Unison	1st - C
1	Minor second
2	Second	2nd - D
3	Minor third
4	Major third	3rd - E
5	Perfect fourth	4th - F
6	Augmented fourth or Diminished fifth
7	Perfect fifth	5th - G
8	Minor sixth
9	Major sixth	6th - A
10	Minor seventh
11	Major seventh	7th - B
12	Octave	8th - C

Which one do you prefer ? Names or numbers ? I personally find the names of the intervals quite confusing. When someone talks about a "fifth", I hear 5, so I tend to think that there are 5 semitones or 5 tones, when there are in fact 7 semitones in a fifth... Admittedly, there is a logic behind these names: they are named in relation to the position of the note in the C major scale.

 Now, I am not going to change the way millions of musicians call intervals, but as a coping strategy I memorized the table above and mentally translate an interval name into its number of semitones.

Application to the guitar fretboard

The guitar fretboard is divided in semitones, which means that we can again do simple arithmetic to work out how to play the major scale in any key.
The standard guitar tuning is E A D G B E, from the 6th (biggest) string to the 1st (thinnest) string. Using the clock diagram above, this tuning translates into 4 9 2 7 11 4 in the clock notation.
By subtracting these numbers in pairs, we can work out the intervals between the strings:

• Between 6th and 5th: 9 - 4 = 5 semitones
• Between 5th and 4th: 2 - 9 = -7, -7 + 12 = 5 (when we obtain a negative number, add 12)
•  ...
• Between 3rd and 2nd: 11 - 7 = 4 
• Between 2nd and 1st: 4 - 11 = -7, -7 + 12 = 5
• 4 5 9 5 2 5 7 4 11 5 4

We can see that the interval between two strings is 5 semitones, except between the 3rd (G) and 2nd (B) strings, where we have 4 semitones. This is why some people prefer to tune their guitar in "All fourths" (5 semitones between every string), or in "Major thirds" (4 semitones between strings).

Using this knowledge, let's play the 5 (F) major scale on the guitar. As mentioned above, the intervals of the major scale are 2212221.

• We need to find the root note of the scale, 5 or F or Fa. The 6th string of the guitar is the note 4 (E). So if I press the 1st fret, I am going to play a 4 + 1 = 5 (F)
•  The second note of the scale is 2 semitones above the root, so I play the 1st fret + 2 semitones = 3rd fret
• The third note is again 2 semitones above the previous note. I can play the 3rd fret + 2 semitones = 5th fret on the 6th string. But since the 6th and 5th strings are separated by 5 semitones, I can also play the 5th string open, because 5th fret - 5 semitones = 0
• The fourth note is 1 semitone above the previous note. We play it on the 0th fret + 1 semitone = 1st fret of the 5th string.
• ... and so on. When your fingers are stretching, you can play a note on the following string if you subtract 5 to the number of frets, or 4 if you are going from the 3rd to 2nd string. 

The result is the following:

This means that you no longer have to rote learn the major scale in all keys and in all positions. You just have to remember three things and you can work it out !

• 2212221, the intervals of the major scale
• 4 9 2 7 11 4, the standard tuning of the guitar
• When you speak with other musicians or read partitions, you need to know the correspondence between note letters and note integers: C D E F   G A B C => 0 2 4 5   7 9 11 0

As an exercise, try to use this trick to play a 8 (G#) major scale, starting on the 4th string.

In my next post I will explain how to find and play chords using the clock notation.