A dozenal notation for western music

Why a new notation?

I'm currently a bit dissatisfied with the notations we currently use, espcecially the one - tabs - that guitarists use. I think we can do better.

In my view, a good notation for anything has the following qualities:

1. It's concise,
2. It's practical to read and write,
3. It makes the important obvious and hides the irrelevant,
4. By embodying appropriate concepts, it gives us a powerful mental model to what it represents. In other words, it helps us think about what it stands for.

Therefore we need to ask: what is important when interpreting musical notation? In my view, it boils down to a few elementary questions:

1. Given a note, where does it fit on the scale ?
2. Given a set of notes (a chord, a scale, a musical phrase...), what are the intervals by which these notes are related ?
3. Given a base note, what is the note that is at some interval of this note ?

Some notations are especially bad at this. For example, a guitar tab doesn't help you answer question 1 at all. Music sheet makes it easier, but not as easy as possible, because it's irregular.

Once you represent notes and intervals as numbers (like computers do), you realize that answering those questions translates to very basic mental caculations, like addition and subtraction.

Here are some examples:

This is where the notation can help you: by using a few mental caculation tricks, you can build a mental map of music much faster than if you just learned by heart the intervals between every pair of notes. Cf the "counting with notes" section below.

Counting with notes

Scale notes

In many respects, notes that are exactly one our several octaves (12 semitones) apart can be considered the same, e.g a Sol4 is considered the same as a Sol5, and we just call it a 'Sol'. In Dozenal notation, instead of writing it 47 or 57, we just write it 7.

Likewise, an interval of 3 semitones is considered the same as an interval of 15 semitones or an interval of -9 (i.e going down the scale) semitones, and we call it a 'minor 3rd'. In Dozenal notation, we write it 3.

From this simplified view, the set of all notes (and intervals) forms a cycle:

We call the 12 notes in this cycle the scale notes.

This cycle representation is useful for displaying sets of notes (such as chords and scales) as visual patterns, as shown in the figure below:

For instance, you can see that all major chords form the same pattern with various rotations. Minor chords form a different pattern. In musical terms, a 'transposition' translates to a rotation.

Once you have learned how to add and subtract scale notes, it becomes very easy to add and subtract notes and intervals. So we'll start by practicing that.

If you have a good memory, you can simply write the addition and subtraction tables - just like in elementary school - for all 12 scale notes and learn them by heart. They are given below:

If you already feel confident with this, you can train with the following exercise:

Counting trick: the complement

Personally, I have a very bad memory, so it would be very difficult for me to learn all 248 cells of these tables by heart. Fortunately, there are several mental calculation tricks you can use to avoid remembering everything.

The first one consists of learning the complement of each note in the scale. For instance, 3 is the complement of 9, because going 3 steps up the scale is like going 9 steps down the scale. Similarly, 7 is the complement of 5, 2 is the complement of a, and 6 is the complement of itself.

It's musically important to know the complement of each interval, but it's also practical for the Dozenal notation. You can make many calculations much easier by rewriting them using complements. Here are some examples:

You can practice with complements here:

Counting trick: the cycles

The complement trick can get you a long way. Another useful one is noticing that, if you add+3 repeatedly, you end up with a cycle of 4 notes:

You then realize that these cycles divide the 12 notes in 3 distinct groups, as shown in the exercise below:

Something similar happens when adding +4:

Practicing harder

As you may have noticed, some of these calculations are easier than others: for instance, 7 + 2 or 5 - 3 are much easier than 9 + 2, 3 - 6 or 3 + a, because they're more familiar. Therefore, in order to train efficiently, it's better to focus on the harder ones. This is what the following exercise provies:

For guitarists

As a guitar player, I find the Dozenal notation especially interesting. When a guitarist reads music, she may be interested in 2 things:

• A mechanical understanding of the music ('where do I put my fingers to play this chord?')
• A harmonic understanding of the music ('what note of the scale am I playing here?', 'is this a minor or major chord?', 'how does this note relate to this chord?', etc.)

Traditionally, the notations used by guitarists are the tablature (which makes the mechanical aspects obvious, but obscures the harmonic aspects) and the sheet music (which obscures the mechanical aspects, but makes the harmonic aspects obvious.)

Most guitarists (myself included) are lazy and want to play the music more than they want to understand it, and so we choose tablatures. As a consequence, we end up being somewhat illiterate about harmony - we think of the pieces more visually than musically.

But it turns out that writing tablatures in Dozenal notation gives us the best of both worlds. This is because a dozenal number is good at representing both a note (being dozenal) and a position on the fretboard (being a number).

As an example, consider the following 'traditional' tablature:

It turns out that these are just 4 different ways of playing the same chord (D major), which is not obvious at all from the notation used. On the other hand, using dozenal notation makes it evident, because then all notes end with the same 3 digits:

All it takes to translate from this Dozenal tablature to fingers placement is a little bit of mental calculus like we saw above. The following 2 exercises will help you practice it:

This may be a bad idea

I'm but un amateur and uneducated musician, an as such cannot claim with certainty that Dozenal Notation is strictly better than mainstream notations and that everyone should adopt it. The point of this website is not to evangelize musicians to Dozenal Notation, rather to get feedback from them.

In particular, although this notation seems promising to me, I can imagine various reasons why it may objectively be a bad idea:

• Maybe I have a weird brain. Maybe I'm unusually bad a remembering notes, and unusually good at mental calculus, and so this notation is not adapted to most people.
• Maybe irregularity is beneficial. Conventional music notation yields irregularity by not accounting for the inherent symmetries in music, but it's possible that this irregularity actually fosters learning and creativity.
• Maybe notation is not that important. Maybe reading music is an insignificant concern compared to other mental processes that accomplished musicians go through, and so there's not much to gain by improving notation. Maybe I should just go back to playing music.