A rational approach to music theory

Written music conveys information from the composer to the performer who then presents it to the listener. Some of this information is essential and some is extra. In simplest terms, essential information is ‘what to play’ and extra information is ‘how to play it’. ‘What to play’ can further be divided into two (and only two) essential pieces of information:
⦁ TIME – when does each note start and stop?
⦁ PITCH – exactly what note is required?

A performer, or for that matter a full orchestra, that plays all the right notes at the right time and for the right duration has ‘correctly’ performed the piece, meaning that the essential information has been conveyed. All the rest is ‘extra’.

This extra information is often the difference between a mechanical and a great performance. Some of it is specified by the composer. Some is provided by the performer or the conductor. A long time ago, composers provided very little extra information. Performers were expected to know ‘how to play’ and only had to be told ‘what to play’ by the composer’s score. With the coming of the Romantic Period, composers began to add more and more ‘extra’ guidance to their scores, most of which can be ignored until the essentials of the piece are mastered, namely the TIME and PITCH of every note. In other words, first get it right, then make it nice.

The rest of this article deals only with the two essentials, TIME and PITCH. The extra information will not be discussed further.

Time in music has been well understood for centuries. Music and movement were inseparable, especially in marching and dancing. It is therefore no surprise that early composers evolved a sensible and almost unambiguous way to denote time in musical notation. There are some words and conventions to learn, as there are in every field, but they are very learnable, because they make perfect sense. The system of time signature, bars, note and rest durations accurately conveys the relative start and stop time of every note (which is the only requirement), and if an absolute speed is wanted, the composer can also specify a metronome setting. Job done; the Rationalist is happy!

Pitch is not so easy. In fact, the terms we use to describe musical pitch are very confused. This confusion arises because, with the introduction (in Western music) of equal temperament, we had an opportunity to devise and adopt a new nomenclature for pitch that would accurately reflect the reality of equal temperament, but- we did not rise to the challenge. Instead we soldiered on for the next 300 years forcing our historic nomenclature to fit into an equal temperament environment to which it was simply unsuited.

The root of the problem is that our historic nomenclature allows for only seven ‘natural’ pitches, A, B, C, D, E, F, G, whereas Western music requires eleven distinct pitches. Worse, the seven ‘natural’ pitches are not equally spaced; we have tones and semitones to deal with and a complex system of ‘sharps’ and ‘flats’. For example, the pitch between A and B is called either A# or Bb, depending on context. Yet, in equal temperament, it is the same pitch, struck by the same key on a piano. Logic says it should have one name, not two. A# (or Bb) is not a special kind of A (or B). It is a pitch in its own right and should have its own unique name.

In equal temperament, the ‘octave’ is divided into 12 geometrically equal steps (the chromatic scale). The word ‘octave’ is historical and unfortunate. Technically, it is a doubling of frequency (the first overtone of the fundamental). Historically, it is the eighth step of a ‘diatonic’ scale (like C-major or A-minor). It is also the twelfth step in a chromatic scale. It is helpful to visualise the twelve equal tempered pitches as a clockface:

The absolute pitch of note 0 doesn’t matter for now. Also, let’s replace the term ‘semitone’ with the simpler ‘step’. Starting from 0, we can proceed clockwise, step by step, in an ascending (chromatic) scale, or anticlockwise, step by step, in the descending scale. An even better visualisation (though harder to draw) is a ‘spiral’ (helical) staircase, one note per step, twelve steps per octave, so that after twelve steps you are vertically above where you started.

Returning to the clockface, there is much to be discovered from simple inspection. For example, instead of proceeding stepwise, we can proceed by twos:

Conventionally this would be called the ‘whole tone’ scale, but a more logical name (avoiding tones and semitones) might be ‘scale of twos’. (The Rationalist is not trying to impose a new nomenclature; he is simply thinking aloud!) By inspection, there are exactly two scales of this type – the even numbers and the odd numbers. (Remember we have not specified absolute pitch yet). Similarly, we can proceed by threes:

Conventionally, this is called a ‘diminished’ arpeggio, of which there are exactly three. ‘Diminished’ is one of the historical terms that is no longer required if we base our nomenclature on the reality of equal temperament in which every pitch has equal status. And of course we can proceed by fours:

Conventionally, this is called an ‘augmented’ arpeggio, of which there are exactly four. ‘Augmented’ is another historical term that we could do without, for the same reason. For completeness, there are two more ways to go round the clock in equal steps. The less interesting is six steps. This takes you from 0 to 6 and back to 0. It divides the octave in two and of course there are six versions of it. Conventionally it is called a ‘tritone’ (three whole tones) but as we have dropped tones and semitones in favour of steps, we should simply call it a six-step. And now, the five-step sequence:

As five is not a factor of twelve, we have to visit every single pitch once, and only once, before we complete the cycle. For the classically trained musician, this should be ringing bells of recognition. Haven’t we just reinvented the ‘Cycle of Fifths’ or ‘Key Cycle’? Yes, we have. But we have extracted it from the inherent symmetry of the equal-tempered 12-step scale, without recourse to sharps & flats, or major, minor, diminished & augmented intervals. Please also note that the 5-step sequence and the 7-step sequence are identical. Only the order of visiting each pitch is reversed. The Rationalist is feeling encouraged, thus far!

Here is a table of intervals (two pitches) comparing their conventional or historic names with how they could be named using the equal temperament 12 pitch scale.

The list of conventional intervals is not exhaustive but it includes those most commonly encountered. The list of equal-tempered intervals is complete, and simple. There are exactly eleven named (and playable) intervals, discounting the unison and octave.

Naming the inverted interval is also far simpler in ET-12 than in conventional nomenclature. For example, conventionally, the inverse of a major third is a minor sixth. In the suggested ET-12 nomenclature, 4-step inverts to 8-step. (4 + 8 = 12). Conventionally, the inverse of an augmented fifth is a diminished fourth. In ET-12 nomenclature, 8-step inverts to 4-step (8 + 4 = 12).

The following diagram shows (in blue) a conventional major scale starting on pitch 0 (which we have still not defined):

As the pattern is not symmetrical, it follows that there are exactly 12 possible major scales. Of course the same applies to minor scales and all the standard modal scales. Notice also that the five pitches that are not included in the major scale above form the ‘pentatonic’ scale (in red):

And again, there are exactly 12 possible pentatonic scales as it can start on any pitch.

Alternatively, instead of aranging the clockface as a clockwise ascending chromatic scale, we can proceed by 7-step intervals, which are the conventional ‘fifths’, as explained above. This is in fact the way a piano tuner creates the ET tuning in practice, by very slightly flattening each progressive ‘fifth’ (7-step), so as to arrive after 12 such small adjustments at a perfectly tuned octave. If we then plot out the diatonic major scale (blue) and the pentatonic scale (red) on this version of the clockface, we get:

If the pitch at ‘0’ is a conventional ‘C’, then we find all the piano ‘white keys’ map to the blue polygon and all the ‘black keys’ map to the red polygon. There is also an interesting axis of symmetry in the line between pitch 2 (D) and pitch 8 (G#/Ab).

Here is a table of common chords (as used by guitarists and others), in generic form, i.e. without specifying the root, which can be any of the 12 pitches. Remember we have still not assigned an absolute pitch to the ‘0’.

The conventional chord names are usually abbreviated for convenience. For example, taking C as the root we have: C, Cm, C7, C°, C+, Cmaj7, Cm7, Cm maj7. Again, the list is incomplete. Guitarists, especially jazz guitarists, are expected to learn up to a dozen conventionally named chords in every key, which is no mean feat.

Thus far, we have discussed the ET-12 chromatic ‘clockface’ in relative terms, i.e. without specifying the absolute pitch of “0” (and hence all subsequent pitches). The most obvious approach is now to assign letter names to each pitch. Lower case can be used to avoid confusion with conventional pitch names:

The pitches then relate to conventional pitches as follows:

As an example of one possible approach to the new naming of conventional keys, C-major becomes d4, A-minor becomes a3, E-major becomes h4, C#-minor becomes e3, etc.

Piano keyboards – these are not designed for an equal temperament mindset. Although modern pianos are tuned for ET-12, they are physically bound to the system of seven natural (white) notes and five ‘accidental’ (black) notes. Interestingly, it is only the keyboard layout and its mechanism that makes this distinction; the strings and hammers are ‘colour-blind’ and undifferentiated in this respect. One could argue that the piano is a simple ET-12 instrument made complicated by its diatonic keyboard. The guitar, on the other hand, is much more compatible with ET thinking. Its fingerboard is a matrix of frets and strings with a note at each crosspoint. There is no differentiation between ‘natural’ and ‘accidental’ tones.

Notation – this is the most difficult area to ‘rationalise’. We have seen that it is possible to dispense with key signatures and with ‘accidental’ sharps and flats. We have also stated that the terms ‘major’, ‘minor’, ‘diminished’ and ‘augmented’ become redundant on adopting unique letter names for all 12 pitches. So far so good. However, conventional notation provides one great boon which, at first sight, may not be matched in any ET-12 notation, namely, Vertical Compactness. Simply BECAUSE a natural note, its sharp and its flat all exist at the same vertical height on the staff (e.g. D, D#, Db all live on the 4th line of the Treble staff), a typical choral score of Soprano, Alto. Tenor, Bass is accommodated in a mere two staves of five lines each (plus the occasional leger line for the extremes of bass and soprano). Of course, this is by design and predates equal temperament by several centuries. The Rationalist expresses grudging admiration for the inherent ingenuity of this tradition!

Author’s note: I wrote this for my own entertainment and to think through a few ideas about music theory and notation. I am not seriously proposing any changes, nor would I expect to be taken seriously if I did. Also, I am not claiming originality for any of the ideas presented here. Thousands of people must have had the same ideas. My aim was simply to present the topic as clearly as possible, in the hope that it may be of some interest.

Thank you for reading!