Relative Music Theory

Cyril Monkewitz

Section 1) A brief history of the 12-note system
Section 2) The Relative Music Theory
Section 2.b) Scope of application | examples

Western music is based on the 12-note system since around 600 years. Currently the keyboard is tuned to an "equal temperament" making the octave the only playable "perfect interval" on the piano. We are limited to writing 12 distinct frequencies due to this, or 36 if we consider sharps(#) and flats(b) (60 with double flats/sharps).

Hereby is proposed a musical notation system that will allow us to access all existing frequencies from within the infinity of possible frequencies. It permits the use of perfect intervals whilst respecting the harmonic nature of sound as well as writing notes on the Pythagorean scale. It is able to define any desired temperament and is able to transcribe traditional music in a precise manner amongst other things.

Section 1) A brief history of the 12-note system

To understand why it might be desirable to expand on the 12-note system it is important to understand how it appeared, why it was chosen and what it's limitations are.

The tempered system is based on natural harmonics, specifically on the octave and the fifth interval. Let us quickly examine the natural harmonics, and how their frequencies would be expressed mathematically. The harmonics are the fractional divisions of the base wave, we may then bring the waves back into the same octave by coming down as many octaves as possible.

divisions	illustration	the interval	frequency	frequency normalized to the octave
1		- the base* wave*	f=Basef	f=Basef
2		- the octave	f=2×Basef	f=2×Basef
3		- the 5^th* + octave*	f=3×Basef	f=(3/2)×Basef
4		- the double octave	f=4×Basef	f=2×Basef
5		- the major 3^rd* + 2 octaves*	f=5×Basef	f=(5/4)×Basef
6		- the 5^th* + 2 octaves*	f=6×Basef	f=(3/2)×Basef
7		- the minor 7^th* + 2 octaves*	f=7×Basef	f=(7/4)×Basef

The first natural harmonics are also the strongest and will appear most often in nature. Interestingly there is a major chord with minor 7^th in the divisions of each sound. We now understand why almost all traditional music is based on fifths and octaves as well as why Pythagoras proposed the 12-note system. In fact the cycle of 12 fifths results from an attempt to put natural fifths inside octaves.

Pythagoras noticed that if we chain 5^ths we will never fall on an octave of the base note since the dividend of (3/2)^x=3^x/2^x is odd whereas 2^y is even. However, he also noticed that if we choose x=12 and y=7 we are pretty close. 2⁷ / (3/2)¹² is called the Pythagorean comma. It is a frequency ratio of 524288/531441 = 1/1.0136432647705078125. One possible approach to improve the precision of the natural intervals would thus be to search for x where the (frequency ratio)/(number of 5^ths) would be smaller than the (Pythagorean comma)/12.

Below is a small application which will give you the frequency ratio for any given number of notes. You may try 12 and will receive the Pythagorean comma. We notice that all the numbers of know scales tend to be close, 5(pentatonic scale), 7(heptatonic scale). This is most likely because they were the previous attempts at putting 5^ths into octaves. One very notable division is 665, which returns a ratio of 1/1.000043655... for 665 fifths and 389 octaves. Evidently a scale with 665 notes would be quite impractical. Accordingly, it was decided to not take any further step in this direction and an entirely different approach is advocated in Section 2).

(number of 5^ths)		(results)
	number of 5^ths:
	number of octaves:
	frequency ratio:

Now we must take a look at temperament, as well as equal temperament. Historically, temperaments allowed keyboard players to play perfect intervals, since the Pythagorean comma can be negated selectively from the 12 notes, leaving some 5^ths perfect. Later tuning the piano to the equal temperament became the norm (removing 1/12 of the Pythagorean comma from each 5^th). This also had the adverse effect of removing our capacity to play any perfect interval other than octaves. Also, it is important to note that we have been only interested in 5^ths and octaves up until now and created the 12-note system solely upon them. The other natural harmonics haven't been considered, and all are represented with varying degrees of falsehood as the closest 5^th.

The 12 notes of the equal temperament are defined by this formula:
2^n/12*Basef , with n being an integer between 0 and 11.

Finally, here is a comparison of a major chord with minor 7^th written with the equal temperament as opposed to the true harmonic fractions, derived from a 440Hz Base frequency.

**Equal temperament**
A = Basef	C# = 2^4/12*Basef	E = 2^7/12*Basef	G = 2^10/12*Basef	A(octave) = 2*Basef
440Hz	554.365Hz	659.255Hz	783.991Hz	880Hz

**Harmonic fractions**
A = Basef	C# = (3/2)⁴/4 C# = (5/4)Basef	E = (3/2)*Basef	G = (4/3)² G = (7/4)Basef	A(octave) = 2*Basef
440Hz	*556.875Hz 550Hz	660Hz	*782.2̅ Hz 770Hz	880Hz

*Although in all examples here we use the natural 7^th if translating western music using the double inferior 5^th will be a better bet. It is also slightly lower than the equal temperament. This is due to the fact, that the natural 7^th hasn't been explored. We should thoroughly explore the natural 3^rd first.

It is important to note that a musician may very well play perfect intervals despite the music being written on the 12 tone system. For example, a violinist can tune his string perfectly around his A string, although this may create noticeable disparities in his G string compared to the piano. On this stunning view of 4 prime numbers resulting in a linear progression of frequencies we conclude our brief tour of the 12-note system. The harmonic fractions shown above are a fitting introduction for Section 2) and almost exactly the language proposed.

Section 2) The Relative Music Theory

Current music notation allows the composer to define notes. A note, withstanding appended text/symbols contains 3 pieces of information. The pitch of the note, or frequency. The time at which it begins. The length of the note.

The key to this theory is that instead of defining the pitch of the note, we will define it's relationship to another note. This relationship would be a function, a fraction to define perfect intervals or any other formula such as the one for equal temperament. Hence the "Relative". The rest of this paper, is simply put one of the plethora of ways of attempting to make this idea useable and demands to be improved upon.

The way in which a note contains information is re-examined. In the proposed system each note will contain any number of variables, and each of these variables may contain a function and pointers. To be considered playable, a note must contain at least (frequency, start_time, length). Unplayable notes are not useless and can be used to define measure bars for example. Although it may be interesting to eventually allow any type of function, in the scope of defining musical notes a fraction and simple math should suffice. It is possible to use the same approach for the other necessary pieces of information required for a note to be playable. Let us see what such a note might look like, in this case a 5^th with fictive values for the other variables:

note{
   frequency: (3/2)*Other_note.frequency
   start_time: Beat_bar3.start_time
   length: Beat_bar3.length/2
   amplitude: (121/100)*Other_note.amplitude
   instrument: violin
...
}

Although we haven't specified a frequency for the note, we may specify a note that defines the tonality of our piece for example. In this manner any piece is already transposed, and only one link must be changed to swap tonalities. In contrast a piece we would want to transpose from one tonality to another in the current system would require adding sharps(#) or flats(b) as well as changing the notes' heights. This is one of the most notable qualities of the new system. It could prove to be useful in computer generated music as well as streamlining the composers actions. In fact, we will only need to write a major chord once, then simply link it to any base note and move around the notes with octaves as desired. The major chord would become a kind of blob or "module" that would be linkable to any note. A "module" wouldn't need be a chord either, it could be a theme from a fugue for example, generally speaking a group of notes all linked to a single source.

We can already see that there is a need to group multiple notes together. There are multiple ways to group notes together. The simplest would be a "cloud" of notes, in no particular order. A cloud could be the result of a composers actions, a computer generation or any other source. We can imagine a more structured version, which we will call "module". In such a group of notes, all dependencies in the functions would be resolved, and it could be linked to any other note. These modules could be used in the toolbar of a composing application. Drag-dropped onto the target note, and then the appearing notes edited through other modules. Simple modules would include chords, chromatic scales and intervals. More complex modules might include melodies, chord progressions, etc. Since any composition can become a module, the toolbar would be a collection of elements the composer is using and rearrangeable.

It is important to render this notation system readable, comprehensible and usable. Although making a user friendly GUI is beyond the authors capacity, here are some pointers on how we could build one. First we have the problem of showing the frequency of the note. It is proposed to render notes much as a graph. The y axis will present log(frequency) since this will make all octaves equidistant, and all like intervals will represent the same distance. The distance may not be enough to recognize an interval, but we dispose of other means such as the colour of the note, it's thickness, background gradation, etc. The x axis is time.

Below a simplistic illustration of the chord progression V⁷ - I , in this case A Major⁷ - D Major

Finally, we will show some examples using the Web Audio API. It is very important to note that the sound quality is secondary in these examples. We are using simple sine wave oscillators. There may be pops due to cutting off notes without ramping the amplitude, etc. These examples are a simple proof of concept, and are destined for comparison between the Relative Music Theory and the Equal Temperament with the same audio quality. You may choose the base frequency, presenting the functionality of a module as it would be used in a composing application. Below is transcribed the music from the graph above in the first 2 examples to compare RMT and Equal Temperament (the frequency is purposefully set high so you can hear the problem, both are horrible but one is also terrible). The third and fourth are chaining the V-I chord progression 3 times, to show what is possible with RMT. (V-(I=V)-(I=V)-I)

-Relative music theory: Frequencies are defined by the fractions shown in the illustration above.

-Equal temperament: Frequencies are defined with 2^n/12*input_frequency, with n chosen accordingly

-Relative music theory, V-I looped (3 times): The RMT module is chained 3 times without any modifications. This is analogous to the composer drag dropping the module onto the D note twice.

-Equal temperament, V-I looped (3 times)

The Relative Music Theory having been exposed as best as possible, we may now examine the possible applications of this novel musical language.

Section 2.b) Scope of application | examples

1) The most urgent task ahead is to make a usable, practical Graphics User Interface. This will allow composers to explore the new possibilities offered by RMT and will make all the other applications of this theory easier. Such an application is absolutely necessary for this new language to take hold in the musical world. It will also allow composers to usher in a new era of musical creation. We may possibly be able to devise a hand written version later on.

2) This musical notation language allows us to transcribe any and all music from around the world. Currently, the 12-note system dominates traditional music. We can imitate other musical languages with the 12-note system, but the true reasoning behind the notes is lost since we cannot write pure harmonics. Using RMT we can conserve all traditional music systems from around the world so our understanding of them would not be lost. This is a tremendous task and will require the participation of many musicians.
[ See Bach's "Neverending Canon" implemented as an RMT module ]

3) Given that we are defining notes with more precision than before, we will encounter instances where a note previously written as a single note on the keyboard will become ambiguous in RMT. We must develop our understanding of harmony and chord progressions to be able to resolve these ambiguities. Also, there are an infinite number of new scales and harmonics to explore. As an example, let us consider the Augmented chord, as well as listen to some new possibilities. There are 2 ways to understand this chord with RMT. We can either have 2 Major thirds (A, A*(5/4), A*(5/4)²) or consider it the chromatic division of an octave into 3 parts (A, 2^1/3*A, 2^2/3*A).

It is surmised that eventually the 12-note system will join the other chromatic scales and be considered apart from harmonies. All chromatic divisions are interesting in their own right, and those that can be played are at the correct frequencies on the equally tempered keyboard. (the 2-chromatic scale: A - Eb), (the 3-chromatic scale (augmented chord): A - C# -E#), (the 4-chromatic scale(diminished 7th chord): A - C - Eb - Gb), (the 6-chromatic scale(the whole tone scale): A - B - C# - D# - F - G).

-V-I chord progression chained eternally, base goes down an octave if it's higher than an octave over base frequency. Theoretically the base will always be different so you can listen until the end of time without the music ever repeating. (limited by the computer's capacity and bad programming)
[ View standalone version using bigInt fractions instead of floating point numbers ]

-perfect Base 3 chord (1, 5/3, 7/3, 3) -> D Major
-Base 3 perfect chord downwards (1, 7/9, 5/9, 3/9) to a Base 5 chord (3/5, 4/5, 1, 6/5)
-a different progression: minor->Major +double 5th -> 7th -> IV=I
-9 note Base 3 'Tesla' scale (he liked numbers 3,6 and 9). (1, 11/9, 13/9, 5/3, 17/9, 19/9, 7/3, 23/9, 25/9) to Base 3 chord (1, 5/3, 7/3, 3)
-10 note Base 5 scale (1, 7/5, 9/5, 11/5, 13/5, 3, 17/5, 19/5, 21/5, 23/5) to Base 5 perfect chord (1, 9/5, 13/5, 17/5, 21/5, 5)
-pentatonic chromatism (1, 5^(1/5), 5^(2/5), 5^(3/5), 5^(4/5), 5)

4) Library of harmonic relations. Point 3) will eventually bring us to defining all harmonic relations as modules. It would be a good idea to group all possibilities in a general music library. We could attach adjectives to each relation to describe what feelings they evoke, thus making the library very useful for point 5).

5) As stated before, RMT will be very practical for computer generated music. Any module can directly plug into another note. Using the Library of harmonic relations 4) and associated adjectives, we can imagine a music generator that would select modules in accordance with cues sent from a program. Integrating the modules although not straightforward will be facilitated by RMT.

-Automatic improviser based on "Kids' Games, Fanfare" by Gérard Massini. It's a small library of melodies, with a simple randomizer and some random objectives, showing the possibilities points 4) and 5) would offer.
[ View improved standalone version with tempo control ]

-I dedicate this work to my son Zénon Yota