Music By Numbers

The following post is an edited version of the first of four chapters from my honours thesis, originally written in 2013. The thesis as a whole acts as a kind of “how to” guide for composing in a few different styles, each of which somewhat removes human aspects of music composition, at the same time exploring ideas of musical universals – those aspects of music that seem to be ubiquitous across all cultures or even found to be in common across different species! This chapter details the creation of an alternate tuning system for musical notes which is based on phi, a mathematical relation found in art and nature.


 

Chapter 1: Music By Numbers

Music is numerical by its very nature. The subdivision of time into rhythms, beats and pulses; the subdivision of frequency into myriad distinct scales and tonal systems, the mathematical principles underlying the combination of frequencies and notes in the creation of harmony. Many aspects of music are based on numbers, but is there a correlation between the numbers that define musical theory, and those that can be used to define natural phenomena? If such a correlation exists, implications could be drawn regarding the development of musical systems, as well as the ideas of musical universals.

There are several ways in which number patterns can be used to describe natural phenomenon. The self-similarity of fractals can describe the growth of plants, logarithmic curves and Fermat’s spiral can be used to describe the shapes of spirals in ram’s horns, the arrangements of leaves on plant stems, and the spiral curves in nautilus shells; chaos theory can explain the flow and shape of rivers, and the shapes of seashells can be described through the use of cellular automata. One pattern that can be found in nature as well as art to a great extent is the Fibonacci sequence.

Given two ‘seed’ numbers, the Fibonacci sequence can be derived by adding the previous two numbers to generate the next. By definition, the two seed numbers used to derive the Fibonacci sequence are 0 and 1, although nearly identical sequences can be derived with the seed numbers 1 and 1, and 1 and 2. The first few numbers in the sequence are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34… (Sigler, 2002).

The recursive addition found in the Fibonacci sequence describes a set of numbers which can be found in nature; in the petals of flowers, the branches of trees, the curvature and shape of seashells, the number of seeds in the spirals of a sunflower, and the dimensions of the human body. The Fibonacci sequence can also be found in artificial works; in design, art, architecture, and music. Even more prevalent than the Fibonacci sequence in man-made works is the golden ratio, also known as Phi (φ), which is related closely to the Fibonacci sequence. By taking any two consecutive Fibonacci numbers and dividing the larger by the smaller, the result is an approximation of Phi (The ratio of Fn to Fn-1 approaches φ as n increases). Roughly equal to 1.618, Phi has been regarded as aesthetically pleasing in many contexts, particularly in the form of the golden rectangle (Fig. 1.1), which has been used to define proportions in visual art and architecture for thousands of years. Le Corbusier presents the golden ratio as a “natural rhythm, inborn to every human organism” (Jovanovic, 2003, unpaginated).

Golden Rectangle

Fig. 1.1: A Golden Rectangle with square section.

(a+b)/a = a/b ≡ φ

Phi was first written about by Greek mathematician Pythagoras, however its aesthetic quality has been shown to exist in artefacts from Europe, Africa and Asia which long predate Pythagoras’ writings (Mainzer, 1996). Phi‘s ubiquity between cultures is perhaps indicative of some primal predisposition to its aesthetic forms in the human mind.

Phi has also been used in music composition, perhaps most notably by Béla Bartók in Music for Strings, Percussion and Celesta, as analysed by Lendvaï (1991). Other works  which show evidence of the use of Phi and the Fibonacci sequence include Erik Satie’s Sonneries de la Rose+Croix, Claude Debussy’s Reflets dans l’eau, and several of Frédéric Chopin’s études (Carter, 2007, pp. 17-20). In each case, Phi has been used to dictate changes in form, with sections of compositions adhering with great accuracy to golden ratio proportions of the entire length, or recursively – sections within sections, the turning point of each being 61.8% of the larger length. See Madden (2005) for a lengthy survey of the use of the golden ratio in constructing musical form.

The most predominant area of music in which ratios are used is in tuning systems, within which each interval can be described precisely by a ratio. The organisation of music into tuning systems, while the systems themselves are varied, is considered a universal element. A rule-based organisation of musical pitches is required for many basics of a musical system such as pattern recognition, repetition, and standardisation. I created a tuning system based on the ratio of Phi, with twelve notes per octave in order to make it compatible with modern western music tradition and notation.

 

Creating a musical tuning system based on Phi

The standard tuning systems that have been used in Western musical traditions have each been based on the harmonic series of a given starting note. Several methods of ‘Just Intonation’ have been based on this rule, however, because of the nature of overtones in the harmonic series, no scale can be comprised purely of ‘natural’ intervals. Just intonation scales either favour fourths and fifths or thirds and sixths, with each key within the scales being comprised of slightly different intervals. This is known as as tonal temperament. Pythagorean tuning is based on a recursive use of harmonic fifths (taking another fifth above the new note created, and so on), which results in an ‘octave’ that is a quarter-tone sharp. This has been circumvented by reducing one of the fifths within the scale to make the octave ‘fit’, but which relegates that fifth to being more dissonant than the others. See Fig. 1.6 for a comparison of the above tuning systems.

In order to create a musical tuning system based on the Fibonacci sequence and Phi, the golden ratio, one must take a similar approach to the on used to create Pythagorean tuning system. Taking a starting note (in this instance, A = 440Hz was used), I created a series of harmonic fifths above, but rather than using the Just Intonation ratio 3:2 to describe a fifth, I used Phi, generating a series of ‘golden intervals’ above the fundamental, each in a golden ratio with the frequency of the last. In aiming to create a tuning system that was compatible with modern 12-tone equal temperament (12-TET, also called 12-EDO, equal division of the octave), I took each frequency that was outside of the octave (above 880Hz), and halved it to find its sub-harmonic frequency (Figs. 1.2 & 1.3 show the results). This method of halving is also used to create Pythagorean tunings and their variants. Many sub-harmonics were halved several times to find a frequency that was within the desired octave.

Tuning systems need not be based around a 2:1 octave. The Bohlen-Pierce scale (Pierce, 2001), which uses a 3:1 ‘tritave‘, Wendy Carlos’ Alpha Scale and Beta Scale (Carlos, 1989-96), as well as Gary Morrison’s 88-cent scale, which is divided into 14 equal steps over a 1232-cent pseudo-octave (Sethares, 1998) all feature stretched octaves as a primary part of their tonal quality. However, the resonance and common overtones shared by octaves, as well as its near-universal quality between many widespread cultures have led to this Phi tuning to be based around a 2:1 octave.

 

fig. 1.2 Frequency Doubling Table

Fig. 1.2: Frequency halving table. Each harmonic (left) is halved until a subharmonic within the octave was found. Within-the-octave pitches are marked in bold

 

fig. 1.3

Fig. 1.3: Each within-the-octave pitch arranged in ascending order of frequency

 

The resulting frequencies show a surprising regularity: the ratios of all but three of the intervals are ~1.059, the three others being ~1.102 (between the 4th and 5th chromatic degrees), and ~1.0407 (between the 8th and 9th, as well as between the 11th and the octave). Within the system of equal temperament the ratio between each of the semitone intervals is212or ~1.0595, the semitone divisions found within the Phi scale are markedly similar to 12-TET tuning of semitones, although the three aforementioned differences result in three separated tetrachords making up the entire chromatic scale, each group of pitches having a different offset pitch (shown in cents) from the standard 12-TET equivalents, shown in Figs. 1.4 and 1.5.

 

1.4 Phi based tuning system nearest pitches

Fig. 1.4: A Phi-based tuning system, chromatic scale (nearest pitches)

 

1.5 Phi based tuning system played pitches

Fig. 1.5: A Phi-based tuning system, chromatic scale (played pitches)

 

There are several ways to use this tuning system in composition. The first is to utilise it as a replacement for 12-TET, using either the chromatic scale as described above, or defining ‘major’ and ‘minor’ equivalents using a subset of the above pitches. In this way, the Phi tuning system is used in a similar way to an exotic scale, and its character comes from the differences in pitch when compared to those which people are more familiar with hearing.

Another way to utilise this tuning system in compositions is to take each tetrachord and use these as subsets of the entire chromatic scale. Within each tetrachord, the intervals are extremely close to 12-TET intervals, yet the interest in the system comes when switching between subsets of the three tetrachords, or from intervals that cross the boundaries between them.

In either case, most acoustic instruments used would require retuning to accommodate playing in the new system. Instruments in the string section, trombones, and the human voice would be able to play these intervals without adjustment, due to a continuous pitch-producing element (the fingerboard on stringed instruments, the slide on a trombone, and vocal folds). Some training would be required in order to have the musicians play with correct intonation for these intervals, however. Electronic instruments, and digital facsimiles of acoustic instruments, are able to be adjusted for the new tuning system with ease.

 

Notation

There are two ways of notating pieces composed using this system in standard notation. The first is to notate pitches as above, marking with accidentals the microtonal adjustments required for correct pitches every time those pitches are notated. The cent markings are not required in the body of the notation, as each adjustment flat or sharp is the same (35 cents flat or 32 cents sharp). The tuning system should, however, be explicitly described and notated so as to avoid confusion.

The second method of notation is to write each pitch as it corresponds to standard western notation. The second tetrachord above would be written as C#, D, D#, E, whereas in terms of frequencies the notes are closer to the semitone above. In this instance, players must be well acquainted with the tuning system in order to play a notated E as a sounding F-35 . In this instance, the tuning system, as well as the notation method being used, must be adequately described before the start of the piece.

When describing the notes in writing, subscript and superscript notations of the cents will make each note (and the tetrachord within which it is located) easily discernible at a glance. As such, the entire chromatic scale in A is as follows: A, A#, B, C, D-35, D#-35E-35, F-35, F+32, F#+32, G+32, G#+32.

Because of the irregular nature of the tuning system, a chromatic scale beginning on any note is different from one starting on another note. This is what 12-TET and other equal division of the octave (EDO) scales seek to avoid, and allows transposition without chords or melodies sounding different in different keys. In the Phi tuning system, there are 12 individual chromatic modes, each containing a slightly different set of intervals (Fig. 1.7).

 

Composing music using this tuning system

As synthesisers were originally conceived to be capable of playing music outside of the standard 12-TET system (just as rhythm and drum machines were originally conceived to play music in very complex time signatures and subdivisions), a synthesiser ensemble was my first choice when writing music in this system. An ensemble of five synthesisers and one digitally retuned piano was used, but to create timbres that worked with the tuning system, some mathematics had to be used to re-tune constituent harmonics in the sounds.

As described in depth in Sethares (1998), any set of notes can sound consonant or dissonant, depending on the timbre of the notes played. As such, the constituent notes of many scales throughout the world are defined by the sonic spectra of the materials that the instruments used to play them are constructed of. Sethares speaks of spectra of sounds in the same way as light spectra (both light and sound behave as waves, albeit at vastly different frequencies), in that both describe the elements that make up the whole. In terms of light, a spectrum breaks a given colour or colours of light into a set of frequencies, some stronger or weaker than others. Sound is the same, although the original sound can be broken down into a number of sine waves that combine to produce it.

In most naturally-occuring sounds (which Sethares refers to as harmonic sounds), the harmonic series describes the most commonly found set of resonances, of which the octave is the most common, and upon which most naturally developed music theories are based. However, musical tuning systems don’t need to be octave-based. With different timbres and spectra, such as those available in many synthesisers, synthetic scales can sound much more natural than if they were played on physical instruments producing “harmonic sounds”. Similarly, using an octave-based scale with detuned intervals within it can cause unwanted dissonance to occur between detuned fundamentals and the harmonics of another note, for example (in the Phi tuning) the note E-35 and the second harmonic of A will have a dissonance that causes the notes to “beat” against each other. This makes the tuning system sound “out of tune” to a far greater extent than it otherwise might.

To remove this unwanted dissonance, the harmonics of each synthesiser must be controlled or re-tuned to match the scale. Very few synthesisers are capable of re-tuning each of a note’s harmonics, but it is more common to be able to limit the harmonics of a sound to purely consist of octaves, or to limit them at a certain point, with no higher harmonics being synthesised. I limited the harmonics being produced to be only octaves, as the Phi tuning system is built around octave equivalence, and as such no extra dissonance will be added to intervals because of higher harmonics.

To compose music in this tuning system, I first created a MIDI Tuning Standard (MTS) file with a software tool called Scala, specifying each interval as it should be re-tuned. Importing this tuning file into two VSTi software synthesisers (Freely available IVOR and Xenharmonic FMTS), the notes played were re-tuned to suit the Phi system.

 

fig. 1.6 Comparisson

Fig. 1.6: A comparison of Pythagorean, Just Intonation, and 12-TET tuning systems

 

fig. 1.7 Phi Modes

Fig. 1.7: Each of the chromatic modes found within the Phi tuning system

 

12-note Phi Tuning System

Phi and the Fibonacci sequence are very prevalent in art, architecture and nature alike. Both have been used in music composition, though usually used to define section lengths and melodic material. The former seems to me like a legitimate use of Phi, in a similar way to its use within architecture – to define proportions in a way that is aesthetically pleasing. However, using the Fibonacci sequence, or any mathematical series of numbers, to inform melodic choices of notes from what is essentially an unrelated or arbitrary set of pitches (usually the 12-tone equal temperament tuning system).

In my point of view, Phi (expressed most naturally as the ratio 1:1.618) is much more suited to describe a tuning system rather than using its numbers arbitrarily. As such, I created a tuning system from a series of intervals, each 1.618 times the frequency of the last. Each of these “golden intervals”, stacked vertically in frequency (pitch) far exceeded the octave, which I wanted to preserve in order to keep the system “compatible” with modern 12-TET tradition, pieces, and instruments. I took the twelve golden intervals and halved them in frequency if they were beyond an octave, resulting in a single-octave, 12-tone tuning system based on Phi:

 

 

 

Realising this tuning system with instruments

Some instruments have a continuous sound-producing element, such as the fingerboard on stringed instruments, or the slider on a trombone. These are capable of playing within the Phi tuning system, provided that the musician is familiar and capable of playing the different intonations. However, for my purposes, real players were not a possibility.

I turned to software synthesisers and samplers, but it became clear early on that quantising the pitches to microtonal shifts of standard 12-TET was going to be difficult. I took three separate approaches to the problem:

 

  1. MIDI splits and global shifts

The first approach was applicable to this scale fairly easily because of the nature of the scale: it is broken into three tetrachords, each shifted by a certain degree:

A A# B C C# D D# E F F# G G#

Normal pitches +65 cents +32 cents

Using software plugins inserted in the MIDI chain before the synth, I split the MIDI input above every C, E and G#, sending all instances of each tetrachord to a separate MIDI channel, each routed to a different synthesiser. So all of the A A# B C notes on the input keyboard were being routed to channel 1, all the C# D D# E to channel 2, and so on. All that was required then was to re-tune the individual synthesisers (channels 2 and 3) to their appropriate cent-shifts. Playing the keyboard, the musician wouldn’t notice any of this back-end routing, and the keyboard would play in the Phi tuning system.

There are definite benefits of this method of re-tuning, such as being able to re-tune not only synthesisers but sample libraries. I was able to re-tune any instrument in this manner. However, because three instances of any synth/sampler are required for each instrument being played, it quickly becomes unwieldy for computer systems with modest processing power. Also, each instrument required 24 instances of splitting and routing plug-ins in order to split the signal for a full 88-key keyboard into the requisite tetrachords.

While this method works well, it is a fairly heavy workaround (both in terms of CPU processing and fiddly work), and I was sure there was a better way to implement the tuning.

 

  1. A DIY re-tuning plugin

I had experimented with re-tuning on-the-fly with pitch bend controller modifiers in the past, and this seemed like a good solution to the problem.

Using a built-in scripting language in REAPER, I wrote a small plugin that sent MIDI pitch bend data (status byte 0xEc) with each note that was processed. Within the plugin, the user can specify to what degree each note is shifted, and as such it is usable for all 12-tone tuning systems.

While this method is a much more elegant solution, and easily adjustable after-the-fact, there were several drawbacks:

  • For each instrument, the plugin is only capable of monophonic lines
  • If the line is played legato, the plugin’s MIDI output data would need careful editing afterwards to achieve a legato sound with smooth pitch shifting
  • The plugin is incompatible with certain instruments depending on the instrument’s implementation of MIDI pitch data. In EWQL Symphony Orchestra, a second pitch-bend status byte is sent with each note-off, resulting in a perfectly re-tuned note only until the note-off message, at which point the release tail sample is called, at the original unchanged pitch. As such, this method is unusable for some virtual instruments.

 

  1. MTS re-tuning

A small number of synthesisers are capable of interpreting MTS (MIDI Tuning Standard) data files, which send a bulk retuning of 128 notes and then retuning for each note as it is played.

While it is true that few synthesisers support the use of MTS files, this is by far the most usable solution, as it is a) a simple, all-in-one method, and b) it allows playing of chords and legato lines within the tuning system (or any other tuning system).

Using an open source Linux software package called Scala, I created an MTS file of the tuning system. This required some editing to make it compatible with Xen-Arts software synths Ivor and Xenharmonic FMTS.

 

Scala Tuning File: Phi.tun

!
Phi
12
!
99.27089
198.54151
297.81267
466.18059
565.45135
664.72237
763.99326
833.09030
932.36119
1031.63207
1130.90296
2/1
!

 

Audio

 


Chapter 1 references:

Carlos, W. (1989-96). Three Asymmetric Divisions of the Octave [Online]. Available: http://www.wendycarlos.com/resources/pitch.html [Accessed 6/08/2012].

Carter, G. (2007). Piano Mannerisms, Tradition and the Golden Ratio in Chopin and Liszt. Wendleydale Press, Ashfield, NSW.

Jovanovic, R. (2003). Zeisung and Le Corbusier [Online]. Available: http://milan.milanovic.org/math/english/golden/golden6.html [Accessed 30/08/2012].

Lendvaï, E. (1991). Béla Bartók: An Analysis of his Music. Pro Am Music Resources, New York.

Madden, C. (2005). Fib and Phi in Music: The Golden Proportion Musical Form. High Art Press, Salt Lake City.

Mainzer, K. (1996). Symmetries of Nature: A Handbook for Philosophy of Nature and Science. Walter de Gruyter & Co., Berlin.

Pierce, J., (2001). Consonance and scales, In: Perry R. Cook. Music, Cognition, and Computerized Sound: An Introduction to Psychoacoustics. MIT Press, Massachusetts.

Sethares, W. (1998). Tuning, timbre, spectrum, scale. Springer, London.

Sigler, L. (2002). Fibonacci’s Leber Abaci. Springer-Verlag, London.

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