The Harmonic Sequencer: Mapping the Harmonic Series to a Sequencer

Abstract:

This paper discusses a software tool created by the author in MAX. I have created a sequencer that mimics (in reverse) the relationship between the harmonic series and the equal temperament tuning in terms of temporal timing. I will discuss the theory underpinning the software and its implementation.


Introduction

This concept of the Harmonic Sequencer has its origins in my thinking about the relationship between equal temperament tuning and the harmonic series. It is common in certain fields of contemporary music practice to experiment with tunings that deviate from equal temperament. Although Harry Partch, LaMonte Young and Pauline Oliveros are some of the most recognised, the list of composers using just intonation or other tuning systems in the last 100 years is as long as it is illustrious and varied. However, over the past three centuries equal temperament has become dominant and all different temperaments and tunings tend to be regarded as deviations from this norm.

The harmonic series — also sometimes referred to as the natural overtone series, (giving rise to just intonation or ‘Helmholtz's scale’) is a simple formula yielding the frequency of the overtones of any given fundamental note. [1] This series is a naturally occurring phenomena of and not a subjectively determined idea. [2] It is generally agreed that equal temperament was developed independently in both China and Holland in the 16th century. [3] Arguably this was devised in order to accommodate evolving instrumental and compositional practice. The new tuning system sacrificed the perfect consonance of certain intervals for an acceptable compromise consonance across a wider range, thereby extending the use of all intervals across all keys. [4]

Working recently on a project with the Berlin-based composer Arnold Dreyblatt (b. 1953, NYC, USA) who works with a tuning system based on the harmonic series, I started speculating about what could be the reason that the harmonic series has continuously intrigued composers. The consideration that kept coming to the fore was the impurity of equal temperament with its carefully calculated compromises. It is the mathematical simplicity and elegance of Pythagoras’ original concept of a tuning based solely on the 3:2 ratio that is so appealing in which exact doubling of the frequency gives the next pitch in the series. There is something pure and appealing as a composer about this tuning as its inherent simplicity yields an increasingly complex harmonic system.

As a composer that uses their voice as the primary source for all of their work and has studied singing techniques from many cultures, I am used to many different means of vocal production and intonations, having no preference for any one system. The music that interests me demonstrates that the world is nothing if not abundant with local, historical and personal systems of intonation. Furthermore, as every practicing musician knows, compromise is very much part of our everyday practice. Our intonation changes, depending on which other instrumentalists we are playing with. Certainly as singer, I find micro-intonation to be one of my most sophisticated tools of expression (even disregarding my extensive work with overtone and throat singing, which relies on very specific use of the overtone scales and all the intonational issues involved in that).

The thinking behind the Harmonic Sequencer is a conceptual development of taking equal temperament as a standard unit and other temperaments as deviations from this but expressed now in the domain of rhythm. Such thinking has its origins in Stockhausen’s well-developed ideas about the continuum of duration and pitch (Stockhausen, 1957). [5] A personal motivation in exploring the fluidity or continuum of rhythm that the Harmonic Sequencer facilitates is due to the fact that I have always been distinctly annoyed by metronomically regular beats or pulses. I have always found the clockwork regular beats of, say, dance music to be musically claustrophobic. I am not trying to imply some sort of hierarchy of rhythmic feel, where say a ‘natural’ drummer is ‘better’ than a drum machine (after all, drum machines can be programmed in very sophisticated ways, as well). This is merely an aesthetic preference, the way I might tend to prefer certain music to other music, or wool to nylon. As composers we all make subjective decisions and for me, this is one of them. Conceit is at the root of the whole idea of the Harmonic Sequencer, which is in fact partly inspired by my desire to humour my same conceits about metronomic pulse.

Conceptualisation

In conceptualising the Harmonic Sequencer my ideas was that if four-to-the-floor metronomic regularity is the ‘norm’ akin to equal temperament then perhaps there is also an arch-pulse, a pure pulse, a ‘...and God created Pulse’. What if the relationship between the harmonic series and the equal temperament holds true also for pulse/rhythm, but in reverse?

Each note in equal temperament tuning deviates by a particular amount of cents from its equivalent in the harmonics series. If we use a’ (440 Hz) as an example, then the next harmonic is a’’ (880 Hz) and the third harmonic will be the fifth, e’’’ (1320 Hz). However, in the equally temperament system the same e’’’ is tuned to 1318.51 Hz, a difference of 1.49 Hz. This difference can be expressed as the ratio: 1320 / 1318.51 = 1.001130. In cents, the harmonic e’’’ is 1.9 cents higher than the equal tempered e’’’. By calculating this ratio for a series of notes and then ‘transposing’ them into ‘phase-durations’ as Stockhausen terms them, we can reconstruct a proto-pulse or proto-rhythm. Using this conceit of proto-timing, I have constructed a sequencer that will govern the timing of events according to the ratios of deviation of the harmonic series partials from the equal temperament equivalent notes. [6]

In the design process for the Harmonic Sequencer I had to make decisions about how to construct the sequencer itself as a tool useful for my compositional process. One of the problems with mapping tuning-based systems onto rhythm is that they do not work in the same way from a perceptual perspective. Two notes with specific intonations create a harmony or a beating between frequencies. Although two rhythms create a polyrhythm, due to the small ratios involved, perceiving a regular pulse and subtle shifts either side of this is much harder. Despire this, I found that the most pertinent way to construct this proto-sequencer was along the lines of traditional sequencers as my intended application of the Harmonic Sequencer is to use it for on-the-fly live situations where it is an advantage to have visually straightforward interfaces. It was the factor of its being used as a live tool that had a strong bearing on the design of the Harmonic Sequencer. I wanted to construct a tool that had a strong conceptual underpinning but also one that I could use musically and intuitively.

Ratios

Table 1 shows all the ratios calculated from the two sets of pitches. I calculated the first 64 harmonics starting at a’ (440 Hz).

Table 1: The harmonic series and the equal tempered tuning in Hz and as ratios.

Table 2 shows the ratios as they could be laid out in a traditional 16 x 4 sequencer. This version wraps the 64 harmonics into four parallel timing sequences, so that there is an ‘overlapping’ of the different ratios. It is acknowledged that this arrangement is a subjective decision derived from my initial concept, but it is one that I find compositionally stimulating.

Tab02.png

Table 2: Visualisation of one possible setup as imagined in a 16-step 4-channel sequencer.

Looking at Table 2, it quickly became obvious to me that actually there are two ways to construct and run this sequencer. The first is to let the ratios of deviation determine the length of each step, the second and slightly more complicated way is to let each ratio determine the offset of that particular note from an underlying ‘ideal’ regular step. The value at the end of each row shows the total length of each row when implementing the first option. [7]

I decided that both versions have their uses and I implemented them both. With the first option, the rows go out of synchronisation after one cycle and then simply continue to drift further. This can be used to create slowly evolving non-repeating rhythmic patterns. In the Harmonic Sequencer I call this mode ‘WILD’. In the second option, where the ratios determine the deviation instead of the length of each step, the sequencer is actually running in synchronisation, so to speak, and each cycle will always be the same. This is due to the fact that there is an underlying ‘ideal’ step that is ticking away inaudibly in the background and we only hear the individual offsets. I call this mode ‘CYCLIC’.

Examples

To illustrate the output of the Harmonic Sequencer below are four examples using simple clicks:

Sound Example 1: metronomic clicks (60 BPM)

Sound Example 2: harmonic clicks (row 1, 60 BPM)

Sound Example 3: metronomic and harmonic clicks superimposed (row 1, 60 BPM)

Sound Example 4: metronomic and harmonic clicks superimposed (row 1, 30 BPM)

As is evident in these examples, the difference between the the metronomic and harmonic sequences is very small. In fact, one would have to listen closely to hear the deviations when the metronomic clicks are not superimposed. This is because the ratios calculated from the differences between the harmonic series and equal temperament are actually very small (see Table 2). Sound Example 4 is the same as Sound Example 3 but played at a slower tempo. This makes it a little easier to hear the differences in the various steps.

The Harmonic Sequencer runs the 64 ratios in four parallel streams making it possible to pick different steps from the four rows to emphasise the offsets.

Sound Example 1: metronomic clicks (60 BPM)

Sound Example 5: harmonic clicks (selection of rows 1-4, 60 BPM, see Figure 1)

Figure 1: Sequencer Settings for Sound Example 5.

While this is one possible way to use the Harmonic Sequencer, I consider that this could be quite limiting compositionally as only certain combinations of the 64 steps would be used. This implementation adheres to my initial concept but is still bound by the traditional sequencer model. Even in Sound Example 5 the timing deviations are still quite subtle. Although metronomically correct the results were not as rhythmically interesting from a compositional perspective as I had envisaged. One solution that I implemented was the idea of proportionately increasing the ratios that determine the timing deviations. To this end, I have implemented a ZOOM slider whereby every ratio in the sequencer increases exponentially by decimal values ranging from 1 to 61.

Sound Example 6: harmonic clicks, zoom 1 (row 3, 60 BPM)

Sound Example 7: harmonic clicks, zoom 2 (row 3, 60 BPM)

Sound Example 8: harmonic clicks, zoom 3 (row 3, 60 BPM)

Sound Example 9: harmonic clicks, zoom 5 (row 3, 60 BPM)

Sound Example 10: harmonic clicks, zoom 13 (row 3, 60 BPM)

Admittedly this exponential scaling is a move away from the ‘pure’ harmonic series-based rhythmic implementation I originally conceived. However, it turns my conceit of a given proto-timing into a tool whereby I can create non-linear patterns ranging from very subtle to perceptually random which nonetheless repeat. [8]

Sound Example 11: harmonic clicks, settings a (rows 1-4, 60 BPM)

Sound Example 12: harmonic clicks, settings b (rows 1-4, 60 BPM)

Sound Example 13: harmonic clicks, settings c (rows 1-4, 60 BPM)

Sound Example 14: harmonic clicks, settings d (rows 1-4, 60 BPM)

The Patch

Figure 2 shows the Harmonic Sequencer as a MAX patch in presentation mode. In the artistic practice for which this tool is intended, I work with live voice performance. In the current implementation of the Harmonic Sequencer there is a live sampler that collects timings of loudness peaks in a recorded file. At the top of the patch is a gain bar for the input signal and a peak indicator so that one can set the sensitivity appropriately. If the input signal level is low and the peak indicator is not reacting then there will be no timing values for the sequencer to play from. The audio is recorded to two buffer files called ONE and TWO and the two waveform~ graphs show the recorded buffers. Rows 1 and 2 take peak values from buffer ONE and rows 3 and 4 from buffer TWO. So, one needs at least 32 peak values in each recorded buffer in order to populate all the steps of the sequencer fully.

Figure 2: Harmonic Sequencer as a MAX patch in presentation mode.

WILD and CYCLIC are the two available modes as described above. Below the output gain indicators there are two sliders that set the sequencer’s BPM and ZOOM (exponential factor of the ratios).

The row of green LEDs shows the current step in the sequence and the four rows of red LEDs below are where the steps of each row of the sequencer are activated. [9] The sequencer’s number of steps is also adjustable [10]  (The values 4, 8, 12 and 16 are just quick choices. The Harmonic Sequencer enables steps from any number between 2 and 16.) After each row of sequencer steps there are some sliders and toggles that allow the length and pitch of the respective row’s buffer playback to be adjusted. The length is dynamically related to the four envelope function graphs (one for each row). Finally at the bottom of the patch there is a collection of sequencer presets.

Some Final Thoughts

The Harmonic Sequencer is a rhythmic sequencing tool for those who – like me – find regular pulse-based rhythm patterns constricting and want to explore other, non-linear repeating rhythmic patterns. Using the Harmonic Sequencer to time material that is more complex than the clicks of my sound examples as well as judicious application of the ZOOM slider can yield some very interesting non-linear, repeating rhythmic patterns. The timing of these patterns will not be based on arithmetic manipulations of a common time denominator (no matter how complex), but will be directly related to the non-linear relationship of the harmonic series to the equal temperament system.

The Harmonic Sequencer can easily be expanded beyond the harmonic series/equal temperament ratio relationship by implementing other sets of ratios based on other tuning systems. In fact, any set of ratios centred around 1 could be used. Future planned development of this work is to use real-time data from Adrian Gierakowski and Paulina Sundin’s MAX implementation (Sundin, Adkins and Gierakowski, 2013) of William Sethares’ dissonance curves (Sethares, 2005). This way, one could time events based on the frequency content of the incoming material, exploring in the rhythmic domain the relationship between tonal consonance and dissonance according to Sethares’ ideas.


Acknowledgements

I would like to express special thanks to Dominic Thibault and Alex Harker for their MAX advice during the programming of the Harmonic Sequencer. Its implementation is far more elegant as a result of their advice.


References

Sethares, W. A. (2005) Tuning, Timbre, Spectrum, Scale. 2nd ed. London: Springer-Verlag.

Stockhausen, K. (1957) ‘... wie die Zeit vergeht …’ Die Reihe 3, pp.13-42. Translated from German by Cornelius Cardew ‘... How Time Passes …', English ed. of Die Reihe 3 (1959): pp. 10-40.

Sundin, P., Adkins, M., Gierakowski, A. (2013) Beyond Pythagoras. Proceedings of the International Computer Music Conference Proceedings 2013. San Francisco: International Computer Music Association, pp. 383-386.


Notes

[1] Overtones can also variously be referred to as partials and formants. The latter is particularly common in vocal practice.

[2] Essentially an overtone is a straightforward doubling of the previous overtone’s frequency: 400 Hz, 800, Hz, 1600 Hz, 3200 Hz and so on.

[3] Zhu Zaiyu (1536-1611) in 1584 and Simon Stevin (1548-1620) in 1585.

[4] This is a simplistic description of equal temperament tuning. However, for the purposes of this article, I am not interested in exactly defining the equal temperament tuning conceptually — I am content to simply use its mathematical properties.

[5] To rephrase and recapitulate Stockhausen’s concept: frequencies are numbers. These numbers will obey the same rules regardless of their magnitude. 100:50 is mathematically the equivalent of 2:1. So if the pitch relationship/ratio of two pitches is X, it follows that when you change the frequencies proportionally, the ratio will remain the same. E.g. the ratio of 523.251 Hz and 1046.502 Hz (1:2) is the same as that of 2 Hz and 4 Hz. Crucially though, the first pair we would hear as two notes (c’’ and c’’’, one octave higher). The second pair would be heard as two steady pulses, the second twice as fast as the first (120 BPM and 240 BPM, respectively).

[6] This measure, the ratio of deviation, is sometimes known as inharmonicity and actually I first intended to call the sequencer the Inharmonic Sequencer. But since the timing is based on the harmonic series (in a convoluted way) and also has pretensions or using an UR-rhythm, it seemed more appropriate to call it the Harmonic Sequencer.

[7] In a normal sequencer, each row would add up to exactly 16, each step having the length of exactly 1.

[8] The fact that the patters are repeating is essential and was the chief reason to use a sequencer as a model for this project.

[9] There is no visual step counter when running in WILD. It would become increasingly irrelevant, as the four rows run on different time cycles.

[10] E.g. if you set the number of steps to 4, you will be playing the first 4 steps in each row and you will therefore be using ratios 1-4, 17-20, 33-36 and 49-52 of the overtone series.


About the Author:

Girilal Baars is a composer, vocalist and live electronics performer. His work seeks to extend folk music and extended vocal techniques derived from many different cultures around the world by the use of electronics and improvisation. His paper presents his work on the ‘harmonic sequencer’ - a MAX instrument designed to subvert the rhythmic regularity of traditional sequencers by using ratio differences between the harmonic series and equal temperament to create a sense of constant rhythmic flux.