Fun with Emulator X: Bohlen 833 cents scale and harmonics

I have been experimenting lately with alternate tunings and scales. A couple that have particularly piqued my interest are the Bohlen-Pierce scale and the much-less-used Bohlen 833 cents scale. The latter is intriguing in that it is based on properties of the fibonacci sequence and the golden ratio (although Bohlen admits he did not have those concepts in mind when he stumbled upon this scale).

Based on the golden ratio (1.618034…), one can construct a harmonic series as multiples of 833 cents that has a very distinct timbre. This can be easily implemented in Emulator X as a series of sinewave voices (or voices of any other harmonic single-wave sample) tuned multiples of 833 cents above the fundamental:

The series above consists of a fundamental, three golden-ratio harmonics, followed by the octave above the fundamental (traditional first harmonic 2:1 ratio), and the three-golden-ratio sub-harmonics of the octave.

Using these and other harmonics, Bohlen was able to construct the following seven-step scale between the tonic and the tone 833 above.

Step
Ratio (dec.)
Ratio (cents)

Diff. to previous step (cents)
0
1.0000
0
1
1.0590
99.27
99.27
2
1.1459
235.77
136.50
3
1.2361
366.91
131.14
4
1.3090
466.18
99.27
5
1.4120
597.32
131.14
6
1.5279
733.82
136.50
7
1.6180
833.09
99.27

Emulator X does not have editable tuning tables, although it does have a 36ET tuning (36 divisions of the octave). Bohlen suggests that playing specific steps out of the 36ET scale yields a good appoximation of the 833 scale:

Step (just)
Cents (just)
Step (36/octave)
Cents (36/octave)
0
0
0
0
1
99.27
3
100.00
2
235.77
7
233.33
3
366.91
11
366.67
4
466.18
14
466.67
5
597.32
18
600.00
6
733.82
22
733.33
7
833.09
25
833.33

Combining the Bohlen 833 scale and harmonic series, which are both based on the golden ratio yields a new tonality. Although it is quite different from the traditional Western tonality based on integer ratios, it is nonetheless “harmonic” with respect to its own overtone series. This is perhaps a simple counter-example to to the Monk's Musical Musings from an earlier article.

But how does it sound? To that end, I provide the following audio example consisting of the scale played on the 833-timbre in Emulator X, along with some additional intervals. Because this is only an approximation using 36ET, things aren't perfectly “harmonic,” but I think one can get a feel for the tonality. I particularly like the “tri-tone” (600 cents above fundamental) here.

The next steps are to come up with a more musical timbre based on the harmonic series, as well as short composition using the scale…






Professors, Monks, Imbalance, Pattern, Harmony and Noise

A fun, far-reaching flight of fancy for this evening's post.

I opted to enjoy a quiet day off in my yard rather than fight the inevitably nasty Santa-Cruz-area traffic. It's actually been quite productive, a lot of cleaning in the garden as well as some much needed maintenance work on the outdoor sculptures. In particular, rust management on the metalworks, and cleaning off the accumulated grime from my own fountain sculpture entitled Imbalance. I don't use a lot of chemical treatments in the water because a lot of local critters wander through and drink from the surface, notably neighborhood cats and the hummingbird that is flittering about the fountain as I write this – or rather, was around the fountain until I pulled out the camera. Anyhow, here is a post-cleaning photo (I do need to figure out something to hide that electrical cord):

In keeping with the work's title, the various columns have shifted and tilted in relation to the ground below and the weight of the stone elements.

After a mid-afternoon's hard work, I settled down to relax, enjoy a refreshing beverage and read for a bit. I am currently reading Metamagical Themas: Questing for the Essence of Mind and Pattern by Douglas Hofstadter, who is best known for his earlier book Gödel, Escher, Bach. It's actually not as heavy as the name implies. It's a series of pieces Hofstadter did for Scientific American in the early 80s, covering a wide variety of issues including patterns, creativity, language, etc. The two articles a read this afternoon dealt with the pattern and aesthetics of the music of Chopin, and transformations on simple “parquet floor” patterns as a form of visual music, respectively. While the latter was more interesting to me personally, it is the former that I wish to write about. While I admire the musicality and technical skill of Chopin as both a composer and pianist, I can't say that I've ever been a “fan.” Indeed, his music is about 180 degrees from my own aesthetically. However, I was struck in particular by one passage Hofstadter wrote:

That there are semantic patterns in music is as undeniable as that there are courses in the theory of harmony. Yet harmony theory has no more succeeded in explaining such patterns than any set of rules has yet succeeded in capturing the essence of artistic crfeativity. To be sure, there are words to decribe well-formed patterns and progressions, but no theory yet invented has even come close to creating a semantic sieve so fine as to let all bad compositions fall through and to retain all good ones. Theories of musical quality are still descriptive and not generative; to some extent, they can explain in hindsight why a piece seems goodm, but they are not sufficient to allow someone to create new peices of quality and interest.

I was reminded of an article that I read last week entitled A Monk's Musical Musings: Musical Philosophy. The author, Huchbald, attempts to argue (with all the style and sophistication usually found in right-wing political bloggers) that everything right and good in music derives from the “god-given” harmonic series, and anything that eschews baroque-era diatonic voice leading rules is somehow not music at all. In the process, he dismisses atonal music (and probably a lot of other music) as “noise.”

There are numerous ways to refute his claims (other than simply celebrating noise as music), perhaps the simplest being the rather casual way he dismisses everything other than his voice-leading rules as “simply rules based on taste which can be left to the discresson [sic] of the composer.” Well, as Hofstadter eloquently points out, this discretion and not the rules is precisely what makes for the best music. It was what separates a genius like J.S. Bach (admired by both authors discussed here) from a typical student in a first-year class on music theory. The sieve is simply too coarse, and “accepts” both the good and bad equally. One need only consider what Bach was able to do contrapuntally with the chromatic theme of A Musical Offering to see how much more there is to even baroque music than basic harmony. There is something in Bach's music that can be described and informed by harmonic theory, but it doesn't tell nearly the whole story, nor explain how he can work with both harmonicity and chromaticism with such ease.

But back to the god-given harmonic series. Simply put, the harmonic series as a set of frequencies that are all integer multiples of the lowest, or fundamental frequency. That is, for fundamental f, the harmonic series is (1)f, 2f, 3f, 4f and so on. Starting on a really low C, i.e., the bottom C of a piano, one can approximate the corresponding harmonic series as follows:

Note the use of “approximate”, we'll get back to that in a moment. The harmonic series does indeed play an important role in acoustics, the timbre of musical instruments and are perception of musical harmonies. For those who would like play with the harmonic series, a good example can be found in the “additive_synthesis” tutorial of Open Sound World – in OSW, simply go to Help:Browse Tutorials, select the “audio” subfolder and open “additive_synthesis.osw”. You can increase or decrease the contribution of different harmonics and hear the effect on the timbre of a sound. The low harmonics (2,3,4, etc.) do indeed contribute to a constant timbre, though some of the higher harmonics start to get a little “squirrelly.” As one gets into harmonics that are not simple powers of two or multiples of three and a power of two (e.g., 6, 12, etc.), the harmonics appear to play less of a role, even when they can be approximated by notes in the western diatonic scale. Moreover, these are approximates that differ from the standard note degrees in western music, the divergence is illustrated in in this chart and elsewhere. One can preserve harmonic relationships using so-called “just intonation”, which is easily to do on electronic instruments, but would require our friend to retune his guitar whenever he changed keys.

Even if one accepts the harmonic series as central to making music, there are numerous ways to use it besides diatonic voice leading. Consider the first few harmonics, which form octaves and perfect fifths. Octaves and perfect fifths are the most consonant intervals – any popular or contemporary musician will immediately recognize them as “power chords.” Prior to the baroque era, such power chords were used quite often in western music, both serious and popular, as the consonances and cadences. In serious music, there were also the Greek modes, which initially did not include the Ionic mode corresponding to our modern notion of a major scale. Indeed, one of the more common modes was the Dorian mode, which can be found on the piano by playing the white keys starting on D. It is a minor mode that can be found in some of my favorite pre-baroque music such as Josquin Des Prez's Missa Mater Patris, and is the foundation for the blues scale that informs American jazz and popular music. Despite violating most of the rules Huchbald puts forth as inherent in music, minor modes sound quite “natural” and moving to most people.

And what of music beyond the harmonic series? Many (most?) acoustic instrument timbres have overtones outside the harmonic series, and indeed some instruments (e.g., bells) can be very inharmonic. Such inharmonicity can lend itself to different ideal tunings and scales than western just intonation, and indeed we see different tunings in other musical traditiions, such as Middle Eastern, South Asian and Southeast Asian (i.e., gamelan) music. Even where we don't hear the western diatonic scale and direct allusions to the harmonic series, we can nonetheless recognize the music as music, and appreciate it in many levels, from simple enjoyment to deep spiritual understanding.

As modern composers and musicians, we often work to subvert these traditions, and indeed I found myself experimental with alternate tunings, such as 19-tone and Bohlen 833 (Golden Ratio). They have tonicities that can be quite different from what we are used to, but a good composer should be able to immerse himself or herself in them and use knowledge from other musical experiences to produce something interesting.

Well, that's enough on the Monk's philosophy and my opinions to the contrary. In subsequent articles, I would like to touch more upon alternative tunings as well as some more of Hofstadter's writings, which certainly deserve more time.