Thummer Plays the Blues

What is the blues scale?.

In Africa and the Blues, Gerhard Kubik describes the blues scale as arising from two overlapping harmonic series, one starting a perfect fourth higher than the other. He shows this using a diagram showing only the 5th through 9th partials (harmonics) of each note's harmonic series, which I have modified as shown below (at right).

In the figure at right, the lower-pitched harmonic series is Do (outer ring, with harmonics as black-filled circles), the upper-pitched one is Fa (inner ring, harmonics as unfilled circles). Partials 5 through 9 are shown for each. There's very good alignment between Do's 8th partial and Fa's 6th partial (and of course the octaves thereof), and also between Do's 6th partial and Fa's 9th partial. These well-aligned pairs are a perfect fifth apart.

An alternative way to visualize two overlapping harmonic series is shown below.
In the figure at left, the harmonics of Do are in black while those of Fa, offset a perfect fourth higher, are in grey. At 700 cents above Do's fundamental, Do's 3rd, 6th, 12th, & 24th harmonics align with the 9th & 18th partials of Fa -- hence aligning the Do6 and Fa9 partials as in the circular figure above. Likewise, Do's 1st, 2nd, 4th, 8th, & 16th partials (far left, at 0 cents) align well with Fa's 3rd, 6th, 12, & 24th partials (far right, just past 1199 cents) -- hence aligning the Do8 and Fa6 partials as in the circular figure above.

At the top of the figure above and left, scale degrees are shown. The 3rd and 6th scale degrees are underlined, with each underline joining two stacks of harmonics. The third degree joins Do9 and Fa7 (and their octaves), while the sixth degree joins Do7 and Fa5 (and their octaves). According to Kubik (if I understand his section of his book correctly), these not-quite-aligned Do9/Fa7 and Do7/Fa5 pairs enable the tunings of these scale degrees to be flexible within a fairly wide range.

Another point raised by Kubik is that the 7-edo scale -- also known as the "equiheptatonic" scale, dividing the octave into 7 intervals of equal width -- is common in some parts of Africa. A "third" in 7-edo tuning is 343 cents wide, which is right in the middle of the range of the 3rd scale degree in the above-left figure, providing yet another source of instability in this range.

Does the Thummer suit the blues?

Played in today's standard Western 12-edo tuning (i.e., an “equal division of the octave” into 12 pieces), the Thummer should be at least as blues-capable as the piano or guitar. Its expressive controls (thumb-operated joysticks & electronic motion sensors) allow the user to play the blue notes "between" the notes of 12-edo as a guitarist can do by bending strings, and which a pianist simulates by "crushing" adjacent keys.

However, the Thummer's real potential as a blues instrument arises from the tuning invariance of its isomorphic keyboard, which gives it the same fingering in any tuning of the syntonic temperament, which includes both 12-edo and 31-edo (i.e., an “equal division of the octave” into 31 pieces, in which the tempered perfect fifth is 696.8 cents wide -- only 3.2 cents narrower than 12-edo's tempered perfect fifth).

The use of septimal (7-limit) ratios for the blue notes is explored by W.A. Mathieu in his excellent book Harmonic Experience (Chapters 17 & 33).

In 12-edo tuning, the augmented second (A2) has the same width as the minor third (m3) -- 300 cents -- so they are often treated as if they were "the same" interval.

However, in any other tuning, including 31-edo tuning, the A2 and m3 have different widths, each signifying a different just interval, as does the M3.

• 31-edo's A2, at 271.0 cents, is only four cents narrower than the just septimal minor third (7/6 = 266.9).
• 31-edo's m3, at 309.7 cents, is only 5.9 cents narrower than the just minor third (6/5 = 315.6).
• 31-edo's M3, at 387.1 cents, is less than one cent wider than the just major third (6/5 = 386.3 cents).

This harmonically-relevant distinction between the A2, m3, and M3 gives musicians a choice of three different notes to play across the range of blue 3rds. They can use the A2 to signifiy the 7/6 ratio, the m3 to signify the 6/5 ratio, or the M3 to signify the 5/4 ratio. All of these notes provide a better match with the harmonic series when using 31-edo than when using 12-edo.

Perhaps even more importantly, 31-edo distinguishes the augmented sixth (A6) from the minor seventh (m7).

• 31-edo's A6, at 967.7 cents, is only 1.1 cents narrower than the septimal minor seventh (7/4 = 968.8), making it well-suited for the harmonic seventh, also known as the "babershop seventh," chord.
• 31-edo's m7, at 1006.5 cents, falls almost exactly between just intonation's Pythagorean minor seventh (16/9 = 996.1) and diatonic minor seventh (9/5 = 1017.6), making it well-suited for use in a dominant seventh chord.

Hence, musicians can use the A6 to signifiy the 7/4 ratio, or the m7 to signify the 16/9 and/or the 9/5 ratio. Either way, musicians get a better match with the harmonic series when using 31-edo than when using 12-edo. (Well, actually, 12-edo's m7, at 1000 cents, is a better match with the Pythagorean m7, but it's a worse match with the diatonic m7 and is completely useless as a septimal m7.)

According to this reference, the tonic, subdominant, and dominant chords (I7, IV7, and V7) should be played as harmonic seventh chords (i.e., with an A6) except for the V7 at the turnaround, which should be played as a dominant seventh chord.

If I understand this correctly -- which I very well may not -- then in C, using the tuning described above, the I7, IV7, and V7 blues chords would be played in 31-edo as:

• I7: C-E-G-A#
• IV7: F-A-C-D#
• V7: G-B-D-F (turnaround) or G-B-D-E# (otherwise)

31-edo supports free modulation as well as 12-et does, with a caveat or two. For example, you can't play Coltrane's Giant Steps in 31-edo, becuase Giant Steps' chord progression relies on the fact that 12-edo's way-too-wide 400-cent major third tempers out the diesis , so that a chord progression through three M3's will return to the same pitch class. 31-edo's major thirds are almost perfectly just at 387.1 cents, so progressing through three of them will bring you to a point that's 38.7 cents short of the starting pitch class. Oops! In 31-edo, Giant Steps is a giant stumble, because it relies utterly on the unique structure of 12-edo -- whereas most blues relies on a structure that mixes 7-limit intervals with 5-limit intervals, at which 31-edo excels.

So, how does the Thummer compare to other blues instruments?

• On the piano keyboard, making a distinction between the A6 and m7, or the A2 and m3, is impossible. Both intervals share the same key. Pianists can fake it by crushing adjacent keys, but that's clearly a kludge, which does nothing to distinguish among the relevant harmonic ratios.


• On the guitar, it's possible to play perfectly intoned notes -- through string bending -- but difficult. In this YouTube video lesson from Berklee, the instructor states (2:09 in) that "String bends are kinda tricky. They take a little getting used to...like YEARS, actually, to develop a good sound."


• On the Thummer, you can learn to play perfectly-intoned blue notes in minutes. To paraphrase Bach, you just press the right button at the right time, and the instrument plays itself. In C, for example, if you want the A2, you play the D# button; if you want the m3, you play the Eb button; if you want the M3, you play the D button. You can use the portamento controller to slide smoothly from one precisely-tuned note to another, with no guesstimation involved. Or, you can play any of the notes above, and use pitch bending to slide 'em around at will.

31-edo is good for other musical styles, too. It is nearly identical to 1/4-comma meantone, which dominated the early centuries of Europe’s Common Practice Era, of which the use of augmented sixth chords was distinctive part.

31-edo may also be well-suited to klezmer and gypsy music, which use many augmented intervals.

If 31-edo is so cool, why hasn't it been more widely used? Apparently, because it was thought to require the use of an instrument with 31 keys per octave, for which there could be no mass-market in a 12-edo-dominated world.

This barrier may have been eliminated by the recent discovery of tuning invariance, combined with electronic transposition. Despite having only 19 buttons per octave, it appears to be true that all of the tonally-relevant 5-limit and 7-limit intervals of 31-edo fall on the Thummer's keyboard. (The above-mentioned neutral third does not, because it's an 11-limit interval.) Add to this the Thummer's expressive potential (controlled by its thumb-operated joysticks and internal motion sensors), and the Thummer could be a very credible blues instrument.

However, it's important to realize that 12-edo and 31-edo are just points along the syntonic temperament's tuning continuum, along which the Thummer can be retuned dynamically in real time. On the one hand, that means that a Thummer player can slide smoothly back and forth between 12-edo and 31-edo (or Pythagorean, or 7-edo, or whatever) in real time, choosing the tuning that best fits the current note or phrase. On the other hand, this real-time flexibility enables entirely new musical effects such as expressive polyphonic tuning bends, tuning progressions, and temperament modulations.

My collaborators and I have barely scratched the surface of the possibilities of tuning invariance -- our initial scientific paper was published just three months ago -- so we're not yet sure how big the creative opportunity really is.

Now, if y'all could please point out the errors I've undoubtedly made in this blog article, I'd appreciate it, and will update the article accordingly.

Thanks! :-)