04.29.2015 § Leave a comment
Here’s a teaser to get you excited about the potential of my musical idea “Yaw“.
This is what the main riff of The Beatles’ Daytripper looks like on a piano roll:
And this is what that sounds like. I’m sure you’re familiar with the riff, but this is to get you accustomed to this piece of software’s sound:
Now if we take this section of sheet music and rotate pitch against duration we can get some interesting new music. This is the basic idea of musical yaw.
Here’s Daytripper rotated 90 degrees:
And this is what that sounds like:
Pretty interesting, right? Despite being a chromatic run of 10 consecutive of 12 pitches with 2 missing, it doesn’t sound too bad, and even includes an inverted major chord (B D# F#) and a power chord (D A) expanding into a major sixth dyad (C# A#). Also, rhythmically, we have a stable set of quarter notes at the end (by virtue of having been whole tone steps when as pitch), and the strongest moment – the chord – is an eighth ahead of the second bar, in the same exact position as the strong point in the original riff, the fifth note in the string of five consecutive notes.
The question is though – how did I arrive at this particular mapping of pitch onto duration? Of course we know that pitch ultimately derives from very small durations – frequency – but the durations of rhythms are too many orders of magnitude removed to directly exchange. Fortunately, like most music, Daytripper quantizes both its pitch and duration information.
- Daytripper, like most esoteric and popular Western music, uses the 12ed2 tuning system. In other words, it divides the octave – or harmonic ratio 2:1 – that’s the “2” in 12ed2 – into 12 equal steps – that’s the “12”. Because of octave equivalence, the phenomenon by which most humans perceive harmonic ratios of 2:1 to be so essentially the same note that we give them the same letter, we can say that pitch space repeats itself every 12 steps in this system.
- Also like most esoteric and popular Western music, it uses divisive rhythm, and like much of it binary-style, meaning that it recursively divides time into smaller and smaller halves of itself. The smallest power of 2 required to express Daytripper is 16: it lasts 16 eighth notes, spanning two bars of 4/4. Because Daytripper is a riff/ostinato, by definition it is understood to repeat itself in place a number of times, so we can say durational space repeats itself every 16 steps in this system.
If 12 = 16 we’d be set, that is, if the 12 units of repeating pitch space equaled the 16 units of repeating durational space, rotating would be a trivial matter because we’d have a square grid: just rotate the filled points within it and don’t worry about redrawing the scale at all. Unfortunately we don’t have that.
We have two best options at this point: we can either equivocate the periods of repetition, or the interval units. Let’s try one at a time.
- If we equivocate the periods of repetition, we’re saying that once the riff has rotated 90 degrees, and all pitch information has become durational and vice versa, then the octave (pitch’s period of repetition) will be divided into 16 equal steps (instead of 12 as it was at 0 degrees), and the two rhythmic bars (duration’s period of repetition) will be divided into 12 equal steps (instead of 16 as it was at 0 degrees).
- Now as for the latter half of this, the 12 equal rhythmic steps, this is perfectly normal in the Western canon; this still qualifies as a divisive rhythm, just not a binary one – it’s a ternary rhythm, that is, one that features a single 3 in its prime factorization along with the other 2’s.
- However, as for pitch, 16 equal steps is uncommon. By dividing every set of three semitones into four equal pitch intervals, it creates a very different tonal palette, one that would strike the average listener as bizarre. Personally I am a fan of 16ed2, but with this experiment I was striving for something slightly more accessible.
- So let’s try equivocating interval units instead, i.e. one semitone of pitch treated as equivalent to one eighth note of duration.
- This turns out to be much simpler. We simply draw a square grid of 16 units by 16 units and rotate it.
- You may notice that the Daytripper riff spans more than a mere 12 semitones. If we had equivocated periods of repetition, the result would have been that the three notes in the Daytripper riff with pitches greater or equal than one octave above the lowest pitch would end up overlapping the beginning of the next durational repetition of the riff once rotated 90 degrees. This is just the same as how they essentially overlap the beginning of the next pitch repetition vertically in the original rotation. However, anyone can hear that while the octave higher E’s are in some sense the same notes as the octave lower E, the power of that octave difference is non-negligible; in fact, the resolution from the high version of the E to the low version from the last note to the first note gives the riff a lot of its power!
Finally, I decided to anchor the initial note into the pitch E, so that we’d be grounded in that root no matter how we rotated.
Here’s Daytripper rotated 180 degrees:
And here’s what that sounds like:
Note that unlike the other two rotations here, this one already sees widespread use, because it is equivalent to the inverse retrograde (that is, the upside-down backwards version of itself). This traditional counterpoint technique arrives at the same result by two flips in place of my effect’s rotation.
Finally, here’s Daytripper rotated 270 degrees:
And here’s what that sounds like:
Also quite interesting (though, again, simply the inverse retrograde of the 90 degree version).
To realize the full potential of musical yaw, we have to hear what Daytripper sounds like as it rotates between these four even alignments. I am working on that now. Normally a pitch circular environment is required to achieve musical yaw, since the pitch/duration plane needs to be tiled infinitely to rotate within, both vertically and horizontally, since vertical and horizontal can switch; however in Daytripper’s case, because of the importance of that E octave difference, I will need to create my first pitch curved environment, which you can read more about in the chapter of my Fun Musical Ideas book draft called Extended Pitch Circularity.