Trying to push my Push skills up a peg with Peg and Steely Dan’s gorgeous µ harmonies.

Trying to push my Push skills up a peg with Peg and Steely Dan’s gorgeous µ harmonies.

Very satisfying to receive this series of books from OUP at long last. Very pretty looking academic books, if you can believe that. My chapter with Eugene looks quite cool including all those brain bending Coltrane Cubes, M-Space and improvisational fields.

Available here

*Swing friction* is a term I coined in my PhD thesis and is defined as the differential of swing values between individual performers (or groups of performers). If the swing friction is significantly large and consistently maintained, it may form a characteristic of ensemble feel.

Chuck Berry’s *Johnny B. Goode* (Berry 1958) provides an instructive example of swing friction. Berry, often considered the father of rock n’ roll, was instrumental in ‘straightening out’ the blues 12/8 shuffle rhythm into the archetypal electric guitar riff. Johnny B. Goode features this ‘straight 8^{th’} guitar rhythm, as well as equally straight lead playing juxtaposed with a stubbornly bouncy drum, bass and piano feel. Heavily swung quaver values occur in the ride cymbal pattern, often near the 67% mark, a significant deviation of over 52ms from the straight quaver at 170bpm. The guitar rhythm part however remains resolutely straight rarely venturing beyond 52% swing. This already large 15% discrepancy of swing value is exaggerated with the guitar part often sitting on top of the beat (ranging between 0% and -4% latency) leading to a mean separation of about 17% (≈60ms). The lead guitar is equally straight, although not pushed, and occasionally falling behind the beat. Piano interjections are loose but quavers are generally quite swung, mainly in the 60-67% range and repeated quaver triplets prevail. The bass plays mainly crotchets, with the occasional quaver (usually ≈67%). A representative extract from the track can be heard here:

Figure 1 shows a composite two bar template for the lead, rhythm, bass and drum parts, with time-feel components added. There is a huge gap between the swing values of the guitars and bass and drums. The vocal track tends to fall in between these two extremes. In order to hear the effect of swing friction, This example contains electronic sequences of this section with varying time-feel values: 1) as from Figure 1, 2) all instruments at 67% 3) all at 52% 4) all at a middle ground of 60% and 5) back to the ‘true’ values for comparison.

Figure 1. Composite swing and latency values for guitars, bass and drums in *Johnny B. Goode*.

The sequences have been rendered with MIDI instruments on purpose; although the section would sound better with human performers, but the elimination of the inflection they would inevitably provide allows focus on the power – and limitations – of the SLW model. Mean values for swing and latency have been provided, but the standard deviations of these values introduce the component of looseness or tightness, again different between players. Weighting elements also occur, (the cymbal has a slight emphasis on offbeat quavers for example,) with both mean and standard deviations). This extract plays the sequence first as Figure 1 then with swing, latency, weighting standard deviations from Figure 2 introduced, which add a clearly-defined randomness to each of three time-feel elements, and instruments, individually. There is a subtle but appreciable difference between the sequences; attention to the cymbal pattern, for instance, will reveal a slight offbeat emphasis and looseness.

Lead Guitar | Rhythm Guitar | Bass | Drums |

µ_{s}= 51 ∂_{s}=1.5 µ_{l}= 2 ∂_{l}=
µ |
µ_{s}= 52 ∂_{s}=1
µ µ |
µ_{s}= 67 ∂_{s}=2
µ µ |
µ_{s}= 67 ∂_{s}=2
µ µ |

Figure 2 Mean and standard deviation values of swing, latency and

weighting (measured as dB level)

An averaging out of time-feel components over the entire track runs the risk of over-generalization and may incorrectly group specific mechanisms that occur only occasionally. There are for example, brief moments when the bass seems to join with the rhythm guitar’s straight quavers. There is also the assumption, with a single matrix per instrument, that all beats of the bar are the same, which ignores the emphasis on crotchets 2 and 4 in the drums. Matrices could be provided for beats 1 and 2, and beats 3 and 4 separately, or even weighting at the crotchet level, for greater sophistication when needed.

Despite these acknowledged limitations, the discretionary use of this type of analysis allows for an instructive and parsimonious description of ensemble time-feel elements.

The most succinct analysis I can make of Arvo Pärt’s Spiegel Im Spiegel (a stunning and elegant work). Listen here: https://open.spotify.com/track/3rlqTqUOzu0zDwQFJe44gk

I’ve had this insight about palindromic scales, modes, modal groups, and the Euclidean (and sometimes maximally even) distributions of 7 notes in a one octave scale. This is what it looks like in my head. I really like that the ‘central’ scale of a group is not the traditional figurehead (Ionian, melodic minor etc.) but the palindromic parent. Nice to see the patterns emerge diagrammatically, and I will aim to use Melodic Phrygian and its modes in composing/improvising!

It’s hard (for me at least) to start talking about any aspect of musical theory without talking for an hour or writing several thousand words and/or producing many diagrams. So I’m going to try and just provide some vignette posts, little moments of insight in a hugely complex field of study.

Here let’s look at the harmonic series (the pitches that emerge by multiplying a frequency by the set of integers (1,2,3,4 …) and its relationship to 12-tone equal temperament (the ‘standard’ division of the octave into 12 equal divisions).

The following diagram courtesy of the incredible programmer Dean McNamee (and Mathematica) shows how the pitches generated through the harmonic series differ from 12-tone equal temperament.

The X-axis names the harmonic number (ending at the 20th harmonic) and the Y-axis is the deviation from the nearest even-tempered note (where for example -0.2 is 20% of a semitone flatter than equal temperament). If you look carefully, you’ll see that the fundamental and all the octave equivalents (harmonic numbers 1,2,4,8, 16 etc.) align perfectly with the even-tempered equivalents. In fact these powers of 2 are the only point where the harmonic series meets up (exactly) with 12-TET. These points are circled in red below.

The ‘non-octave’ harmonics are indicated with different coloured arrows (showing in which way they differ from equal temperament). The 3rd harmonic (indicated in orange) is about 2 cents (a mere 2% of a semitone) sharper than the equal tempered equivalent (a perfect 5th), and so of course are its octave equivalents the 6th, 12th, 24th etc. The 9th harmonic is about 4 cents sharper (as it is ‘affected’ by two 3rd harmonics), and is indicated with 2 orange arrows. This ‘stacking’ (and unstacking) of the 3rd harmonic is what is used to create the Pythagorean scale, also known as a 3-limit scale, since we only allow the 3rd (and 2nd) harmonic as tools to reach musical notes.

The 5th harmonic is significantly flatter than the 12-TET equivalent (and the cause of the many historical temperaments that have emerged in polyphonic music. The 10th harmonic has the same ≈14 cents discrepancy. The 15th ends up about being 12 cents flatter, since 15 = 3*5, it uses the 3rd harmonic (which makes it ≈2 cents sharper than 12-TET) and the 5th harmonic (≈14 cents flatter) resulting in an overall drop of about 12 cents (note the contrary orange and yellow arrows). Incidentally, the use of the 3rd and 5th harmonic (together with the trivial 2nd harmonic), creates the 5-limit system of tuning, with its extensive and beautiful use in Hindustani music.

You’ll see that the other prime-numbered harmonics get their own coloured arrows (7th=green, 11th = blue, 13th = purple etc.). The 11th is about as far from 12-TET that is possible, which is why Harry Partch stops there in his 11-limit 43-note universe. Since the primes never end so the harmonic pathways of tuning are (theoretically) endless.

Another way to see the relationship between the harmonic series and 12-TET is to visualise the harmonics resting at points on an infinitely expanding spiral. Each sweep of the spiral represents an octave, and the 12 equally spaced slices of the octave pizza represent each 12-tone slice. This has the advantage of indicating where in the octave each harmonic lies, as well as the octave relationships between sets of harmonics (e.g. between the 3rd ,6th, 12th and 24th harmonics etc.). Here’s a beautiful rendering by Skye Lofander (whose website musicpatterns.dk has many awesome visualitions of musical concepts and is highly recommended).

You’ll notice that the harmonics get closer and closer together, and essentially ‘sweep’ through the 12-TET discrepancy in an ever (but never completely) flattening upward slope. Here’s Dean’s rendering which shows the same phenomenon emerging with the first 500 harmonics:

So along the way the 12-TET and harmonic series will get ‘close enough’ most certainly for human perception (which is a couple of cents on a good day) and acoustic instruments (which waver considerably in temperature and human execution). Even a slight head-turn will invoke enough doppler shift to make fractions of a cent meaningless in real terms.

So which is ‘better’ 12-TET or tuning derived from the harmonic series (also known as just (i.e. ‘true’) intonation)? And what incantation of the harmonic series is more ‘pure’? Is 4 jumps up the 3rd harmonic more ‘natural’ than one jump up the 5th harmonic? They both roughly approximate the major 3rd (the first a ‘Pythagorean third’ ≈8 cents sharper than 12-TET, the second a ‘just 3rd’ at ≈14 cents flatter). Some have attempted a measure of harmonicity (see Vals and Monzos), but its relationship to subjective experience is sketchy at best. Can one think of 12-TET as existing way, way up the harmonic series (so still harmonic), is it a ‘good enough fit’, or wholly incompatible with the harmonic series? What’s inate and wha’t encultered? Why 12 equal divisions and why not some other number?

Just intonation enthusiasts can get heated about its superiority to 12-TET, but there is no doubt that the limiting of notes to 12, close-to, or exactly evenly-spaced has led to the creation of countless, otherwise impossible, musical creations and activities.

The best answer is that they are different, and both available to every composer, get over it, embrace and enjoy.

P.S. this was 3 times longer than planned. Apologies.

For Guitar Techniques Issue 240, I’ve penned a little thing about different approaches to playing over a simple blues progression. THIS WAS SUCH A CONFUSING THING FOR ME TO LEARN GROWING UP. Why? Because

1) There are several effective approaches, and humans being humans can only give advice on what they know. I received conflicting advice from different *great *players on what to do, leaving me befuddled.

2) Blues playing can be both very simple and intuitive, and hugely complex. Learning to use both intuitive flair and theoretical sophistical takes time (not that I’m done, far from it).

This article to which I owe much to Jason Sidwell for the underlying themes offers 4 different approaches to playing on a 12-bar, 3 chord progression. I found it very useful, I hope you might too.

I asked my friend and many-time collaborator Anna Tanczos to visualise Villa-Lobos’s *New York Skyline Melody *for a recent lecture-presentation. The results are fantastic (I predict 1000s of views), and you can see exactly how Villa-Lobos translated the New York Skyline into a solo piano work (note the multiple voices with the foreground and background buildings). This piece has been a big inspiration to me the field of data sonification. For more on *New York Skyline Melody *and similar works* *see here, and for all things Data Sonification here.