Learning & Teaching Lectures & Presentations Music Theory Research

GFA Keynote

It’s privilege to present the Keynote for the Guitar Foundation of America convention this year.

The event runs June 26, 2022,5 PM – 9:30 PM UTC+01 on Zoom (register to join)You are warmly invited to join us for a series of discussions with scholars from Brazil, the United States, Switzerland, and England. Attendees will have the opportunity to speak informally with the presenters or simply listen. Registration is required. Abstract below:

Diamonds, Abaci, and Hexagrams: Exploring the Pitch Surface of the Guitar Fretboard

In this presentation, I will analyze the idiosyncrasies, challenges, limits, and affordances of the guitar’s “pitch surface”: its fretboard and tuning. Whatever the context—staff notation, music theory, improvisation, performance, and other arenas of guitar activity—the layout of the fretboard exerts a profound influence on how guitarists learn, play, and compose. Inspired by De Souza, Goodrick, and Martino, I will also explore the benefits of creative constraints: scordatura, capo, MIDI tools, and other ways of remapping the guitar’s pitch matrix. These constraints can be used to break the guitarist’s tight auditory-motor link, revitalizing the fretboard as a primary musical space and insightful musical “abacus.”

Full Schedule

MiltOnNotes Music Theory

MiltOnNotes: The 4 Chord Loop(s)

Music Music Theory Publications Writing

J.S.Bach in Aeon

My essay on J.S.Bach’s crafty brilliance and its implications for us all is now published in the wonderful Aeon digital magazine. Many thanks to Nigel Warburton (author, philosopher and podcaster) for the commissioning, editing and constant support.

Guitar Music Music Theory Musical Illustrations Publications

Led Zep IV at 50

It was fun writing this track-by-track musicological analysis of Led Zep IV (50 this year ahem) for the cover feature of Classic Rock Magazine. Limited by copyright and terminology, it’s challenging and instructive to communicate deep theory without dumbing down. Also I now know how to hear the beginning of Rock n Roll and the bridge of Stairway. Get it here and all reasonable shops.

Music Theory Musical Illustrations Publications

Arvo Pärt Analyses in Print

Musical illustrations of Arvo Pärt’s Spiegel im Spiegel and Beatitudes are now published in Arvo Pärt’s Resonant Texts: Choral and Organ Music 1956–2015 (Shenton 2018) and Illiano, R., & Locanto, M. (Eds.). (2019). Twentieth-century music and mathematics. Brepols.

These unpack the musical mechanisms of Arvo Pärt’s music as outlined in this lecture below.

Guitar Lessons Music Theory Publications

Playing in the Key of Sea

A breakdown of the musical science of the Sea Shanty in Total Guitar Issue 344

Guitar Music Theory Publications

Anatomy of a Guitar Solo

Originally commissioned for Total Guitar, this examination of the reader-voted top 50 guitar solos is now publicly available on Guitar World.

Notes per bar profiles
TImeline vs Popularity revealing ‘Mount Late 70s’
Guitar Jazz & Improvisation Lectures & Presentations Live Electronics Music Technology Music Theory Research

Digital Self-Sabotage

A live video presentation at the fantastic 21st Century Guitar Conference “in” Lisbon, March 2021, hosted by the wonderful Amy Brandon and Rita Torres. ‘Digital Self-Sabotage’ explores we guitarists’ deep and twisted engagement with the fretboard, and how technology can expand and disrupt this bond for learning and insight.

Digital Self-Sabotage Lecture
Guitar Lessons Jazz & Improvisation Music Music Theory Publications Research Rhythm

Time-feel in Soundboard Scholar

Soundboard Scholar (No.6) features my paper. “Monitored Freedom: Swing Rhythm in the Jazz Arrangements of Roland Dyens” examines the time-feel in the performances and scores of Roland Dyens, in particular reference to his arrangement of Nuages – and Django’s performances of this piece. Working with the genius Jonathan Leathwood is always a privilege and joy, and I am very grateful that my illustration is used as the cover image to the journal. Available here.

Guitar Lessons Jazz & Improvisation Music Theory Press & Interviews Rhythm

The Deep Theory & Technique of Time-feel – The Guitar Hour Podcast

A pleasure to indulge in full-tilt nerdery about micro-timing with the annoyingly talented and annoyingly nice hosts of The Guitar Hour Podcast. \m/ \m/

Music Theory Publications Research

Arvo Pärt Analyses published by Cambridge University Press

Pleasing to see my ‘postcard’ analyses of Arvo Pärt’s Beatitudes and Spiegel Im Spiegel published in Andrew Shenton’s fine new work Arvo Pärt’s Resonant Texts (Cambridge University Press).

Here they are without explanation.


Lectures & Presentations Music Music Theory Press & Interviews Research

Radio 3 Music Matters – Ancient Greek Modes

A lovely discussion with BBC Studio Manager (and Radiophonic scholar) Jo Langton and presenter Tom Service on Radio 3’s Hidden Voices series on Music Matters. Kathleen Schlesinger’s The Greek Aulos, Ancient Greek Modes, microtonality, the work of Elsie Hamilton and its legacy today. 

Listen from 34:54

Composition Learning & Teaching Music Theory

Tonal Harmony Flowcharts (Major & Minor)

By (heavily belated) popular demand here’s a stab at ‘common-practice’ chord progressions in major and now minor keys!

Lectures & Presentations Music Theory Skepticism

Truth Hertz at QEDCon

Really looking forward to being on this fabulous panel at the fabulous QED conference in fabulous Manchester. Panel on 14th October 2018 Time TBC

Music Technology Music Theory

Steely Dan’s Peg on Push.

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




Jazz & Improvisation Music Theory Publications Research Writing

Music & Shape

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


Guitar Music Music Theory Research

Swing Friction in Johnny B. Goode

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 8th’ 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
µs= 51 ∂s=1.5  µl= 2    ∂l=

µw= -3 ∂w= 2

µs= 52 ∂s=1

µl= -3 ∂l= 1

µw= -3 ∂w= 2

µs= 67 ∂s=2

µl= 0    ∂l= 1

µw= 4   ∂w= 2

µs= 67 ∂s=2

µl= 0    ∂l= 0

µw= 2   ∂w= 1

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.

Composition Music Theory Research

Spiegel Im Spiegel on a Postcard

The most succinct analysis I can make of Arvo Pärt’s Spiegel Im Spiegel (a stunning and elegant work). Listen here:

Spiegel Analysis

Music Theory Research

Palindromic scales and their modal groups

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!

Palindromic scales


Music Music Theory Research

Harmonic Series vs. 12-Tone Equal Temperament

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.

Annotated Harmonic series vs 12-TET

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 has many awesome visualitions of musical concepts and is highly recommended).

Ligedeling vs naturtoner

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.

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