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
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.
It’s possible to gain an incredible fluency at music reading, absorbing large sections of music at a glance. I remember as a youngster being in awe at those who could instantly transform squiggles into beguiling sounds, as if there was no barrier between the symbols, the musical brain and the appropriate finger movements. Even without an instrument to hand, they could somehow hear the page. However, music reading seems to differ from word reading in that it can be quite easily tripped up. With word reading you can read wrds wtht vwls frly cnfdntly, or when they are uʍop ǝpᴉsdn, or even ndsᴉpǝ poʍu ɐup qɐɔʞʍɐɹps. However the odd enharmonic or dotted rather than tied note in music notation, can momentarily derail even an experienced reader.
What I’ve found puzzling is that there is no one unanimous convention for notation, nor a system that will please all the people all the time. So in this short article I will attempt to reveal some of music notation’s hidden conventions and reader preferences.
Ok so Figure 1, shows a series of numbered extracts, from which the subject indicates a preference (or no preference).
Figure 1: The Test
You may want to make a note of your preferences at this point, before we look at the results from our cohort of readers.
We’ll break these down in sections so we can discuss the various implications.
Examples 1-3 test the concept of the imaginary barline – the idea that we should notate as if 4/4 was actually 2 consecutive 2/4 bars, separated by an invisible barline. Of course, semibreves (and dotted minims) violate this regularly, so we might consider them honorary exceptions to this rule. But this begs the question, at what point does the imaginary barline ‘kick in’?
Figure 2 presents the results of the first 3 exercises
Figure 2 Preferences for Examples 1-3, testing the limits of the imaginary barline. Notice how preference is eroded by the weaker ‘bridge posts’. (NB NP= no preference and please note also that due to rounding errors, the percentages in this – and other examples – may not add up to 100% exactly, please get over it.)
You’ll notice that in all 3 cases, a central minim is preferred over an imaginary barline. At this point I’m going to suggest some terminology, let’s call the formation with a central minim (covering beats 2 and 3) as a minim bridge. It seems that readers tolerate this well. Note that positing the minim bridge allows us to talk of – among other things – crotchet bridges on beats 1& to 2& and their implications in sight reading.
Notice that the preference for the minim bridge reduces as we subdivide the material on beats 1 and 4). 1a has a minim bridge well supported by a crotchet post on either end, clearly presenting beats 1 and 4. 1b and 1c on the other hand have weaker posts, the quavers and semiquavers respectively, slightly masking the clarity of beats 1 and 4.
We could in fact imagine minim bridges of different strengths. While the majority may prefer a minim bridge to an imaginary barline (IB), as the material at beats 1 and 4 become more complex, perhaps more people prefer the IB over the bridge presentation (see Figure 3)
Figure 3 Hard and weak minim bridges with imaginary barline equivalents, and a suggested preference curve. (Click to embiggen)
The limited data supports this, but it’s only a tentative suggestion. Furthermore, I suspect that there will be a significant proportion of readers who will always prefer (or at least tolerate) a minim bridge regardless of the content in the rest of the bar.
So far we’ve seen that there is a general preference for minim bridges over imaginary barlines, less so perhaps as the minim’s supporting posts on beats 1 and 4 become more complex. But what of the minim bridge itself? How robust is it to any complexities? The answer, it seems, is not very. Figure 5 shows that when the minim bridge is divided into a quaver-crotchet-quaver pattern (a crotchet bridge formation), preference swings dramatically to the imaginary barline presentation, and slightly more so when the beats 1 and 4 posts are made more complex. So a fractal//Inception-style bridge-within-a-bridge is too much for many readers to bear, although a good 1 in 4 readers actually preferred it over IB. I suspect that some readers, don’t need any IB at all, and are happy to see a sequence of rhythmic values with no visual indication of beat 3. Let’s call these readers who can tolerate (or prefer) a sequence of rhythmic values over IB presentations as ‘sequential readers’.
Figure 4. A minim bridge collapses for most people when it is subdivided into a crotchet bridge. Still, those 1 in 4 sequential readers prefer it to IB.
So we’ve dealt to some degree with minim bridges and their relationship to the imaginary barline, and their kryptonite (the fractal bridge), but what of bridges on beats 1 and 2, and beats 3 and 4? In other words, do the same principles scale down from the whole bar to the half bar?
Figure 5 Testing the Crotchet Bridge (or crotchet IB)
Examples 6 and 7 address this question. You’ll see in Figure 6 that the majority prefer the crotchet bridge to (beat 2 and 4) imaginary barlines (6a over 6b) although this preference is less common than at the whole bar level (1a over 1b). However the crotchet bridge (surprisingly to me) seems far more susceptible to weakening than the minim bridge. Notice that as soon as we weaken the posts on the first and last quavers, preference sways over dramatically to the crotchet IB (7b over 7a). This was just a tentative study (quiz really) but if I knew of this phenomenon I would have interrogated the extent and nature of the fragile crotchet bridge more thoroughly.
Let’s look now at the quaver bridge (semiquaver-quaver-semiquaver). Figure 6 shows that a similar ratio of readers prefer quaver bridges to minim bridges (8a over 8b, as compared to 1a over 1b). However some subjects reported (without solicitation) that 8b was ‘horrible’ or ‘horrifying’. Why this ‘over-information’ created such a negative reaction (as compared to 1b) is not clear. It could be that it is simply the case of 4 instances over 1, or the shortened time scale is less tolerant of unnecessary fuss. I suspect it’s the sequential readers kicking up the fuss, I’ll check.
Figure 6 How to piss off a sequential reader.
The quaver bridge also allows us to check if the standard IB (splitting 4/4 into 2 2/4 bars) is sufficient once semiquavers are introduced. You’ll see in Figure 8 that in fact a slight majority of readers prefer to have imaginary barlines on every beat in the presence of semiquaver syncopation. A good portion preferred standard IB but one who didn’t was upset by the crotchet IB violation. He’ll recover in time.
Figure 7 Half vs. quarter imaginary barlines with semiquaver syncopation
Let’s now test sequential vs. standard IB vs crotchet IB in a syncopated semiquaver passage (Figure 9). The majority preferred crotchet IB (essentially 4 imaginary bars of 1/4), but a significant portion were happier with either no IB or standard IB presentations (Incidentally the former upset some readers presumably because it was the opposite of patronizing). Why did more prefer 10a to 10b (given 10c’s majority) is a bit puzzling. Perhaps there is an in-for-a-penny-in-for-a-pound sentiment in operation here (“if I don’t need one tie over beats 2 and 3, I don’t need any other”).
Figure 8 No vs. standard vs. crotchet IB with semiquaver syncopations.
Aside from minim bridge and the trivial semibreve and dotted minim cases, there are relatively common violations of standard IB, in the context of additive meter. To test this, let’s compare preferences for 3+3+2 vs. 2+3+3 vs. 3+2+3 quaver groups. All of these violate standard IB but will any be preferred over it. In every case standard IB is preferred (11b, 12b and 13b), but around 20-30% went for the ‘full additive’ version. This was particularly true in the 3+3+2 formation rather than the other permutations (one of which was ‘offensive’ to some). Why would this be the case? They all violate standard IB so what else differentiates them? It may be that 3+3+2 is the most commonly seen of these meters (Butler 2006) or – and perhaps because – it has a clear beat 4 ‘post’ to anchor the reader. You’ll notice that readers clearly preferred standard IB over crotchet IB (11b, 12b and 13b over 11c, 12c and 13c).
It perhaps hits the sweet spot between rhythmic clarity and patronizing over information.
Figure 9 Additive meter vs. imaginary barlines
This was a limited, uncontrolled study glorified quiz with a small number of participants, all of whom could see each other’s responses. Still, useful (and surprising) concepts emerged. In particular, I think there is mileage in terms of the bridge concept, and a more flexible concept of the imaginary barline which adapts dynamically to subdivision and syncopation, as well as the sweet spot between fussy information and pure sequentualism. I’m also going to investigate preference patterns and consistencies (or otherwise) within (rather that between) subjects’ responses.
The preferences here may be merely suggestive of experience and convention (although how these conventions emerge is interesting of itself). It’s unlikely that our preferences are the results of evolutionary pressure – quick-read-this-syncopated-passage-to-distract-the-tiger scenarios – but it may tell us something about our music reading faculties, and whether an inexperienced notator’s (and Logic notation) can be so baffling . Whether these preferences are internally consistent or optimal may be questioned, but it’s useful to actually know what a majority of readers prefer when preparing scores. So let’s ask them (ourselves).
Many thanks to those who gave up their time and composure in completing this test. Very interested to hear your preferences, and any feedback below.
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.
Have you noticed the music of Charlie’s Brooker Weekly Wipe?
It’s a lovely theme by one Nathan Fake which captures the perfectly appropriate ‘twisted newsiness’ vibe. Take a listen.
There’s a little moment at the end of the theme that seems to pull the auditory carpet from under your ears. (at around 0:26).
What’s going on?
Well, we humans are excellent at gleaning a pulse from a piece of music, that’s to say a time subdivision which a sufficient number of important musical events satisfies. Our sense of pulse is related to how we might clap or tap our feet along to music, it’s based on mathematical principles of prediction but for many of us it’s a perfectly natural, innate skill.
Now a pulse can be further split into various levels of ‘subdivision’, and pulses may also be grouped in at various ‘higher levels’ of rhythmic organization given rise to the phenomena ‘meter’, ‘beat groups’ and so on.
Our ability to group (and regroup) rhythmic events can be exploited for expressive gain and/or musical surprise.
In Weekly Wipe, there is a pulse sensation of 192pm grouped in 3s (1, 2, 3) with hihats occurring on beats 2 and 3 and a repeating melodic motif, that splits these 3 beats into 6. So we can see the melody group as cycles of 6 beats, with an underlying pulse of 3 sets of 2-subdivisions (see Pulse Perception A in the top portion of the diagram).
What happens at 0:26 is that the melody suggests a different grouping of subdivisions. The 6 subdivisions (which were previously split as 3 sets of 2), is regrouped into 2 sets of 3. (See Pulse Perception B on the lower portion of the diagram).
It sounds disruptive because our predictive faculties are forced to recalibrate soon after 0:27 when are expectations of another 2-subdivision group is extended to 3 members. The tempo doesn’t change (in fact the original accompaniment continues albeit quieter) but our experience has shifted.
Sometimes regroupings can lead to tempo changes whereby the pulse is reinterpretated to make sense of the groupings. This is known as metric modulation. (See Elliot Carter, Bill Bruford’s Earthworks and many others for examples). The effect in Weekly Wipe is not a tempo shift but certainly surprising, particularly to find such a technique in a theme tune.
Now go away.
This is a very interesting audio example (and site). The continuity illusion in optical illusions is perfectly paralleled in the sound world. Strangely it didn’t work the first time on me, and now does consistently. How was it for you? D d y u he r t e g ps?
The Continuity Illusion | Auditory Neuroscience.