Categories

## 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: https://open.spotify.com/track/3rlqTqUOzu0zDwQFJe44gk

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## 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!

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## 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.

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.

Categories

# Patterns in Sight Reading Preferences

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.

Some disclaimers:

• This is a ‘study’ of a mere 54 participants, ranging from intermediate to highly experienced note readers. All from Facebook, and most of whom are my friends, so make of that what you will.
• We are limiting ourselves to 4/4, single notes and no rests. A major restriction, but one must start somewhere.
• This is very lazy research. I spent a few seconds googling relevant terms, but basically wanted to follow through these ideas prima vista. I’m sure if I looked for 7 seconds longer I’d find Ideathief & Twatamaholey’s comprehensive and beautifully argued 1973 thesis on the topic. So be it.
• I have no affiliation with Big Quaver.

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.

### The Extent of Imaginary Barlines

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.

### Bridge Stability

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.

### Crotchet Bridges and Imaginary Quarter-Barlines

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.

### Quaver Bridges, Patronizing Over-information and Even More Imaginary Barlines

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.

### Additive Meters vs. Imaginary Barlines

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

# Conclusions

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.

Categories

## The Modal Compass

NOTE: This Modal Compass has gone through some generations. The latest versions (©2024 Milton Mermikides) – as featured in Gresham Lecture ‘The Colour Spectrum of Scales and Modes’ are here:

And ‘left-handed’ Dark to Bright (©2024 Mermikides)

Created by Skye Løfvander (musicpatterns.dk) (2015 Løvander/Mermikides).

2015: Original. A way of visualising the relationship between the modes of the major scale. When I have time I’ll expand to other modal groups (and fix the dodgy alignments).

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## 4 Levels of the Blues

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.

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## New York Skyline Melody Visualised

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.

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## Changes Over Time

Here’s the theoretical section of my PhD Changes Over Time (2010)

And the practice portion:

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## PulseWipe: Pulse Prediction in the Weekly Wipe Theme Tune

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.

http://youtu.be/5ohqspB-yHY?t=07s

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.

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## Filling in the gaps

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.

Categories

# Hertz so Good?

### The resonating delusion of the 432Hz movement

@miltonline

[tl;dr Kindly read at least the following 5 points before commenting:
1) I am pro-choice in terms of music, and actively encourage experimentation and challenge of the status quo in music, both in my practice and my teaching. I am also well-versed (but always learning) in terms of the science, history and multicultural practice of music.
2) This article examines all the various ‘scientific’, ‘historical’ and ‘cosmic’ arguments I have found for the superiority of the 432Hz over 440Hz tuning reference, and find them to be fallacious, contradictory, misinformed and/or unevidenced. I also see no evidence for 440Hz superiority to 432Hz (or any other over any other for that matter).
3) While there are some subtle physical world differences of 432Hz to 440Hz, I suggest that it is not a very significant musical argument to have, and there are many far better music areas of development and cultural shift (including temperament, microtiming etc.)
4) I use different tuning references, I’m very happy for anyone to do so (or anything they want to musically of course), I just am yet to find any good argument or evidence for 432Hz superiority.
5) Despite accusations to the contrary, I am not a shill of some shadowy musical establishment, Illumaniti or lizards to challenge 432Hz. Believe it or not, I’m not being paid to write this. Heaven forbid.

I welcome any critical feedback, but please keep it civil, and threats and insults to me and my family will not be tolerated. I can’t believe I have to write this about a music theory post. ]

There’s been a recent musical movement by a small but impassioned group of people advocating a change from 440Hz to 432Hz musical tuning. Proponents claim that tuning music to this frequency results in a more sonorous and ‘natural’ sound which will ultimately make every one of us happier, peaceful and healthy. Tips of how to retune music libraries and instruments abound and the benefits of the adjustment are zealously extolled by advocates. These recommendations are often accompanied by claims that the prevailing 440Hz standardization has negative effects, as well as links to Nazi Germany. The Illumaniti have also been implicated (presumably in the guise of a political powerful and wealthy musical academia) who are hiding this ‘musical truth’ which ‘they don’t want you to know”. Any popular interest in musical analysis – the enquiry into the complex and beautiful mechanics which make music work – is rare and welcome, and new ideas, subversions and revolutions are the lifeblood of musical progress. All music traditions however fossilised today are built upon revolutionary ideas of the time, so this – as any other – movement which challenges the homogeneity in musical practice, wherever they arise deserves serious consideration.

But in order to understand the effects of such a move, some basics of frequency and pitch need to be discussed, so here’s a very succinct introduction.

## Frequency and Pitch

An instrument, voice – or any ‘excited’ object – will disturb the molecules in the air surrounding it, and as that object vibrates – imagine the oscillating prongs of a struck tuning fork – the molecules in the surrounding air (or whatever medium) oscillate sympathetically. Thus a wave of changing pressure propagates from the object, spreading its energy out in the surrounding air, perhaps to fall on some local ear drums (or microphone diaphragms) which in turn oscillate sympathetically. The amount the molecules are displaced is known as the amplitude and is related to our experience of volume, but here we are more concerned with the frequency of the waves: the time for each vibration cycle of the object to be completed (for example how often the prongs of our tuning fork move from their rightmost extent, bounce back and forth and return to the rightmost extent). The frequency of this oscillation is usually measured in Hertz (Hz.) the number of these cycles per second (or kHz for 1000s of cycles per second) and when these fall in the approximate range of 20Hz-20kHz we humans experience them as sound, with the frequency value correlating to pitch. The lower the frequency, the lower the pitch, the higher frequencies are heard as higher pitches. Crucially, this frequency is continuous – we can have theoretically create any frequency (and pitch) at whatever increment (440Hz, 440.02Hz, 893.3482Hz whatever) it is not stepped – although there is a perceptual limit to fine difference. It’s also important to understand that our experience of frequency is logarithmic, we hear multiplications and divisions (rather than additions and subtractions) of frequency as similar musical intervals. For example, a doubling (or halving) of a frequency creates a musical interval known as an octave, which has a sense of musical equivalence (this phenomenon is sometimes referred to as the Law of Octave Equivalence). Incidentally, doubling (or halving) a frequency is equivalent to halving (or doubling) a string length and that’s why for example you’ll find an octave exactly halfway along a guitar string.

Almost all musical cultures share the concept of the octave and give the same note names to pitches whose frequencies are multiples (or divisions) of 2. So if concert A is at 440Hz, we’ll find an infinite number of theoretical As (although only 10 or so audible ones) at 880Hz, 1.76kHz, 3.52kHz etc. above and 220Hz, 110Hz, 55.5Hz etc. below.

## Slicing the Octave

So by halving or doubling we can create octaves, but other musical intervals can also be created by multiplying (or dividing) by numbers other than 2. In fact by multiplying a frequency by various frequency ratios the octave itself can be sliced into any number of subdivisions. How an octave is divided in music theory and practice is hugely interesting, complex and musically relevant with an early history that includes Ancient Babylon, Pythagoras and the Arabic world resulting in a staggering number of systems from the 22 Hindustani shrutis, 55-division Classical tuning, Javanese Slendro and Pelog, the 15(+) divisions of the Arabic maqam, Blues microtones, Harry Partch’s 43-note universe and innumerable other intuited or theoretically grounded systems.

The system many of us have inherited – at least conceptually – divides the octave into 12 equal parts (that is, each increment is based on a fixed multiplication of a reference pitch). In this 12 tone equal tempered (12-TET) system we can calculate the 12 notes by multiplying a reference pitch (usually 440Hz) by this fixed frequency ratio (the 12th root of 2 ≈1.059463) to create a series of pitches (see below). Incidentally these fixed multiplications are associated with corresponding fixed divisions of a wavelength, and that is why you’ll notice the frets of a standard guitar fretboard get closer together as you progress up the string – each fret is placed at about 5.95% of the remaining string – not at fixed distances throughout.  The resulting pitches can be repeated at any number of octaves above (or below) by doubling (or halving) the relevant frequencies.

 12-TET Frequency (Hz) 440Hz Reference Frequency relationship to reference (440Hz) A 440 1 A# or Bb 466.16376 1.05946 B 493.88330 1.12246 C 523.25113 1.18921 C# or Db 554.36526 1.25992 D 587.32954 1.33484 D# or Eb 622.25397 1.41421 E 659.25511 1.49831 F 698.45646 1.58740 F# or Gb 739.98885 1.68179 G 783.99087 1.78180 G# or Ab 830.60940 1.88775 A’ 880 2

Table 1: Absolute frequencies (and frequency relationships) of 12-TET (Concert A to the octave above) in 440Hz. All numbers approximated to 5 decimal places.

12-TET has some specific merits and many shortfalls, simplistically it excels at providing a simple system for handling complex chords and changing keys and hence became standard in the transition from Classical to Romantic music and its ultimate – and perhaps inevitable – ascent (some say descent) into 12-tone serialism. However, 12-TET ignores the spectrum of alternative divisions of the octave including those based on ‘just’ or ‘pure’ frequency ratios such as 3/2, 5/4 (as favoured in the monophonic drone based Hindustani music for example). In fact 12-TET is just one of many possible tuning systems available which have been, are, or could be put into musical practice. It is rarely questioned and many people (including musicians) blindly accept this method of octave division, wholly unaware either that other systems exist or that 12-TET is relatively new in the history of Western Art music, for example we know that Mozart considered enharmonics such as G# and Ab (which are equivalent in 12-TET) to be differently pitched notes. If one is looking for a cultural blindspot against which to rail, 12-TET is the perfect candidate. Other tuning systems are theoretically beautifully, historically and stylistically diverse and startlingly effective and relevant. If you are into a ‘revolution of consciousness’ (as the 432Hz movement purport to offer) then engage with music which eschews 12-TET: the tuning of Hindustani ragas (based on simple ‘pure’ frequency ratios of 3/2, 5/4, 6/5 and the like), the harmonic series exploitation in Tuvan throat singing, Javanese Gamelan, Murail’s spectralism, Lucy tuning’s ‘squaring the circle’ approach to 5ths and 3rds, Partch’s 11-limit universe, Taiwanese 7-TET flute and stretched octave tuning for starters.

So what ‘musical truth’ are the 432Hz proponents offering?

## A New Reference

The 432Hz proponents suggest that the A reference point (often called concert A, the A above middle C on the piano, and the familiar orchestral tuning note) should be tuned not to 440Hz but to 432Hz. It’s about 31.8% on its way down to the next equal-tempered chromatic note (about 32 cents in music tuning jargon – around one third of a semitone). When one ‘tunes to 432Hz’ all the other pitches move down. So for example, the A above concert A which was previously at 880Hz (double of 440Hz, creating the octave above) now moves down to 864Hz, as do all the other 12 notes in every octave (in a 12-tone per octave system). In fact the whole network of relationships, the spectrum of frequency relationships to that reference pitch moves down.

 12-TET Frequency (Hz.) 440Hz Ref Frequency relationship to reference (440Hz) Frequency (Hz.) 432Hz Ref Frequency relationship to reference (432Hz) A 440 1 432 1 Bb 466.16376 1.05946 457.68806 1.05946 B 493.88330 1.12246 484.90360 1.12246 C 523.25113 1.18921 513.73747 1.18921 C# 554.36526 1.25992 544.28589 1.25992 D 587.32954 1.33484 576.65082 1.33484 D# 622.25397 1.41421 610.94026 1.41421 E 659.25511 1.49831 647.26866 1.49831 F 698.45646 1.58740 685.75725 1.58740 F# 739.98885 1.68179 726.53450 1.68179 G 783.99087 1.78180 769.73649 1.78180 G# 830.60940 1.88775 815.50741 1.88775 A 880 2 864 2

Table 2: Comparison of Frequencies and their relationships of 440Hz and 432Hz tuning. All numbers approximated to 5 decimal places.

This is an important point to make – and one many 432ruthers do not seem to understand – everything moves along with the frequency relationships. It’s like moving a Rembrandt painting a fraction of an inch down a wall and claiming an improvement. All the notes have moved, but the frequency relationships between them are identical (Table 2). [Another – perhaps better – analogy might be a small uniform decrease in the saturation of all colours in a painting]. Even though some buzz words concerning tuning systems (temperance, pure 5ths, cycles and spirals of 5ths, fundamental, just intonation etc.) appear in the word-salad of 432Hz proponent sites and posts, few show an understanding of the concept, let alone offer relevant alternatives to 12-TET tuning systems. Certainly transposing everything down to 432Hz alone will not alter any internal frequency relationships – the essence of tuning. Changing a tuning system is significant, changing a reference pitch is not necessarily so. One could argue that a reference pitch (as opposed to a tuning system or temperament) is more about the arbitrary naming of pitches, than the more pertinent interrelationship between them. The 432Hz movement lay claim to a musical enlightenment yet ironically, many accept without the question the highly conventional (and mainly ‘Western’) 12 tone division and 7 note names, and erroneously conflate a poor grasp of tuning systems with that of reference frequencies. And crucially, even when 432Hz recommend ‘just’ intonation or other alternatives to 12-TET, one must remember that 432Hz has no monopoly on these systems, which may (and are) implemented with any other reference frequency (including 440Hz).

## The Arguments for 432Hz Examined

So what difference will changing a reference pitch make, and why is 432Hz specifically promoted? Let’s take some of the key arguments in turn.

1) 432Hz sounds more relaxed/seated/centered/peaceful etc. than 440Hz

Play any higher tone followed by a slightly lower one and you can convince yourself and others that the second is more relaxed and resolved, and less ‘harsh’. This is simply a trick of associating negative connotations to higher pitches and positive ones to lower pitches. A higher pitch could be described as harsh, cerebral, brash, brittle and sterile (as opposed to soft, spiritual, peaceful, relaxed and warm) or strong, thoughtful, brave, optimistic and bright (as opposed to weak, thoughtless, cowardly, pessimistic and dull). Our expectations do the rest. If you value anecdotes, then consider this: I’ve discussed the 440Hz/432Hz issue and played the two tones to an audience, almost all agree that the latter sounds more ‘relaxed’ and ‘peaceful’.  However it’s then revealed that what was actually heard was 432Hz followed by a 424.15Hz tone (a drop in pitch proportional to 440Hz-432Hz).

2) Frequency is important and 432Hz is a special number of cosmic significance

This is the most common argument for the shift to 432Hz but fails to acknowledge some basic errors of judgment. Firstly although frequency is central to musical pitch (as well as rhythm and timbre of course) it is no more important in a 432Hz than a 440Hz setting (or with any other reference point you wish to choose). Secondly, although frequency as a concept is vital, the measuring system we use (Hz: cycles per second) is entirely arbitrary, based on the various measurements available for the Earth’s – not entirely consistent -rotation period. Most recently a specific number of radiation periods of a particular atom at a specific temperature is used as the calculation. There is nothing natural or axiomatic about the number of periods, or the atom and temperature chosen. In addition 432ruthers show a complete obsession with integers, when the decimal numbering system we use is neither natural nor cosmic. Pi and Phi (as ‘naturally occurring’ numbers as one can hope to find) don’t conform to such rational niceties. The onus is on the 432Hz proponent to demonstrate any reason to accept this number over any other and have yet to come close to evidence. Embarrassing arguments about Saturn’s 864 ‘year’ (432×2) procession are approximate, data-mined and irrelevant. Saturn is argued to be the Solar System’s (and hence the Universe’s presumably) time-keeper as it is the furthest planet from the Sun. Which it is of course, not. Links of 432Hz to the ‘Earth frequency’ are examined later.

3) 432Hz is a ‘good’ frequency and tuning to 432Hz makes all the related notes ‘better’

If we accept that 432Hz is somehow better a frequency than 440Hz, then let’s look more critically at what would result from such an adjustment. 432Hz would only be the frequency of the (arbitrarily named) concert A, the other notes would be (presumably 12-TET) multiplications of that frequency, are these special also?

The argument sometimes touted that 432Hz produces more integer frequencies than 440Hz, is patently untrue (see the table above) and irrelevant even if it did. If we are so generous as to suggest that proponents mean that more integers are produced when using the 22-Shruti ‘just intonation’ tuning system, 432Hz fares a little better than 440Hz (12 of 22 vs 10 of 22 integers) but worse than other reference notes like 400Hz, 600Hz or even 622.08kHz the latter of which has all divisions as integers.

It gets worse. Just because we use a 432Hz reference doesn’t mean that that frequency is prominently represented in a piece of music. A piece entirely in the keys of B major, F# major, Eb major, Ab major, Db major among others will have no A notes in them, and other keys will have more prominent frequencies than 432Hz (and its octave equivalents).

And it gets worse still. When a note is played we hear the fundamental pitch, together with a pattern of higher harmonics above it (an electronically generate sine wave is the only instrument which doesn’t exhibit this property) and that contributes to how we say, differentiate a C played on a flute to the same note on a piano. These harmonics are found at clearly differentiated and ordered intervals on melodic instruments, and more smeared across a range of frequencies in percussive instruments and ‘noisier’ sounds. The upshot of this is the following: In reference systems other than 432Hz (including the evil 440Hz), the specific frequency of 432Hz (and associated multiplications) will be heard repeatedly, presumably making it better. Conversely all those nasty frequencies found in a 440Hz reference system will be amply represented in 432Hz tuned music. Most inconvenient. Comparing a piece of music tuned from 440Hz to 432Hz (Figure 1) shows little difference in harmonic content or prominence of the ‘magic’ frequencies.

Figure 1: If you look closely you can probably tell which of these identical pieces of music is in 440Hz and which is in 432Hz. Are there any salient differences in harmonic spectrum?

4) 432Hz links to other auspicious frequencies

Claims that a 432Hz system produces the precise magic integer of 256Hz in a C below, are false. It’s close but that’s not good enough if you want to make the argument from exact integers, which is baseless anyway.

The argument from the ‘cosmological significant’ 108Hz (2 octaves below 432Hz) collapses quickly. Here’s the argument: 108 is cosmological significant because it is the distance of the Earth to the Sun divided by the Sun’s diameter (actually it’s 107.5) and also the distance of the Moon to the Earth divided by the Moon’s diameter (a way off at 110.63), so therefore 432Hz is cosmologically better. Four problems
1) The maths is wrong – or so approximate that you could make the same argument for 430.2Hz (or 442.5Hz)
2) Those 2 very provincial celestial bodies are no more cosmologically significant than the billions of others from which to choose, and the equation arbitrary
3) There’s no reason why a distance/diameter ratio – which would seem to be about gravity and relative density – should have bearing on the arbitrary time-slicing of the Hz.
4) Even if problems 1-3 somehow disappeared, there’s no reason for that number to make better music.

Don’t look behind the curtain.

Another argument – which is perhaps the most interesting – is that 432Hz produces a C# two octaves below at precisely 136.1Hz (very close but not exactly, you’d need a reference frequency of about 432.09Hz for that in 12-TET- more on that later). Why 136.1Hz? Well this is the ‘Earth frequency’  found in the Om tuning fork and is used to tune the drone note of a Sitar in Hindustani music. It is derived from the (first decimal approximation) of the frequency of the earth’s rotation around the Sun scaled up by 32 octaves (let’s not mention the complexities and competing systems of calculating a year). Rather beautiful – it doesn’t employ the fallacy of integers (only the fallacy of single decimal points) or the trappings of manufactured time units. If a reference point is arbitrary you may as well think big. It’s a rather localized geo/heliocentric view of the Universe (we could just as well use the frequency of star formations, galactic spin, cometary orbits and so forth), but if it gives one a layer of awe while engaged with music that’s all good. The implication is that it resonates in harmony (read ‘simple frequency ratio’) with the Earth’s orbit.

But there is a serious issue which arises when attempting to link it with 432Hz tuning.

Remember that there is an ethos here of harmonic rational relationships, from the Earth’s orbit to 136.1Hz. However let’s look at the relationship between 136.1Hz and the proposed 432Hz reference. Sure it’s very close to a 12-TET C# in 432Hz tuning, but this yields a wholly unwieldy 1.26018518518519 ratio (adjusting for octaves) a wasteland – or tenuous approximation – of a ‘pure’ frequency ratio. If we accept that sort of level of harmonic elegance then there are countless other reference points we might choose, and many far better. So 432Hz is linked is 136.1Hz only via the standardized ‘western’ 12-TET system, hardly a claim to an ancient harmonic truth. It’s a henna tattoo of a claim to authenticity. Sure one could use a 432hz reference and 12-TET to ‘find’ a close approximation to 136.1Hz on a C# but then if you wanted to make pure intervals from that fundamental, 432Hz would not be one of those ‘pure’ related intervals. In fact the equal-tempered major 3rd (the interval found between A and C#) is a fundamental problem with 12-TET and why musicians and theorists have for years worked on ways to make 3rds sonorous with a limited set of pitches. As it stands the 12-TET major 3rd is way sharper than the pure third (5/4) which is implemented in musical performance where pitches can be freely altered (Indian classical music, string ensemble, vocal choirs etc.) and flatter than the ‘Pythagorean third’ of 81/64. In short 432Hz and 136.1Hz may be crowbarred together, but it’s a significantly nonharmonic crowbar.

If we’re after sympathetic and harmonic resonance with 136.1Hz (and thus the Earth’s orbit) we could use that (or its octave equivalents) as a tuning reference (as the Sitar does) or if we want something closer to conventional concert A: 408.3Hz (3/1), 435.52Hz (16/5) 453.66..Hz (10/3) are all more elegant – and consistent with claims to harmonic resonance – than 432Hz.  Not that I am advocating any of these, there is as yet no evidence that 136.1Hz is experientially special, nor has it lived up to any of its claims such as healing, being immune to mechanical noise or the like.

But if you enjoy the idea of being in harmonic resonance with the Earth’s orbit (or any Jupiter’s or Io’s or the 49 Bus schedule for that matter) you should at least be consistent in your aims.

5) Instruments and voices sound better when tuned down to 432Hz

This is the only argument which has real world relevance, though still unconvincing. Acoustic instruments are physical objects, and the tension imparted on their structure has a contribution to the nature of their sound – their timbre – so changing the reference frequency will alter somewhat the timbre of the instrument. The timbre will be different, not necessarily better. For example some guitarists like to ‘downtune’ their instruments for the sound it produces and the looser string feel it provides, but there is no reason for this adjustment to be exactly down to (the impossible to achieve with complete accuracy) 32-odd cent differential. Many adjustments are used, and tuning the guitar all the way down to the next chromatic note in 440Hz tuning was favoured by the not un-cosmic Jimi Hendrix. Note that instruments are designed to sound as good as possible, but they are always adjusted to suit the desires of a player or particular style, and there’s no good argument the changes made by this specific change are in any way better than any other tweak, and there’s no reason to think that – given a 432Hz reference – musicians would suddenly stop making tuning adjustments. Some singers prefer lower tuning systems for the relative ease of vocal production, particularly true when repertoire written for lower systems is tuned to modern concert pitch. However in most circumstances transposing music to lower keys (within a 440Hz reference system) can solve any performance range issues, and still there is no significant relevance or benefit to tuning down by those particular 32ish fractions of a semitone

Now adjusting the tuning by this small amount to synthesizers will create no significant timbral change, and adjusting your music library will in no way make the music more ‘natural’: the timbres will not change, save for perhaps some wholly unnatural artefacts of the adjustment process, and any internal tuning system relationships (for better or worse) will remain identical. If you want to do that to all your music, go for it (you could even play your 33 1/3 LP records at about 32.73 rpm) but rest assured you’re not adjusting the music in any meaningfully significant manner, and any perceived improvements are the results of placebo listening. We humans are hugely suggestible and music is an ideal environment to let those biases run riot.

6) It was an ancient practice from a wiser time

The Earth frequency has been discussed, but there is argument from antiquity based on the claim that 432Hz was previously universally employed Western standard, and presumably superseded by an unwise or malicious musical domination. So here’s a very concise (and far more accurate) history: Prior to the 19th Century, people made music with no specific – or highly localized – reference pitches. These varied wildly and frequencies given for concert A, as low as around 380Hz and way higher than today’s 440Hz. So much for ancient universal truth. In fact there was a general rise in pitch forged by instrument builder’s arms race – and perhaps deteriorating (and thus shortening) church organ pipes. But as the demand for a shared music practice grew – and singers complained of this pitch inflation – various attempts at standardization arose. Dozens were proposed in the 19th and 20th centuries (which varied with region, repertoire and even temperature measurement). Among these were the mid 19th century ‘Verdi tuning’ of 432Hz (suggested by the composer as a flattening of the prevailing 435Hz French system), and the fairly close to 432Hz ‘scientific tuning’ of C=256Hz (A≈430.54Hz) with its ‘scientific’ basis of C=2^8Hz but these were not particularly favoured among several others. 440Hz itself was formally proposed as early as 1859, so much for the embarrassing Hitler associations (presumably based around a 1939 tuning conference, for which we have no evidence of his association). Even if he happened to like it (for which let’s repeat there is no evidence) there is as much an argument for associating vegetarianism with the Nazis than tuning forks. 440Hz still took till the 1970s for general acceptance, but is still no means universal: Boston Symphony Orchestra use 442Hz, many German and Austrian symphonies use 443Hz and when playing Baroque music, and many ensembles favour the superlow 415Hz. Even rock and pop music – for technological (intended or otherwise) or ad hoc reasons – vary from 440Hz, although less so these days particular with the use of the music technology’s default of 440Hz. However any decent sequencing software and electronic instrument include the ability to deviate from 440Hz (including 432Hz), so those Illuminati need to work harder.

Figure 2: Illuminati Fail. I can set my tuning on NWO’s Logic Pro X to 432Hz, and also dial in any temperament. No-one stopped me.

In short as regards reference frequency, there was never an ancient truth, just a glorious diversity (including useful standardisations) which still continues today.

7) The argument from pretty patterns

Many 432Hz proponents claim that cymatics – the vibration patterns of a liquid (usually water) driven by a frequency – lends support to 432Hz superiority over 440Hz (and other) tuning frequencies. Stunning videos of the glaring differences, and the aesthetically more pleasing resulting patterns which 432Hz creates. Here’s the fatal flaw. The patterns produced are dependent on the driving frequency AND the geometry of the container. So some containers will have prettier patterns with a 432Hz rather than a 440Hz, and others might have quite the opposite. The differing wavelengths of 432Hz, 440Hz (and any other frequency) will behave differently in various containers. No 432Hz proponent shows a container that responds well to a 440Hz (although they exist just as readily as ‘432Hz-tuned’ bowls) because that doesn’t fit the a priori conclusion. It’s like arguing that a particular shoe size is perfect by showing a video of  one individual happily walking with one pair, and struggling to put on a different size. Shoes for feet. Horses for courses. Bowls for frequencies.

### The Accuracy Issue

There remains a significant pale proboscidea in the room of ‘harmonic’ arguments: There must be some element of acceptable deviation from the harmonic truths, otherwise any microtonal deviation from the pure ratios by unintended human inaccuracy, or intended expression, or even the doppler shift from a slight turn of the listeners head – or temperature fluctuation – will deviate a tuning from harmonic perfection (if ever attainable) of say 3/2 to some crazily irrational proportion ever so slightly higher or lower. I’m a strong advocate of microtonal considerations in music, but I must admit that with a trained musician’s limitation (even in ideal listening conditions) of not much less that 1 cent, along with the laws of physics, means we must take a moderated view to the field. And if one has to make a host of arm-waving concessions to harmonic music, it further undermines any fundamentalist argumentation for the superiority of an arbitrary tuning reference.

## In Sum

There’s no particular detriment to moving all music to 432Hz (or any other) tuning, but it’s also entirely possible to engage with music with a variety of tuning reference points, in fact a quick analysis of my music library finds a host of non-440Hz music including pop (Michael Jackson, Beatles), Classical, Renaissance, Jazz, Folk etc. Why advocate the arbitrary homogeneity of 432Hz  when one could have a broad and colourful diversity? Although it’s probable that some advocates are simply tuning all their music down 32-odd cents without checking whether all of it is actually in 440Hz first, making the farce all the more farcical. Most of the claims of the 432-truthers are dubious or fallacious particularly in reference to any superiority or a Musical Truth. If you want to retune your music libraries and instruments go for it, although time is probably better spent investigating any of the many other more pertinent musical dimensions (including – most relevantly – tuning systems).

Musical experience is subjective so if one enjoys 432Hz for whatever reason (even through the placebo effect), that’s fine. But to make real world claims about what is true and what ought to be in music should invite healthy critique and critical thinking and not gullible acceptance and credulous resharing. Speaking of which, if there is a demand, there are a series of controlled listening tests that I am happy to build to investigating any claims of 432Hz superiority while minimizing cognitive biases.

It’s refreshing to see any sort of popular foray into musical analysis, rather than the usual interest in the particular gyratory behaviour of some artist, or the tortured journalism of some sub-sub-sub-genre as a ‘stunning’ revolution. Unfortunately, even with the most generous spirit there’s practically nothing of substance to be found in the 432-truther movement. It seems the only thing discovered is yet another branch of musical pseudoscience, alongside extortionate speaker cables, crystals on speakers and the brown note.

## Bonus

As a reward for making it through this post, I offer you some alternatives to 440Hz and 432Hz, feel free to tune your music to any of them (and select whether A, C, C# or whatever note name you invent is the most cosmic) and then start a Facebook group (alongside any manufactured history) to advocate its use.

Pi-tuning 402.1Hz

7 octaves up from Pi, this will add a very rounded and cyclical feel to your music, rather than the square 432Hz and irregular 440Hz

Phi-tuning 414.2Hz

8 octaves above Phi, this is a very balanced tuning, unlike the very disjointed 432Hz and 440Hz.

Selfie Tuning

Perhaps the most personal tuning, simply take the reciprocal of your age in seconds, and multiply it up to a sensible pitch for tuning. For example on your 21st birthday, tune to 414Hz. This results in far more subjective, present music than the materialism of 432Hz and 440Hz. Warning, this will rise, and be prepared to sing like Mariah Carey in your eighties.

Olympic Tuning 272.2Hz

272.2Hz is based on the frequency of the Olympic Games, this produces a more triumphant tuning than 432Hz, with a greater sense of unity than the Earth Frequency.

Galactic Tuning 254.2Hz

65 octaves above the age of the Milky Way, 254.2Hz is way more cosmic than the provincial 432Hz.

Universal Tuning 338.1Hz

67 octaves above the age of the Universe, 338.1Hz is the best tuning reference you can use, and if you don’t agree you must be some kind of Nazi.

I’ll Be Back Tuning 332.7Hz

At 332.7Hz this is 33 octaves above the frequency of Arnold Schwarzenegger movies/per year over the last quarter century. More powerful that 432Hz, and works well with accents.

Bloody Cat Frequency 436.9Hz

This is the most irritating of tuning references: 436.9Hz, a mere 21 octaves above the frequency of my cat waking everyone up in the middle night just to tell us that it’s still raining or that the carpet is in the same position on the floor.

Benade, Arthur H (1976). Fundamentals of Musical Acoustics. New York: Oxford University Press.

Danielou, Alain (1968). The Ragas of Northern Indian Music. Barrie & Rockliff, London.

Danielou, Alain (1995). Music and the Power of Sound: The Influence of Tuning and Interval on Consciousness. Inner Traditions; Rep Sub edition.

Duffin, R. (2008) How Equal Temperament Ruined Harmony: And Why You Should Care. Paperback edition. Norton.

Fabian D, Timmers R and Schubert E (eds) (2014) Expressiveness in Music Performance: Empirical Approaches Across Styles and Cultures (Oxford University Press)

Fonville, John (1991) “Ben Johnston’s Extended Just Intonation: A Guide for Interpreters”, p.121. Perspectives of New Music 29, no. 2 (Summer): 106–37.

Gann, K (2016) An Introduction to Historical Tunings. http://www.kylegann.com/histune.html

Hans, J. (2001) Cymatics: A Study of Wave Phenomena & Vibration (3rd ed.).

Raichel, D. (2006) The Science and Applications of Acoustics, second edition (Springer).

Skudrzyk, E. (1971) The Foundations of Acoustics: Basic Mathematics and Basic Acoustics (Springer)

Stephens, R. & Bate, A. (1966) Acoustics and Vibrational Physics (2nd ed.). London: Edward Arnold.

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## Maximally Even Library

UPDATE- Since the publication of Brad Osborn’s Kid Algebra (2014), I’m going to switch to his category of Euclidean rhythms (in their 4 types) to describe the patterns below. In summary, Euclidean rhythms (ER) are rhythms in which k onsets in n divisions are as similar as possible, which essentially means that they will only differ by at most one subdivision each. So in ER the groups are as similar as possible, but the term maximally even we will reserve for ER rhythms where the smaller note groups are as separated as much as possible. For example, (2,2,3,3) and (2,3,2,3) are both ER, but only the latter is maximally even.

This is a library of all the maximally even (including strictly even) rhythms for 2-7 rhythmic onsets within 6,8,12 and 16 beat cycles.
Maximal evenness (M.E.) describes a rhythm which is as evenly spread out as possible given both a number or events (rhythmic onsets), and a number of available slots (beats). Strict evenness (marked with a º) is a subset of M.E. and occurs when the hits are equally spaced. M.E. rhythms are intrinsic to much music making in a wide range of cultures from Sub-saharan Africa, South America to EDM and much in between.
The parenthesised number shows the number of displacements (or ‘rotations’) available for the rhythm in the beat-cycle, and allows for starting on rests. When the number of rotations equals the number of beats in the cycle this is marked with an * and represents maximally independence (MI – a common trait of African timelines and clave patterns). Note that 5,6 and 7 in 12 also represents maximally even pentatonic, hexatonic and heptatonic scale sets e.g. 3,3,2,2,2 represents all the modes of the major pentatonic as well as a 5 in 12 set of ME rhythms. As another example 2,2,1,2,2,1,2 (a rotation of 2,2,2,1,2,2,1) represents both the African standard time-line and the Mixolydian mode. Enjoy.

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## Tonal Harmony Flow Chart

I have been thinking about ‘nutshell’ images for a range of musical concepts. Here’s a work in progress on common progressions in a major key in 17-19th century tonal harmony. Yes caveat, WIP etc, but it seems quite useful. Comments and suggestions welcome! Minor key next, please don’t let me turn them into 3D models for mode mixtures and modulations.

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## The Amazing Metrobird

I have a little place in Greece, on a lesser known corner of the Peloponnese, on a little beach with a derelict and rarely visited acropolis from which the islands of Ψιλι, Πλατεια and (just about) Σπετσεσ are visible.

It’s a magical (and for me painfully nostalgic) place where even when we eventually installed a phone (1996),  modem (2006) and wi-fi (2013) seems eerily frozen (well baked) in time. This part of the world is home to some odd creatures: deafening cicada, scorpions, flying fish, swordfish and a plant with fruit that explode on the lightest touch.

One such unusual animal I have yet to (knowingly) see but I’ve been fasciated by its sound for years. It’s some kind of bird that emits a short tweet at intervals so regular that we use it as a metronome. (It sounds particularly good on beat 4 & in a bossa).

Here’s an unedited audio sample recorded on Tuesday, 7 July 2009 19:32

(non-flash) Metrobird

Notice how (separated by an unmeasured pause) there is a decent metronomic tempo established. Logic Pro X’s transient detector and beat mapping tools reveal that once a pulse is established it tends to stay within a couple of bpm. I’ve played with far worse time-keepers of the human species. Here are the numbers:

To get a feel for it, listen to the same unedited clip with a click track.

(non-flash) Metrobird with Click
Not bad at all. Here’s how it sounds (again completely unedited) in the context of a percussion groove.

Does anyone know what type of bird it is, an what evolutionary pressures gave it such tight timing?

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## The Maths Behind Music

Here’s a short educational video explaining some fundamental concepts of maths and tuning. Nice production by frequent collaborator Anna Tanczos of Sci-Comm Studios.

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## A dozen of one, or not.

It’s tempting to think that it’s only the domain of modernist composers, theorists and ethnomusicologists to talk of anything but 12 notes in an octave. After all if it was good enough for Mozart and Beethoven it should be good for everyone, right? Well, as it happens, Mozart and Beethoven understood F# and G-flat as different notes. A manuscript survives for example of Mozart’s teaching notes to his English student Thomas Attwood showing the difference between a major semitone (e.g. E to F) and a minor semitone (Fb to F). Almost universally considered as identical today, in his they were pitched slightly differently.

Very few musicians are aware that even into the 19th century fingerboard diagrams and scale exercises existed with two types of accidental (e.g. g# as distinct from a-flat) as well as keyboards with split keys so that the player could choose between accidental types.

It’s remarkable how efficiently this has been filtered out of the system so that even professional classical musicians and teachers – let alone students – are unaware of our microtonal recent history.

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## A Glimpse At the Modal Universe

A diagram demonstrating a section of the huge modal universe. You may see how mirroring modes (turning them upside down) can organize them into levels of brightness. It can also identify those modes that are identical in mirror form. These include Dorian (used in a thousand tunes from Scarborough Fair, Shine on You Crazy Diamond, Brick House to The Hitchhiker’s Guide To The Galaxy), Aeolian Dominant (Babooshka) and Double Harmonic (Miserlou from Pulp Fiction).

These are just 3 of the heptatonic even-tempered modes with mirror symmetry parents. There are many others, scales with 2-12 notes, as well as scales with ‘twin’ mode systems. Regardless this technique can be applied widely and is a rich resource for composers and improvisers alike.

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## An Introduction to Popular Music Harmony

[scribd id=52279893 key=key-3u3x1xa6jfgma3er62r mode=book]

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## Melody and Harmony

[scribd id=52240551 key=key-1k1ghlqrhoxb9xs9w7uj mode=book]

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## The Notion of Convexity in Music – Honingh & Bod

Here’s the interesting article examining inherent ster-convex properties of musical scales. Quite intriguing, will give it its due attention when I can.

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## Universal property of music discovered

An overblown title, but interesting nonetheless. Will comment when I have time to explore…

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## Trebuchet – The Interview. Geo and Milt geek out.

A transcript of Milton Mermikides’ interview with George Hrab on his latest album ‘Trebuchet’
Painstakingly transcribed by Kylie Sturgess.
The hours of audio are here: http://www.geologicpodcast.com/2010/07****Warning geeky musical content ****

[scribd id=51447705 key=key-554x9iw362a5ex8r74p mode=list]

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