Google Ngram Viewer shows the frequency of words or phrases in books. This the chart for the words aesthetic and complexity between the years 1700 and 2000:Complexity overtakes aesthetic from 1962, and the frequency of both dip around the time of the two world wars. There’s a peak in aesthetic in the late 1920s and early 1930s, but it doesn’t really get going until the 1850s. The current meaning of aesthetic was developed by Alexander Baumgarten, who used the term to mean “things perceived”, as opposed to “things known”. Baumgarten’s Aesthetica (1750) was written in German. Here’s another chart for the equivalent words in German books over the same period, showing a quite different pattern: The biggest peak for ästhetisch is around the time Kant published the Critique of Judgement (1790). The crossover in frequency to Komplexität is 1965, during a steep rise in usage up to a present-day level that’s very similar to its English equivalent at around 0.0025%.

I’m very happy to announce the publication of a conversation between myself and Roc Jiménez de Cisneros. Our discussion starts with Hanne Darboven’s Opus 17a, a generative piece of music that I translated for EVOL. I’ve written about the technical aspects of that translation process here. In this booklet we talk about the perceptual and philosophical aspects of translating music. EVOL have made a few recorded versions of this piece – Opus17aSlimeVariations, in addition to live performances and art installations. The publication is available via ALKU, printed in neon red ink on good quality paper: http://vivapunani.org/pmwiki/pmwiki.php/Main/HardlyItself

Complexification is a collaborative project with Sun Hammer (Jay Bodley of Portland, Oregon). The project is based on a set of rules:

0. Each make a short, simple piece of music.

1. Swap a copy of the piece with the other person.

2. Modify the copy to make a new piece of music that is more complex (the given piece must be used, but it can be transformed in any way, and new sounds may be added).

3. IF the result is more complex (as agreed by both participants), GOTO 1, otherwise HALT.

The aim of this project is to explore musical complexity through a creative approach rather than an analytical one. The focus is on complexity as a variable musical parameter, not just highly complex music. The process halted on the 10th cycle when it became too difficult to proceed, so there are two parallel threads with 10 tracks each.

Giving ourselves the task of complexification meant having first to agree on what we meant by complexity in music. We didn’t want to be tied down to a specific definition of complexity, but there had to be agreement about how to judge whether one piece of music is more complex than another. We understood complexity as being a function of the quantity, variety and order of musical elements or patterns. So we judged complexity in terms of how hard it would be to describe all those elements and patterns. Comparing two pieces of music, the one that takes a longer and more detailed description is the more complex. Although we understood that the focus was on the complexity of the music itself, as it is heard, and not the processes that went into making it, we found that it’s easy to get lost in the process, and that complicated processes don’t necessarily lead to complex sounds. Computer music pioneer Jean-Claude Risset warned of this in a lecture from 2004, ‘The Perception of Musical Sound’ [PDF]:

It is often a delusion to rely on a physical description for predicting the appearance or the effect of a visual or an auditory scene realized with elaborate technical processes: one should be aware of the complexity of the relation between the physics of the signal and the way it is perceived.

On the difference between subjective and objective complexity, Risset also says:

There is the saying “Beauty is in the eye of the beholder”. What matters in music is not intrinsic complexity, but the complexity we perceive, complexity in us. Some people claim that acoustic sounds will always be more complex than computer-generated ones. This is not true of objective complexity: a signal made up of samples independently chosen at random has maximal complexity, but we cannot cope with that richness of information – white noise sounds undifferentiated.

We developed the practice of using rules and constraints through participation in the Disquiet Junto. Some of the audio processing techniques were developed in Junto projects, and the group’s tradition of documenting the creative process led us to do the same in this project. The technical details about the process are available in a PDF via Entr’acte, which also includes spectrogram images of the tracks and a couple of analytical charts.

An interactive header image on AFX’s http://user48736353001.com/ changes vertical and horizontal pixellation with mouse position. The background image rotates, whilst the pixels remain in the same orientation, producing a coarse-grained Moiré effect. At the highest resolution – with the mouse in one corner of the screen, the number of horizontal and vertical divisions is about equal, meaning that the pixels are about the same proportions as the screen. The rotating image looks like some kind of a ridged, serrated or coiled object. Here’s 10 snapshots taken at different times and mouse positions.

I’ve been investigating the dynamic range of some music, using this plugin for the foobar2000 music player: http://www.pleasurizemusic.com/de/free-downloads. This tool is useful for analysing my own music, for improving my mixes. I also tested the albums that I use for reference, and then a few more.

In general terms, dynamic range refers to the difference between the smallest and largest values of a quantity. In music, it means the difference between the quietest and loudest sounds. Audio compression effects operate on this dynamic range by reducing the difference; expanders do the opposite. Over-use of compression in digital audio has led to the ‘loudness war’. People who care about this stuff have developed various standards and measures of loudness and dynamic range, e.g. EBU, the K-System by Bob Katz, and many more. This particular plugin measures dynamic range (DR) by calculating the difference in decibels between peak and RMS loudness – the difference between the loudest sound in an audio file and its average loudness overall. In foobar2000, it writes this information to a text file (example below) and to audio file metadata, allowing you to sort music by dynamic range. This is for information only; it doesn’t change the sound at all, unlike ReplayGain.

ReplayGain is a system for dealing with variations in loudness in a music collection. It adjusts the playback volume of a track or album based on its RMS loudness. The RG value is the amount of adjustment in decibels relative to a reference loudness value. Whilst this doesn’t solve the problem of dynamic range, it does correct for the differences in perceived loudness. There is some correlation between RG and DR, though, because they’re both based on RMS loudness: tracks that are heavily compressed and very loud will have low RG and DR values, whereas tracks that are more dynamic will have higher values of each. Different kinds of music suffer from lack of dynamic range to different degrees, but in general a higher dynamic range is preferable. In general, DR 10 and above is OK, and less than 5 can be a problem.

So here’s a few selected results of scanning the dynamic range of around 400 albums. Most of them are in the mid range – about 300 of these are between DR 8 and 12. In the list I’ve put a few pairs of albums with the same DR but different styles: Talk Talk’s Laughing Stock – an album often cited as a good example of music with lots of space and dynamics – is equal to the classic rock of Back in Black at DR 13. Manitutshu by Mark Fell is equal DR to the sparse and quiet album Alina by Arvo Pärt. EVOL’s Hyperobject 2 has a surprisingly high DR of 17, given that it’s an hour-long relentless composition based on a single percussive sound (“flangey side stick“). But in that sense, it’s not too dissimilar to Drumming by Steve Reich, at DR 16. The simple drone music of ELEH has a very low DR, but the more complex combination of drones and textures by Hazard (BJ Nilsen) has a high DR. At the top of the list, SND’s delicate stdio album is just below a live recording of a symphony by Arvo Pärt. Symphony No. 4 is a single track that’s mostly very quiet except for a few louder sections, which is probably why it heads the list (high peak – low RMS = high DR). Ben Frost’s industrial music is perhaps over-compressed at DR 5, but Powell’s dance music sounds fine at DR 4. The lowest DR is heavily distorted music – Russell Haswell, Kevin Drumm, and music from Japan.

DR 16: Mark Fell – Manitutshu
DR 16: Steve Reich – Drumming DR 16: Arvo Pärt – Alina

DR 15: Fleetwood Mac – Rumours DR 15: Hazard – Wood

DR 14: Buckingham Nicks – Buckingham Nicks

DR 13: Talk Talk – Laughing Stock
DR 13: AC/DC – Back in Black

DR 12: ZZ Top – Tres Hombres

DR 11: AFX – Analord 01
DR 11: Autechre – Amber

DR 10: Black Sabbath – Master of Reality DR 10: Taylor Deupree – January

DR 9: Aphex Twin – Selected Ambient Works Volume II DR 9: NoMeansNo – Wrong

DR 8: Autechre – Exai

DR 7: Napalm Death – Scum

DR 6: ELEH – Radiant Intervals

DR 5: Ben Frost – By The Throat

DR 4: Powell – Club Music

DR 3: Boris – Akuma No Uta

DR 2: Kevin Drumm – Sheer Hellish Miasma DR 2: Russell Haswell & PAIN JERK – Electroacoustic Sludge Dither Transformation Smear Grind Decomposition nO!se File Exchange Mega Edit

DR 1: NHK – ‘Stomp_1′, from SND/NHK – Split

DR 0: Kevin Drumm – Purge

These DR values are calculated per album; it averages the dynamic range of the tracks. As a result, albums tend to be more centred in the list, whilst the more extreme DR values are likely to be single tracks or albums with fewer tracks. An album with more tracks has less chance of scoring very high or low because it would require every track to have a similarly extreme DR value. For example, the top and bottom albums have just one track each. But in contrast, SND’s stdio (DR 21) has 17 tracks, which makes it more of a rarity.

This isn’t a very scientific analysis, of course. This kind of measurement depends on how loudness is defined and which type of weighting is applied. Perceived loudness varies with frequency, and perception of frequency balance also varies with loudness (equal loudness contour). As albums get remixed and remastered, their dynamic range can change. File format can also affect DR measurements, and in this list is a mix of CD, MP3 and FLAC. If this analysis tells us anything, it’s that when it comes to sound quality, different levels of dynamic range are appropriate for different types of music. A wide dynamic range in music is a generally a good thing, yes, but having a low dynamic range doesn’t necessarily mean that it’s music done badly.

Here’s an online database of albums and DR values: http://dr.loudness-war.info/ It also has links to download tools for DR measurement.

The Online Encyclopedia of Integer Sequences (OEIS) is a useful resource for my Tintinnabuli Mathematica music project. For each sequence, the OEIS lists the first few terms and describes the algorithm that produces it. I use these algorithms to generate number sequences of various types (regular, fractal or chaotic), which I then translate into musical notes and output as MIDI files using a program written in Mathematica.

The OEIS was created by Neil J.A. Sloane, first in the form of a handbook, now a collaborative online project which currently lists over 200,000 sequences. Each entry is assigned a number – for example, A000001 is “number of groups of order n” and the sequence of prime numbers is A000040. Each sequence can also be seen as a graph and heard as a sonification. A thousand of these graphs have been put together as a video showing the first thousand terms of each sequence, with a sonification of Recaman’s sequence (A005132) as the soundtrack:

The sequences in the video were chosen because they were interesting or mathematically important in some way. This interestingness could be intrinsic (e.g., it relates to a known sequence, or it makes a nice pattern) or relative to the other sequences chosen (e.g., it is different to the others). It turns out that this interestingness also affects the OEIS itself as a whole. Using mathematical tools to investigate the OEIS reveals that this ostensibly objective and mathematical project is also shaped by cultural factors.

Philippe Guglielmetti (2009) investigated the frequency N(n) with which each integer n appears in the OEIS. He plotted this as a graph, showing on a logarithmic scale how many times each number up to 10,000 occurs amongst all the sequences in the OEIS. As we would expect, smaller numbers occur more frequently than larger numbers, creating a logarithmic downward curve.

Guglielmetti observed that the integers are divided pretty clearly into 2 groups, noting that there are fewer “interesting” numbers in the upper band, and more common “uninteresting” numbers below. The interesting numbers include the prime numbers (red in the image below), numbers of the form a^n (green), and highly composite numbers (yellow). There are relatively few “slightly interesting” numbers in the middle, and this area is known as “Sloane’s gap” after the creator of the OEIS.

A study by Gauvrit, Delahaye and Zenil (2011) investigated this, noting that “This gap is unexpected and requires an explanation.” The study takes an algorithmic information theory approach, which uses the frequency of occurrences N(n) of each integer n in the OEIS as a measure of a number’s algorithmic complexity. This approach accounts for the overall distribution of the numbers, but not the gap:

The theory of algorithmic information thus provides a good description of what is observable from the curve N(n). That justifies an a posteriori recourse to the theoretical concepts of algorithmic complexity in order to understand the form of the curve N(n). By contrast, nothing in the theory leads one to expect a gap like the one actually observed. (p.12)

Gauvrit et al. hypothesize that the gap is the product of the social aspect of the OEIS, caused by mathematicians’ greater interest in numbers of high and low complexity, and lesser interest in those of medium complexity.

This study also considers which is the smallest number that doesn’t appear in the OEIS – that is, the smallest integer n for which N(n) = 0, or the first number that is mathematically un-interesting. Paradoxically, this makes it quite interesting. Appropriately, the idea of the smallest uninteresting number was discussed in an episode of QI (broadcast November 2011), which pointed out that such a number is culturally, rather than mathematically, interesting. In August 2008, the smallest number not to appear in the OEIS was 8,795, and in February 2009 it was 11,630, changing as new sequences are added.

With a little input from my own work in programming tintinnabuli music, Arve Knudsen has created a real-time generative piece using the ChucK programming language. Although it’s based on a short and simple ascending sequence of notes, the algorithm generates enough variation in timbre and register to keep it interesting: