TOAST THEORY

(see also All The Music)

 

We have proposed the possibility that music production is finite rather than infinite. Even if the latter is the case, we can confine ourselves to a generalisation that places limits on how we engage with music. Tunes can be long, possibly infinitely long, but none of us can listen to the whole of that composition, if the universe ends, the song ends. ‘The Song of a Thousand Songs’ is, sadly or fortunately, never going to be heard by any one individual in its entirety (if it has an entirety). But to concern ourselves with our everyday humming of a tune, particularly one that would meet the criteria of Winnie the Pooh to whom a good tune is one “such as might be hummed successfully to others”, we can, for exploratory purposes, reduce the potentially infinite down to a short ditty, something manageable so that we can encompass all short other ditties before moving on to larger ones, and then to musical things beyond sequences of pitches. So, as we have already focused on a particular pithy phrase in Toast theory in a previous section on removing non-sonic signifiers from the semiotics of endings, we will return to the opening phrase of Beethoven’s 5th symphony as the sequence upon which we will draw our parameters for our attempt at producing ‘everything’. The tune is instantly recognisable, what’s more it is short, and it can be placed within the parameters of the tonal and rhythmic infrastructures from the symphony from which it is drawn. Evidently this is not ‘everything’, but it is everything within the defined parameters, and hence a start in moving towards the wider everything.

 

We set our task on working out how to produce all possible variations of sequences within the parameters derived from the opening refrain from Beethoven’s 5th symphony. One of these mathematically generated variations, set in a logical order from first to last, will be the tune that Beethoven ‘wrote’, that is, the opening two-bar melody appears at a certain point within the millions of permutations and this point is directed by how we proceeded with regards to the logic of the sequence. We put Beethoven ‘wrote’ it in quotations to emphasise our position that if all such variations are already possible, rather than creating them the composer discovers them and then places their flag on that territory. And this takes us back to the central tenet of this book, it maybe that one explorer (composer) has the dominant claim to discovering, and therefore claiming ownership of the territory but on the whole lots of explorer-composers seem to be trying to make their flag fly highest and strongest in the same territories or in overlapping territories that have no defined border except for those that have been drawn in the sand by those with the flags. Returning to our immanent project, our task is to ‘discover’ all the variations within our parameters (within our lines drawn in the sand) and to locate Beethoven’s ‘da da da dah’ tune within them. Consequentially, we will discover all other tunes within those parameters, some of which will already have flags on them. How we will deal with that ‘Monopoly board’ of minding which territory one lands on we are not sure but will speculate on the ideas. Similarly, the temptation to plant a flag on any territory so far unclaimed is tempting but that is not our intent. A life spent battling for ownership in the law courts does not sound very inviting. We will benignly ‘categorise’ the tunes, and, as we are working with computer programmer Neil Ward and have developed an interactive application for this, Beethoven’s tune will be labelled on the BWW (Brady-Ward-Wilsmore) system, like the Bach-Werke-Verzeichnis BWV system of categorising Bach’s works, or the manner in which stars, too numerous to name, end up with impersonal numbers on the Hipparcos star catalogue where Betelgeuse (Alpha Orionis) is merely HIP27989. Each tune in BWW has its own address or set of coordinates.

 

(Here's a screen shot of its original version)                               

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[How it works - The 'tune' is inputted via the data fields which gives the BWW number. We're working on showing the inputted tune within its local matrix (see image below), and also into making the input more user friendly, ie. writing the tune on staves]

 

Method for calculations

The tricky part to actually writing ‘every tune’ is not calculating how many possible tunes there are but in finding a sensible ordering system that will allow us to produce them. Calculating the number of possible permutations can be straightforward when given precise parameters but our task involves the application of the mathematics in such a way as to produce a logical system that can be translated into the tunes themselves; or at least as notes on staves that can then be sounded, the point being that they will be understood as tunes in the manner in which we would normally do so. Hence there is, in one way, no difference between the numbers in the boxes on the vast spreadsheets (that we ultimately produced through applying the maths), to the notes on a musical stave; they are both merely representations of the sound as opposed to being the sounds themselves. It could be said that we are just replacing one notation system with another and in one sense this is true, but the mathematical version we present places emphasis on categorisation, position (a location address) and logical progression, with the order of progression fixed by us in this system from start to finish (perhaps that makes it a composition after all?). As we wish to engage in the everyday world from an everyday perspective, we have at times, for exemplification here, translated our numbers into more familiar musical notation: notes on staves. Perhaps ironically, in order for these ‘notes on staves’ to sound as tunes, they will need to go through a process of conversion to binary so that they can be played from a computer. Unless of course we can persuade ‘live’ musicians to play them. Were they to perform the sequence from start to finish it would last about five years depending on what tempo is set. Our first task is to set out the aim, the parameters and the working methodology. 

 

Aim. Our aim is to produce, in a systematic manner, every possible tune that can be made within limited parameters derived from the opening two bar phrase of Beethoven’s 5th Symphony, and then to locate Beethoven’s melody within that system. Our aim is not merely to count the number of possible tunes within the parameters but to produce a system that can place them in a logical sequence, and that each tune can be given a label within that system.

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As the maths as set out by Phil Brady is complex, we have not included it all here (it is set out in detail in chapter 18 in the book), but rather have chosen to represent the findings here in the following diagram.

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Where is Beethoven’s 5th?

All ‘tunes’ with the parameters outlined above can be classified in a 3-dimensional array (two dimensions [s and l] for the rhythm and a third for the melody). Each tune will be referenced in the Brady-Ward-Wilsmore system as follows:

BWW-[n,s,l](p)-{c or C}number

BWW - prefix to denote part of the Brady-Ward-Wilsmore system.

[n,s,l] - n represents the total number of quavers to fill (this is included to allow the system to be extended beyond our 8 quaver starting point); s the actual number of notes sounded; l the total duration (in quavers) of notes sounded.

(p) - the position in the cell of all rhythms with s notes sounding for l quavers.

{C or c} - upper case ‘C’ represents major; lower case ‘c’ denoting ‘minor’ version.  

number - a number in base 10 which, when converted to an s digit number in octal (base 8), would represent the pitch of each note by a single digit (0=C’, 1=C, 2=D etc.).

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For our purposes the opening two bars of Beethoven’s 5th has overall length 8 (n = 8), 4 notes sounding (s = 4) and 7 quavers of sound (l = 7). It can therefore be found in row 4 and column 7 of the rhythm table; the rhythm 01114 is the first entry in this cell. So the rhythm ([n,s,l](p)) would be [8,4,7](1). The melody (‘minor’ version) is represented by the 4-digit octal number 5553, which when converted to base 10 is 2923 (5×83+5×82+5×81+3×80). So, Beethoven’s tune in our Brady-Ward-Wilsmore system is:

BWW-[8,4,7](1)-c2923

Its position relative to tunes surrounding it is shown in the diagram (below). Since the rhythms sit in an array, we should specify a playing order. The arrow indicates the progression route, for example for a linear ‘performance’, in this case showing progression through the [8,4,7] rhythms. All possible melodies for rhythm [8,4,7](1) are played from c0 to c4095 before moving on to all melodies for [8,4,7](2).

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Fig: "Beethoven was here!"​​

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