Beats to the Bar & Tempo

 A core important aspect to consider of the background to parquet deformation, borrowing from music terminology, is that of the beats of the bar and tempo, which are separate but related aspects. As far as I am aware, these issues have not been discussed as such, or at least in these terms. The beat refers to in effect stages, whilst tempo refers to the predetermined length, or number base. I discuss each in turn below.

I begin with an abstract study of both aspects, to establish the premise, without a parquet deformation. This is then followed by application, with a parquet deformation. Both beats to the bar and the tempo is investigated in a systematic way, in increments of 1- and 2-units. For demonstrative purposes, I then use the same parquet deformation throughout for the sake of consistency of argument, chosen for its aesthetics and ease of drawing.

(i) Beats to the Bar

By beats to the bar (or stage), I mean the repetitive nature of the unit cell. 1-beat would be 1, 2, 3..., whilst 2-beats would be 1, 1; 2, 2; 3, 3... etc., Fig. 1. Of course, even more beats to the bar are possible, although in practice serve little purpose beyond bloating the composition. On very rare occasions, 3-beats are required in order to retain a tiling condition.

Fig. 1. Beats to the Bar. 0, 1, 2 (Upper). 0, 0; 1, 1; 2, 2 (Lower).

(ii) Tempo

By tempo, I mean the length of the composition. The tempo can be of any length, although in practice it is generally favoured around a tempo of 10, for reasons as I detail in ‘Ideal Tempo’ below. Fig. 2 shows a tempo of 10 for both 1-beat (top) and 2-beats (bottom).

Fig. 2. Tempo. 1-Beat to the Bar (Upper), 2-Beats to the Bar (Lower).


I now turn to matters of application, with the same parquet deformation (and is used throughout the discourse for reasons of consistency), based on Fig. 1 and Fig. 2 above, with Fig. 3 and Fig. 4. There are obvious differences here, in two main ways: (i) the physical length of the composition, which doubles, and (ii) the condition of tiling, of which only 2-Beats, i.e. 0, 0; 1, 1; 2, 2 possesses this feature (in alternate columns).

Fig. 3. Parquet Deformation of 1-Beat to the Bar, 0-10.

Fig. 4. Parquet Deformation of 2-Beats to the Bar, 0-10.

There are two main observations:

(i) The physical length of the composition, which doubles.

(ii) 1-beat (Fig. 3), can be seen to lose the condition of tiling, whilst 2-beats (Fig. 4), possesses this feature, in alternate columns with transition zones (this is a necessity of the process, and is not a fault in any way; by its very nature, with successive changes, it is impossible for all tiles to tile).

Whether the loss on the condition of tiling is acceptable is a moot point. Strictly, in his two conditions [*], Huff rules this type out. However, many examples of this type can be seen in the Huff-student inspired work, and so presumably this must have met with his approval. Of the two types, I greatly favour 2-beats. To me, the condition of a tiling is a major factor. A 1-beat can thus be described as curtailed. However, this curtailed beat does have its place. It is perhaps better employed on two-dimensional studies, as this can in practice lead to practical difficulties when drawn out in full, as a 2-beat. Such compositions in terms of the number of tiles may otherwise involve hundreds of tiles, which can quickly become overwhelming. For initial study purposes in general, it is of course quicker to use the 1-beat before the drawing out in full of the favoured 2-beat.

Ideal Tempo

Having thus determined the ideal beat, or at least as favoured, the question of the ideal tempo now arises. But what is the ideal tempo? In short, this is to a degree subjective. As abstract concepts, this is neither too ‘quick’ nor too ‘slow’ but rather somewhere in between. A ‘quick’ example would be where the change is too drastic, of just a cycle of say, two or four stages. This would result in a too abrupt a metamorphosis and would be jarring to the eye. On the other hand, a ‘slow’ instance, of say 20 stages, where the change at each stage is not perceptible, would make for tedious viewing on grounds of excess length. A default tempo of 10 is judged just about ideal, neither too quick nor slow, although there is no firm single choice. Close tempos on either side of this, with 8 and 12, are still deemed permissible within an aesthetic range. However, beyond this point, with tempos of 6 and 14, these become a little too quick and slow respectively, and furthermore, with ever-declining (4 and 2), and increasing instances (16+) even more so, are thus deemed unacceptable. Therefore, the default beat is 2 (1, 1; 2, 2…) with a tempo of 10 (0-10). Many instances (80+) of these favoured types can be seen on the page 'David Bailey'. The above, discussed in abstract terms, is show in application below, for each of 1-beat and 2-beats, with tempos of 0-2, 0-4, 0-6, 0-8, 0-10, 0-12 and 0-20. Note that I show a curtailed range for reasons of expediency, where I show only even number instances (as well as omitting 0-14, 0-16 and 0-18). This is judged sufficient for assessment purposes. Each diagram is then briefly discussed. Further, since drawing large number tempos is relatively time consuming (and arguably is excessive to illustrate the point), rather than show the entire range from 0-20, upon the drawing of 0-12, I then omit 0-14 to 0-18, with 0-20 chosen as the 'large number' exemplar.


Fig. 5. 1-Beat. 0-2, 0-4, 0-6, 0-8, 0-10, 0-12, 0-20 (a-g)


Fig. 6. 2-Beats. 0-2, 0-4, 0-6, 0-8. 0-10, 0-12, 0-20 (a-g) (N.B. 0-20 is curtailed at halfway point for reasons of presentation)


So what can be drawn from this? A lot is subjective, with no clear dividing line between what is ideal, acceptable or a failure. Regarding transitions between the tiles, these should be perceptible, but not to an abrupt or imperceptible degree. Of course, perhaps some people may disagree with me here, especially so of the imperceptible degree types, as shown by the ‘lengthy’ 0-20 here. Undoubtedly, this is a smooth transition, and is not without merit. But a line has to be drawn somewhere. In short, I consider, whatever the beat, 0-2 and 0-4 as a failure; the transitions are far too abrupt. 0-6 is just about acceptable. 0-8 and 0-12 are judged acceptable, although a little quick and slow respectively. 0-10 is judged as the ideal compromise between too quick and too slow a transition. However, even at a slow 0-20 I do not greatly object to it. However, of larger sizes I do. Indeed, I have seen implied tempos of 0-100 on others’ work. Here the change between successive tiles is imperceptible. I consider that the change between the tiles should be noticeable.

Therefore, the default beat is 2-beats, with a tempo of 10.

Page Created 4 August 2021, of essentially a redrawing (and expansion) of a 25 February 2021 page now deleted, being poor in presentation of diagrams in many ways, of a murky grey background, and unnecessary large scale in general (of my early days of drawing in Rhino, hence the shortcomings). Last updated 10 August 2021.