4.1 The Normal Distribution
The fact that, when corrected for pitch-class usage, Beethoven's key usage over his piano sonatas tends towards a normal distribution raises interesting questions about why this might be. Although the dataset collected has limitations about the conclusions which can be drawn from it, two possible reasons for the normal distribution that seem suitable to explore using the current data are: biomechanics and aesthetics.
Other possible reasons are dealt with in a speculative way in An Instrument Is An Egg elsewhere in this journal.
4.1.1 Biomechanical Aspects
One possible reason for the spread of notes approximating a normal distribution may be that playing the keys at the extremes of the piano involves the exertion of more energy than those in the centre, a natural drive for conservation of energy leading them to be played less. This may imply that the smooth s-curve we see may be a function of the similar s-curves found in muscle contraction.
One way of testing whether the spread of notes is due to biomechanical factors can be to measure the standard deviation of notes found on a keyboard with a particular compass and compare it to other keyboards that have larger or smaller compasses. The standard deviation is the variability of the data of a frequency distribution from the mean. The higher the number, the more variable the data, meaning the shallower the normal curve.
If the reason for the normal distribution of key usage is biomechanical, then it is caused by purely physiological factors unrelated to the piano, such as arm span, muscle strength and stamina, that would stay constant irrespective of the size of keyboard the player was sat in front of. This would mean that the standard deviation would stay the same, irrespective of keyboard size, due to the fact that the spread of key usage is caused by the body, rather than the keyboard.
In order to investigate this, the sonatas were grouped according to the size of keyboard they were written for. Only sonatas which appeared to have unambiguous compasses were used. The standard deviation of key usage was then calculated for each of these four groups, shown below:
The size of the keyboard was then plotted against the standard deviation to see if any correlation could be seen between these two factors, using samples covering the five octave range of the smallest piano. If one was expecting biomechanics to be able to explain the normal distribution of notes, one would expect that the standard deviations would exhibit neither a negative nor positive correlation, but remain unchanged.
As can be seen above, there is a positive correlation between the size of the keyboard and the standard deviation, implying that biomechanical reasons are not behind the normal distribution of key usage in Beethoven's sonatas.
A positive correlation exists between standard deviation and keyboard size, with the standard deviation increasing in relation to the size of keyboard. This could be taken as evidence that shows that the size of keyboard had an effect on the use of piano keys in Beethoven's piano sonatas. This is particularly important as standard deviation measures the statistical spread of notes irrespectiveof sample size – thus implying that Beethoven contracted the statistical distribution of his notes when working at smaller keyboards. This evidence goes against the idea that Beethoven somehow found himself compositionally constrained by the size of the smaller pianos, his compositional inspiration constantly stopping abruptly at the brick walls at each end of the limited keyboard instrument. Instead, it implies that Beethoven, consciously or unconsciously, adapted to each keyboard, actually compressing the spread of notes on smaller instruments.
Perhaps the normal distribution could be the result of aesthetic choices made by Beethoven, either consciously or unconsciously. If so, one would expect that his changing aesthetic throughout his life-time would impact upon this aspect of his key usage. The fact that the compass of pianos increases over the time period under examination, means that Beethoven's key usage may simply be a function of a developing aesthetic which moved in synchrony with the chronological increases in keyboard compass. In order to disprove this, one would need to compare the standard deviation of two sonatas, one of which was written for a smaller piano than the other, yet also composed at a later date.
This can be done by looking at two sets of sonatas:
Comparing the 31st sonata with the 29th and 28th, both of which are written for larger pianos and
Comparing the 27th sonata, with its smaller range to that of the 26th, with a larger compass and written earlier.
In both cases, the data shows that these sonatas conform to the pattern of exhibiting a positive correlation between their standard deviation and keyboard size, irrespective of their date of composition and any chronological developments in Beethoven's compositional style.
However, this does not rule out the fact that the normal distribution may be a function of Beethoven's own, idiosyncratic compositional practices, and that the contraction of this distribution in relation to the compass of the keyboards he was using is the result of little more than his own compositional quirks. In order to determine if this is so, Beethoven's key usage must be compared to that of other composers, over a larger time period than just that of Beethoven's lifetime, so that the idea that normal distribution may also be the function of a specific style linked to a chronological period can be dismissed.
An extensive review of other composers' key usage is greatly hampered both by the lack of available MIDI files for composers of the 20th Century – a fact that also makes it difficult to refute the idea that the normal distribution around the centre of the piano may be due to the timbre of piano notes in this region which may allow for the successful functioning of tonality (an analysis of atonal and total serial piano music may be able to refute this). Also, it appears that it is even more difficult to establish the pianos used by other composers writing in the period 1700-1900 than it is to establish Beethoven's, due to the rapid development of keyboard instruments in this period. This means that any concrete conclusions that could be drawn about the effect of keyboard compass on note distribution detached from the composer, chronological period or style are difficult to do, and the research that would be needed to concretely state this is well beyond the scope of this paper.
However, some preliminary research and conclusions can be proposed, with the hope that this will stimulate further research. Below, the key usage over selected keyboard sonatas by Domenico Scarlatti, Franz Joseph Haydn, Muzio Clementi, Wolfgang Amadeus Mozart, and Franz Schubert is presented, with key usage re-adjusted for harmonic key usage:
Domenico Scarlatti (1685-1757) – all 555 keyboard sonatas
Franz Joseph Haydn (1732-1809) – all 62 sonatas
Muzio Clementi (1752-1832) – sonatas 7, 12, 25, 26, 34, 39, 40, 50 No. 3,
Wolfgang Amadeus Mozart (1756-1791) – sonatas 279, 280, 281, 283, 284, 309, 310, 311, 330, 331, 332, 333, 457, 545, 547a, 570, 576, 593-494.
Franz Schubert (1797-1828) – sonatas 42, 53, 120, 122, 143, 147, 164, D459, D537, D568, D664, D784, D840, D894, D958, D959, D960
To open any of the raw data tables in a new tab, please click on any of the links below
As can be seen, key usage by all but one of the composers surveyed approximates the normal curve, showing that this distribution of key usage occurs across five composers whose lifetimes stretch over 147 years. The only exception seems to be Haydn, whose statistically anomalous use of the bass register flattens out his normal curve, although it is not far from being a normal distribution. From this data, it appears that the normal distribution of key usage occurs across composers and time, implying that it is not the result of style, or an individual composer's idiosyncratic composition process. Instead, this data seems to point to the idea that either pedagogy or the piano itself leads composers to move towards this particular way of interacting with the instrument.
Case Study - Domenico Scarlatti:
A selection of 74 of Scarlatti's early keyboard sonatas that seem to be written for a four-octave keyboard ranging from C2 to C6 were analyzed: k001, k002, k003, k004, k005, k008, k010, k011, k012, k013, k014, k017, k018, K019, k021, k022, k023, k025, k028, k029, k030, k031, k032, k034, k035, k036, k037, k038, k039, k040, k041, k042, k045, k047, k048, k050, k051, k052, k053, k058, k059, k060, k061, k063, k064, k065, k066, k067, k068, k069, k071, k072, k073, k074, k076, k077, k078, k079, k080, k081, k082, K083, k084, k085, k086, k087, k089, k090, k091, k092, k093, k095, k098, k100,
This four octave keyboard is one octave smaller than the smallest one used by Beethoven in his sonatas. The best fit line calculated for the relationship between standard distribution and keyboard compass is:
f(x) = 0.098945626x + 6.6758677127
This means that with a keyboard compass of four octaves (49 notes), we would expect a standard deviation of 11.5242033867 based on the data extracted from Beethoven's sonatas.
Calculating the standard deviation for these Scarlatti sonatas gave a mean of 67.0009471577 and a standard deviation of 6.0528163249. This Standard Deviation is clearly much smaller than that which was expected.
To explore why this might be is outside of the scope of this paper and would require extensive research and analysis.
In Beethoven's work, his usage of the keyboard tends towards a normal distribution. It has been shown that the distribution of notes in his work contracts and expands according to the size of keyboard, showing that biomechanical reasons are not the cause for the normal distribution of key usage. Similarly, the normal distribution appears to occur in the work of other composer's keyboard sonatas from different countries and times, implying that this is not the function of general stylistic, or individual aesthetic decisions. It was not possible with the current dataset to ascertain whether the gravitation towards the centre of the piano may be to do with timbral characteristics of the piano within this region which allow a clearer functioning of tonal harmony – this would be an ideal area for further research but would involve a far more extensive MIDI collection involving recent atonal and serial piano works from the 20th Century.
Given the fact that all of the composers analyzed played the instrument and wrote their piano works seated at it, this research gives reason to re-question the role that an instrument has in shaping the performers interaction with it, which extends beyond the use of the piano towards all instruments. Composers and improvisers who interact with the piano by writing compositions whilst seated at it, or improvising at it, may unwittingly be allowing their works to drift into the most banal of all statistical distributions: a normal one.