insightPeter Bradley-Fulgoni‘Pythagoras showed that
there was a ﬂaw innature’s tuning’

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The architecture of theForty-Eight
The Forty-Eight, the colloquial
name for J. S. Bach’s Das
Wohltemperierte Klavier (the welltempered clavier) consists of 48
pairs of pieces, each pair made up
of a prelude followed by a fugue.
They are divided into two books.
The 24 in each book follow the 12
notes, the semi-tones, of thechromatic scale (C, C sharp, D, Dsharp, E, F, F sharp, G, G sharp, A, Asharp, B; then you get back to Cagain) and for each note you have amajor and a minor key. So, forinstance, in Book 1 of the FortyEight, the first pair is in C major,
the next in C minor and so on. Then
you have a second book of the same.
These works, written in the 1720s,
were a manifesto for equal
temperament, or adjustments in
keyboard tuning. Before Bach’stime, temperament was based on
the Pythagorean proportions ofharmonies, in turn based on
observations in nature. In
Experiencing Architecture, Rasmussen
tells the story of Pythagoras hearing
pleasing harmonies as he passed a
blacksmith and returning toinvestigate. He discovered that thelengths of the three hammerheads
were related to each other in theratio of 6:4:3. The longest produced
the keynote; the pitch of theshorter was a fifth above it and thatof the shortest an octave above it. In
the same way, the shortening of avibrating string increases its pitch.
The key harmonic relationships
here are the octave, obtained byhalving the length of a string thus
doubling its frequency of vibration,and the fifth, obtained byshortening the string by a third andincreasing its frequency by 3/2 or
1.5. But by applying mathematicalvalues to sound Pythagoras also
showed that there is a flaw innature’s tuning: its fifths are a little
out with its octaves.
The octave is the perfect interval:
the ratio of the note and its octave
is 2; double the frequency and you
get the same note an octave higher.
Theoretically you could go on
forever like that. On a modern
keyboard, if you move from A to A
through the seven octaves until youreach the highest A, and with every
jump you double the frequency,
you have a progression from x to 2x
through 4, 8, 16, 32, 64, and finallyto 128x. After octaves, the easiest
interval to tune is the fifth (you areforever hearing it with stringed
instruments). On the piano you get
12 fifths, from A to E, then B, Fsharp, C sharp (or D flat), A flat, Eflat, B flat, F, C, G, D and finally totop A. In each case the frequency ofvibration will become one and ahalf times as many. Unfortunately
1.5 multiplied up (1.5 to the powerof 12) does not arrive at 128 (2 to thepower of 7), but slightly more and,on this basis, would result in thetuning of the top A being slightly
sharp. This pitch error (about aquarter of a tone on a piano) is
known as the Pythagorean comma.
While the compass of keyboard
instruments was quite small,things didn’t sound too
disagreeable. But as composers
extended the range of their own
creativity, their creative paletteincreased accordingly and required
instruments to follow suit. Then, inthe higher octaves, one could hear
the Pythagorean comma (andconsiderable historical andarchaeological evidence
corroborates this dilemma of thetime because the tuning of organsof the period was sometimes left
unchanged until our own times).
The necessity arose for aninsightadjustment that ironed out thisnatural discrepancy and that iswhat equal temperament is.
Pythagoras, with only amonochord or a dulcimer, madethe mathematical point about theoctave and the fifth. But if one
thinks of all the other intervals,
tuning by A not only gives two
different readings for top A, it
makes a distinction between manyother notes as well, for example A
flat and G sharp. Of the various
practical solutions proposed, theone that triumphed was the idea oftuning 12 exactly equal semi-tones
in an octave. Every single semi-tone
progression was smoothed over bytaking a given note and (as it were)
multiplying its frequency by the12th root of two (1.05946), thentaking the resultant frequency anddoing the same again and so on.
The results are that there is no
distinction between A flat and G
sharp, which are enharmonicallythe same on a modern piano but
continue to be minutely distinct ona stringed or a wind instrument,
and that all scales are equally out oftune – to most ears imperceptiblyout of tune. Every fifth as one rises
is slightly flat. This resolution has
ironed out a lot of tone colour – asort of acoustic atmosphere thatwas peculiar to specific keys – lessreadily recognisable now as thesound of a particular key is far
blander than it was. In an extreme
case – where someone tunes apiano electronically, multiplying
every frequency by the 12th root of
two to get the next semi-tone up –the instrument sounds pretty
boring. A clever tuner counteracts
this tendency by rule of thumb,perhaps tuning a little sharp in thelower bases. Horowitz, for example,
used to like everything on thesharp side. The object of thistweaking is to enhance the tone
colour by departing from thepurely mathematical and rather
arid experiment that is the essenceof equal temperament.
We have grown up with thesound world of equal temperamentand it sounds right. At the time it
was introduced, however, it wasquite an aberration and muchdisputed. Bach’s set of Forty-Eight
preludes and fugues in every
possible key settled the argument.
Of course it was more than amanifesto or a celebration of agreat idea. He did not write for amore extensive keyboard just toprove his point. Necessity here was
the mother of invention; hewanted to make a musicalstatement, needed a more extensive
keyboard compass for it, andtherefore needed also a tuning thatwas consistent for that keyboard.
One suspects that, like many artists,
he also enjoyed working within –and exceeding – the boundaries thesystem set. And perhaps thisdiscipline is also why so many greatcomposers adopted Bach’s format:
Chopin is just one example; his 24
preludes go through all the keys
and the opus 10 book of his 24
studies follows each major key by
the relative minor.
The general adoption of equal
temperament paved the way for thesonata first movement principle,
an essentially dramatic form inwhich the first and second subject

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groups are in contrasting keys andthe development section is free tomodulate in a more far-reachingmanner. Thematic material could
be more widely spaced on thekeyboard but there was now no fearof dissonance in higher registers.
Bach’s son, C. P. E. Bach, is creditedwith the sonata system which wasthen brought to near perfection by
Haydn, Mozart and, later, with evergreater freedom and iconoclasm,by Beethoven. And with the onset ofthat freeing of the boundaries ofthe sonata principle came
Romanticism. In spite of its
mistuned intervals, equal
temperament made possible thecontinued use of a keyboard with
12 notes to the octave and theJohann Sebastian Bach: writing for a more extensive keyboard …

’more than a manifesto or a celebration of a great idea’… which made audible the Pythagorean comma

The architecture of the Forty-EightPeter Bradley-Fulgoni

3

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development of nineteenthcentury piano music andtechnique in the hands of figures
such as Franz Liszt. Bach was thetrail-blazer.
As for the preludes and fugues
themselves, the prelude could havebeen anything. It is determined byits function, namely to precede afugue. The fugue is determined byits style rather than by its function:
it is a texture with a highly
disciplined format. The
juxtaposition of the prelude’s
rather free-flowing, sometimes
improvisatory music being
followed by something quite
rigorous, strict and disciplined
gives an emotional polarity. But

here they are wedded to sentiment,
feeling and emotion and that iswhat makes Bach’s preludes andfugues not only very interesting
structurally and formally, but extra
special because there is what one
might call an almost Romantic
spirit of expressiveness, anemotional content which seems tocome from disciplined formal
conditions – an unusual andparadoxical dichotomy. Some of
the fugues have more than one
subject. Take the F sharp minor
fugue in book 2: three subjects
blend at some point in the fugue,
like a matrix of genius; three ideas
which, when sampled individually,
seem disparate and diverse, when

J. S. Bach, Prelude in E flat, from book 1 of the Forty-Eight: a more complex fugue than the ‘official’ one that
follows it‘an emotional content that seems to arise from disciplined
formal conditions’

J. S. Bach, Fugue in E flat from Book 1 of the Forty-EightPeter Bradley-Fulgoni

The architecture of the Forty-Eightthey come together don’t justharmonise, they coalesce perfectlyin the structure of the fugue,
making the whole process acrescendo of creative power. It is aremarkable moment. Some of thepreludes are in simple binary form,
but others are fugues inthemselves; an example is the E flat
prelude in book 1, a very
sophisticated fugue, more complex
than the ‘official’ one that follows
it. What provides for all this
precisely is the genius of Bach,using the formal structures of his
epoch to the nth possible degreeand drawing every ounce from aprinciple. He was characterised byhis free-flowing imaginative
process, not a confinement to rule.
His art raises the question ‘what isRomanticism?’ and always suggeststhat the answer can no longer beconfined to a period postBeethoven.
It is also best played on a
keyboard that is slightly out oftune.
Peter Bradley-Fulgoni studied piano inEngland, Italy and Russia. He has mademany broadcasts, given concerto andrecital performances throughout Europe
and beyond and recorded a number ofCDs. Recognised as an inspirational
teacher, a DVD featuring both musicalperformance and discussion will be
released by Sound Techniques in their
Piano Maestros series in Spring 2007(www.soundtechniques.co.uk)