Why twelve notes as one attractive arrangement

From Microtonal

Author: Margo Schulter

License: CC BY-NC-ND 4.0

(contributed article)

Q. Why are there 12 notes per octave on typical keyboards?

A. The form of the typical European keyboard instrument seems to have evolved during the era from around 757, when the court of King Peppin of the Franks received an organ as a gift from the Byzantine Empire, to around the middle of the 14th century, when 12-note keyboards in the famous arrangement with seven diatonic or "white" keys and five accidental or "black keys" were becoming standard.

Here our focus is on the question: how did these keyboards come to have their familiar 14th-century arrangement, still standard in the 12-key instruments of the early 21st century?

An equally important and revealing question, however, is "Why stop at 12 keys?" As other portions of this FAQ discuss from various viewpoints, the best answer may be that a range of musicians over the last six centuries or a bit more _haven't_ stopped at 12, but have designed and often have actually built instruments with anything from 13 to 31 or more notes per octave.

Let's also note that while the history of the familiar 12-note keyboard is largely an adventure in Western European musical styles and tastes, various musical cultures have leaned toward larger tuning systems. For example, some medieval Arabic or Persian traditions favor a 17-note system of a kind in some resembling that advocated in early 15th-century Italy. Traditional Chinese theory recognizes sets of 53 or more notes per octave.

With this bit of perspective, let's return to medieval Europe around the time of Charlemagne and his successors in the 9th century, a period celebrated by one poet as a kind of "rebirth" of Roman culture (scholars sometimes call this the Carolingean Renaissance).

A vital element in the cultural mix was music, with the treatise of the revered philosopher Boethius (c. 480-524) the basis for the learned study of this art. Boethius took a great interest in the theory of consonance and dissonance, and also in the ancient Greek authors and systems, especially those of Pythagoras (as recorded by his followers) and Ptolemy.

Like many world musics, the best-known Western European music of this time was based on a system of tuning in pure fifths and fourths, known in the West as Pythagorean tuning after the Greek philosopher Pythagoras. (Pythagoras, like many of the pre-Socratics, is known to us mainly by repute and by reported quotations or teachings written down by later authors). Scholars have suggested that the ancient Greeks may have borrowed it from a Babylonian tradition.

While both sacred and secular music were practiced in 9th-10th century Europe, and some writers such as Hucbald tell us a bit about popular instruments such as lyres or harps, we know mainly about music for the Church: both traditional chant, and a newly documented technique of _organum_ or "organized" part-music involving the concord of different notes sounding at the same time.

Medieval liturgical chant or plainsong uses a system of eight standard notes: the seven diatonic or "white" keys on a familiar keyboard, plus Bb. In other words, there are _two_ versions of the step Bb/B, and both may occur in certain chants.

In the Pythagorean tuning described by Boethius, and followed as standard practice, we can derive this usual set of eight notes for chant as a chain of seven pure fifths or fourths:

                  Bb  F  C  G  D  A  E  B

Early organs of this era seem often to have had these eight notes per octave, although the term "keyboard" might be misleading: these instruments had devices such as large sliders for opening or closing the wind supply to a pipe, so that two or more players might be needed, and the action was likely _slow_.

If we applied a layout like the modern ones, we might arrange the notes like this, with the "black key" Bb set apart from the others:

                  C    D    E  F   G   A   B  C

However, we will recall, both Bb and B were regarded as regular forms of the same scale step, so a layout with all eight notes on the same row was common, and still in use in some instruments of the 14th century:

                  C    D    E  F   G   A  Bb  B  C

In medieval Western Europe, as in many other world musics, fifths and fourths were favorite consonances: Boethius described their concordant effect, and during the period of around 850-1200, musicians developed more and more complex styles contrasting these stable intervals with a wide range of unstable ones having various degrees of concord or discord. Pythagorean tuning made the stable fifths and fourths pure, or ideally smooth and concordant, and produced an intriguing continuum of tension among the other intervals.

By around 1200, the great composer Perotin and his colleagues were writing pieces in the high Gothic style for three and four voices, using not only the traditional eight notes of most chant but other accidentals: Eb, F#, C#. These notes could also be added to an organ by extending the chain of fifths in either direction, for example in this ten-note chain:

                 Eb  Bb  F  C  G  D  A  E  B  F# C#

At about this same epoch, there was apparently a major technological breakthrough: at least some organs acquired agile keyboards of the modern kind, which allowed them to play the flowing and often ornamented melodic lines favored in the music of the time. A 13th-century poem, the _Roman de la Rose_, tells us that small or portative organs could play either the supporting lower part or the florid upper melody of the sophisticated motets then in fashion, pieces artfully combining voices singing different texts.

Just how quickly and frequently some or all of the extra accidentals became standard on the limber keyboards of the 13th century is uncertain, but by 1325, the theorist Jacobus of Liege tells us that their diatonic whole-steps or major seconds were "almost everywhere" divided into two semitones. This comment become clearer if we look at a possible keyboard layout around 1300, using the 11 notes which had sometimes been in use for at least around a century:

               C#   Eb         F#         Bb
           C      D     E   F     G     A      B   C

Here the seven diatonic notes form an octave with five whole-tones (C-D, D-E, F-G, G-A, A-B) and two semitones (E-F, B-C). Four of the five whole-tones have added accidentals dividing them into semitones: C-C#-D, D-Eb-E, F-F#-G, and A-Bb-B.

Given the special status of Bb as a "regular" note, it might also have been placed in the same row as the diatonic notes, giving us an arrangement of our eight "regular" notes plus three extra accidentals:

               C#   Eb         F# 
            C     D     E   F     G     A  Bb  B  C

By around this same epoch of 1300, a new and compelling argument for these extra keyboard accidentals was at hand: the preference for "closest approach" at cadences or in other directed progressions where an unstable sonority moved to a stable one.

The basic rule, as stated by various 14th-century writers, is that a third expanding to a fifth, or a sixth to an octave, should be major; a third contracting to a unison should be minor. If they are not so naturally, then they should be altered by using accidentals. For example, a typical "closest approach" cadence on D might take two forms; here I use a notation showing middle C as "C4," with higher numbers showing higher octaves:

   C#4 D4    C4  D4
   G#3 A3    G3  A3
   E3  D3    Eb3 D3

Either form features a major third between the lower two voices expanding to a fifth, and a major sixth between the outer two voices expanding to an octave. In the first solution, the two upper voices each ascend by a semitone; in the second, the lower voice descends by a semitone.

While the second form was available using the 11-note set known in 13th-century compositions, the first form called for a 12th note, namely G#.

This first form was in fact much in demand, because in 14th-century style a cadence with ascending semitones was usually considered more conclusive than one with descending semitones, the latter form usually signalling a kind of musical "halfway" point rather than a final cadence.

Since pieces centered on the octave-type or mode of D-D were very common, final cadences with the major third E-G# expanding to the fifth D-A were routine in theory and practice.

Therefore, as a modern software developer might say, G# had found a "compelling application" -- in this case, an application calling for a modest hardware upgrade, the addition of a G# key.

Around 1325, the time Jacobus wrote about the semitones being divided "almost everyone," this 12th note may have already been added on some instruments. The earliest known European compositions for keyboard, preserved in the Robertsbridge Codex with proposed dates anywhere from 1325 to 1365, calls for all 12 notes of such a keyboard, with a chain of fifths very likely tuned like this, from Eb to G#:

               Eb  Bb  F  C  G  D  A  E  B  F#  C#  G#

However, as we might say, the "user interface" wasn't quite yet standardized. The scholar Mark Lindley shows how the 14th-century Noordlanda organ had an arrangement with the eight "regular" notes (including both B and Bb) on one row, and the complement of four "extra" notes on another:

               C#   Eb         F#    G#
            C     D     E   F     G     A  Bb  B  C

At Halberstadt in 1361, however, it appears that the 12-note organ placed Bb in the row with the other accidentals, possibly with earlier unrecorded precedents, and this quickly prevailed as the "standard" arrangement:

               C#   Eb         F#    G#   Bb  
            C     D     E   F     G     A     B  C

Q. We know that there's no need to stop at 12, but why it is _one_ attractive size for a keyboard tuning?

A. Here there may be a quick answer and a more involved answer. Let's take the quick one first.

In his encyclopedic treatise of 1325, Jacobus of Liege mentioned that keyboards "almost everywhere" divided diatonic whole-tones into two semitones by extra accidentals. With a 12-note instrument, as we can see on either the Noordlanda keyboard or the more familiar Halberstadt keyboard, _every_ whole-tone is divided into two semitones, including G-A (by the new 12th note G#). In all we have five such whole-tones, and five accidentals (counting Bb and the four "extra" ones) to do this dividing.

There's a certain cozy symmetry in this arrangement, not to mention that a Pythagorean chain from Eb to G# nicely covers the accidentals typically used in the great preponderance of 14th-century pieces for ensembles or keyboards, including the music of composers such as Guillaume de Machaut (1300-1377) and Francesco Landini (1325-1397).

The more involved answer looks at this feeling of "coziness" or "balance" more closely, and brings into the play an important concept of modern theorist Ervin Wilson: the idea of a Moment of Symmetry (MOS).

Such an "MOS" occurs when a tuning system has only two sizes of _adjacent_ intervals, that is, intervals between the pairs of notes immediately adjacent to each other.

If we look at our 12-note Pythagorean tuning with a chain of fifths from Eb to G# -- we can call this an "Eb-G#" tuning for short -- we find that there are in fact only two such intervals sizes, formed by two varieties of semitones known as "diatonic" and "chromatic," and here marked as "D" and "C," whose distinct sizes we're about to consider:

      C    D   D    C   D   C    D   C    D   D    C   D
    C - C# - D - Eb - E - F - F# - G - G# - A - Bb - B - C

In Pythagorean tuning, as it happens, the diatonic semitones are _smaller_ than the chromatic semitones. To see why they differ in size, we might look again at our chain of pure fifths for this tuning:

                           C = 7 fifths up
               Eb  Bb  F  C  G  D  A  E  B  F#  C#  G#
                         D = 5 fifths down

A diatonic semitone such as E-F is formed from a chain of _five_ fifths; if we start at the lower note of this interval E, we must move five fifths _down_ the chain to reach the upper note F. Note that this relationship also holds for B-C, D-Eb, A-Bb, F#-G, C#-D, F#-G, and G#-A. It is semitones of this kind which are the usual melodic semitones of medieval European music, for example in our sample 14th-century cadences above involving accidental steps (G#-A and C#-D, or Eb-D).

Chromatic semitones such as F-F#, however, are formed from chains of seven fifths _up_. Other chromatic semitones are Bb-B, Eb-E, C-C#, and G-G#, each illustrating this same relationship on the chain. The use of these intervals as direct melodic steps is rather unusual in this era, but Marchettus of Padua and some other Italian composers of the 14th century do it boldly and beautifully.

How large is each of these semitones? One way to compare their size uses the modern yardstick of _cents_, with a pure octave divided into 1200 equal parts or cents. Keeping the math simple, we can take the size for a pure fifth (a ratio of 3:2) as a rounded 702 cents, and of a pure fourth (at 4:3) as a rounded 498 cents. These two intervals add up to a pure 2:1 octave, and their sizes add up to 1200 cents.

A diatonic semitone such as E-F is five fifths down; another way of looking at this is to say that it's five fourths up, starting again at the lower note E and moving to the upper note F:

                   498    498    498    498    498
                E3  -  A3  -  D4  -  G4  -  C5  -  F5

Adding up these five fourths, we get an interval of (498 x 5) cents, or 2490 cents. Actually, in moving up five fourths, we've moved up a semitone plus two extra octaves (E3-F5), as the octave numbers for the notes in our chain show. However, by now moving down two octaves or 2400 cents from F5, we arrive at the upper note of our desired diatonic semitone E3-F3. This interval has a size of (2490 - 2400) cents, or 90 cents.

For a chromatic semitone, let's say F-F#, we similarly move up by seven fifths of 702 cents each, and then down by four octaves:

            702    702    702    702    702    702    702
         F3  -  C4  -  G4  -  D5  -  A5  -  E6  -  B6  -  F#7 

Our seven fifths up (F3-F#7) give us an interval of (702 x 7) cents, or 4914 cents; moving back down four octaves to the upper note of our desired chromatic semitone F3-F#3, we have a size of (4914 - 4800) or 114 cents.

Thus our 12-note Pythagorean keyboard gives us an MOS with two and only two sizes of adjacent intervals: diatonic semitones at a rounded 90 cents, and chromatic semitones at a rounded 114 cents. A diatonic plus a chromatic semitone forms a regular whole-tone (e.g. E-F# from E-F plus F-F#) at around 204 cents:

      C    D   D    C   D   C    D   C    D   D    C   D
    C - C# - D - Eb - E - F - F# - G - G# - A - Bb - B - C
     114  90   90  114  90 114  90  114  90   90  114  90

There is an elegant poise and balance here which makes 12 an attractive number not only in a 14th-century Pythagorean tuning, but in various other historical European tuning systems fitting various eras and styles.

However, one person's ideal "Moment of Symmetry" can be another person's overworn rut.

By the earlier 15th century, only decades after the Halberstadt 12-note design had won out, European theorists were proposing tunings and keyboards based on the next Pythagorean MOS: 17 notes (also found in medieval Arabic and Persian systems). Other such larger MOS systems feature 29, 41, or 53 notes -- Chinese theorists, interestingly, being aware of a special property of 53.

Other approaches to tuning can produce different MOS sizes: in early modern Europe of the 16th and 17th centuries, for example, the prevailing meantone temperaments for keyboards offered such sizes at 12, 19, or 31 notes, and instruments of all three sizes were designed and built, and music written taking advantage of the larger systems.

(It's worth noting, as the earlier medieval instruments with from 8 to 11 notes show, that there's no law requiring that a keyboard size must match some MOS: for example, instruments of 13-16 notes were quite common in 15th-17th century Europe.)

In sum, for 14th-century European musicians and organ-builders, 12 notes was an attractive point of repose; and its symmetrical qualities, as well as familiar keyboard ergonomics, still have their appeal.

However, as musicians have recognized at various times and places in the six centuries and a bit more since, 12 is not the _only_ place to stop, and there are enticing if not so widely recognized reasons to explore larger tunings and instruments.

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