Why you can't play Bach on a piano

musical notes Of course, you can play Bach on a piano, assuming that you've got the technical skill. But, if your piano is tuned in the modern way, to what is now known as an 'even tempered' or 'equally tempered' scale, the note pitches won't be the same as they would have been for Bach's contemporaries. And that's not to mention the difference in note articulation between the modern piano and the keyboard instruments of Bach's day, which is another matter entirely. This article briefly examines the development of musical scales and tuning, and describes the various tuning systems that were available in Bach's day. It then describes the development and properties of the equally-tempered scale, and explains why we don't, in general, hear the music that Bach wrote.

The subject of tuning and temperament is important for an understanding of music composition, because music was composed to take advantage of the strengths, and mask the weaknesses, of the tuning systems in use at the time. There is an intimate relationship between the musical repertoire of a particular day and the science (or art) of tuning as it was practiced. It can even be argued that the reason why a large proportion of the surviving repertoire of pre-classical music is no longer performed is because it simply sounds boring with modern tuning practice.

I should point out at the start that there is a problem of vocabulary when discussing this subject. Because few of us in the Western world have any familiarity with music from before the renaissance (i.e., before the 14th century), or from non-Western cultures, we tend to use vocabulary which is either anachronistic, or culturally out of context, or both. For example, we may talk about musical notes forming an interval of a fourth, or a fifth, but these terms get their meanings from the use of a 7-note scale. It is not entirely helpful to use the term 'a fifth' when talking about, say, a pentatonic scale. But we do. I will try to avoid sloppy use of culturally-determined vocabulary in this article, but it is difficult, and from time to time I will slip up. Sorry.

The importance of the octave

We can make music, of a sort, with sequences of sounds of any combination of pitches, but we are using a scale when we limit our note choices to a selection from a set of particular, well-defined pitches. We usually do this, not only because it's easier to get aesthetically pleasing pitch combinations, but because we can then use a system of musical notation to record our compositions. If music consists of entirely arbitrary pitches, it's almost impossible to record the notes except by writing the exact pitch as a frequency value or something of that nature.

But if we decide to limit our note selection to a scale -- and all but the most experimental music does -- the question arises which particular pitches to use. We could, in principle, just pick a bunch of abitrary pitches, and call them 'A', 'B', 'C' (or Albert, Betty, Charlie) and so on. But we don't really have a completely free choice, because not all pitch combinations are particulary aesthetic, either played together or sequentially. There is also the issue of economy: human hearing is capable of hearing frequencies up to about 20,000 Hz (20,000 cycles per second), or a bit lower if you're an old fogey like me. Assuming we can distinguish pitches that differ only by about 10 Hz, we could end up with a 2,000-note scale. Even if this were aesthetically desirable -- which it isn't -- it would be unmanageable in practice.

It's also unnecessary. It turns out that the pitch ratio 2:1 (which we call an octave these days, at least in the West) is crucially important for the music of all cultures. The reason for this is that notes with pitches in this relationship sound particularly harmonious. In fact, they are so harmonious that it can often be difficult to tell that more than one note is being played. There is a good physical reason for this: most musical instruments produce sounds which are not of one single frequency, but a whole series of frequencies in 2:1 ratios. For example, if you pluck a guitar string, it will vibrate not only along its whole length, but also to a lesser degree along half its length, a third its length, a quarter its length, and so on. Because the frequency of the sound emitted by a vibrating string depends on the length of string that is vibrating, the sound conists of the whole-length frequency (called the fundamental), plus a frequency twice as high, one three times as high, and so on. These higher frequencies are called overtones or harmonics. The highest-amplitude harmonic -- not just for the guitar but for most musical instruments -- is the first harmonic, that is, the frequency twice that of the fundamental.

So the interval of the octave is not just a convention, but something that derives from the the very nature of all musical instruments.

What's important about the octave for our present discussion is that any two pitches that differ by an octave have the same audible relationship, whatever the actual pitches are. This means that two notes with pitches 100 Hz and 200 Hz sound the same in relationship to one another as two notes with pitches 4000 Hz and 8000 Hz, or 600 Hz and 1200 Hz. And this means that when we devise our scale, we really only need to concentrate on notes whose pitches are less than an octave apart. Because of the importance of the octave relationship, it is almost inconceivable that we would use note pitches in the range (say) 2000 Hz to 4000 Hz which are not in 2:1 ratio with the pitches we use in the 1000 Hz to 2000 Hz range. If we use 2000 Hz for one note, we will almost certainly want to use 4000 Hz for another, and not, say, 3992 Hz.

All this means that we are mostly concerned, when devising a scale, with the pitch intervals within an octave. We can then repeat those notes at frequency multiples of two, four, eight, and so on, to form an extended scale.

How many notes?

So we've established that, in practice, we only have freedom to choose note pitches within the octave. We will then span the audible frequency range by successively doubling these pitches.

But, within the octave, we still have a lot of freedom. For example, do we divide the octave into three notes, or five, or seven, or fifty? There is something to be said for three, five, and seven-note scales, among others. With fifty notes to the scale, we run into the problem of economy again, quite apart from the difficulty of actually manufacturing musical instruments that are able to form fifty different pitches in an octave.

The problem of deciding on the number of notes is, of course, intimately connected with the problem of deciding on the pitches of those notes, which is the subject of the next section. For now, I will just point out that the 7-note major and minor scales which are the foundation of almost all post-renaissance Western music, are supported by considerations of accoustic physics -- they aren't entirely cultural, as is often claimed. Nevertheless, there is a certain intuitiveness about a 5-note scale that can be derived by successively increasing pitch in the ratio 3:2 (more on this below). This is one form of pentatonic scale, but not the only one. Even with our 7-note tradition, much Western music is pentatonic in character, particular in the jazz and blues repertoire. A good example is Gershwin's summertime, the melody of which is pentatonic but for one note. This kind of pentatonic music nearly always uses note pitches that are a subset of the pitches in our common 7-note scale, but there are pentatonic scales in Eastern music that use somewhat different pitches. For example, the slendro pentatonic scale of Indonesia uses pitch intervals that are equally spaced within the octave, and whose notes therefore are not playable with precision on a Western fretted or keyboard instrument.

So it is quite likely (but not certain) that there is a tradition of 5-note scales that long predates the 7-note scales we are now so familiar with. There are also 4-note and 6-note scales, which may or may not approximately match pitches within the 7-note scales we now use; but it isn't entirely clear whether they developed before or after the 7-note scales in the West.

Of particular importance in modern Western music, of course, is the 12-note chromatic scale. The pitches in the modern chromatic scale are selected so that we can play 12 different 7-note scales, with the intervals approximately the same in each case. But that isn't necessarily the only way to choose the pitches of the 12 notes. In addition, although 12 notes to an octave would seem to most people be to be an ample sufficiency, even some modern Western music uses notes which cannot be represented in our standard 12-note scale. For example, many blues performers sing note pitches that fall between the 4th and 5th or 11th and 12th notes of the scale ('blue notes'), and some modern composers have made a reasonable case for a 24-note scale.

Note pitches within the octave

In short, the number of notes within the octave is influenced by both cultural and physical factors. In order to understand why certain note pitches have become widespread in many different scales, while others are quite singluar, we must first have an understanding of consonant and dissonant pitch relationships. A consonant pitch relationship is one that is variously described as 'stable', 'settled', 'harminous' or 'at rest'. A dissonant pitch relationship is one that sounds unresolved, or in tension. In Western mustic we usually expect a dissonant relationship to resolve into a consonant one fairly soon after it is played but -- these days, at least -- we feel that we need at least some dissonant intervals in our scale, as a generator of musical tension.

Now the 'most consonant' pitch relationship, as already discussed, is the octave. The question that naturally arises is: are there significant consonant relationships within the octave? If there are, can we use these pitch relationships as the basis of our scale? To some extent it has to be recongised that consonance and dissonance -- apart from the octave which is ubiquitous -- are at least partly a matter of cultural expectation rather than physics. For example, pitches in the ratio 5:4 (which we generally call a 'major third' these days) are now extremely important in all Western music. But it seems this interval was rarely used before the Renaissance, and may even have been heard as a dissonance. Happily, there are at least a few pitch relationships that have nearly always been considered consonant, in nearly all musical cultures.

Pythagorean tuning

Most importantly, it has been recognised for thousands of years that the pitch ratio 1.5:1 (also expressed as 3:2 and called a perfect fifth in modern usage) is one such relationship. In fact, this relationship was described by the Greek mathematician Pythagoras (although he probably was not the first to recognise it). It's not immediately obvious why this ratio should be so important because, unlike the octave, frequencies with this particular relationship to the fundamental are not necessarily common in the sound produced by musical instruments. But the human ear is very good at analysing sounds in terms of octaves, and most musical instruments produce sounds with frequencies three times that of the fundamental (and higher), as well as twice that of the fundamental. A pitch ratio of 1.5:1 is what you get if you take the pitch ratio of 3:1 -- which is widely present in instrumental sounds -- and transpose the interval down an octave.

Whatever the psycho-accoustic explanation, the interval 1.5:1 is widely recognised as consonant, and will usually have a place in any musical scale. In modern Western music, the interval between the notes C and G (which we call a fifth, because there are five white notes from C to G in modern keyboard tuning practice) is almost exactly 1.5:1 -- almost, but not exactly. In equally-tempered tuning, the ratio of C to G is actually 1.4983:1 and that's why, although it is a fifth by definition, it is not a perfect fifth, although that term is widely (mis-)used. Similary, it isn't strictly correct to use the term 'a fifth' to describe a tuning interval of 1.5:1 except when we have a 7-note scale. But we very frequently do, because it's easier to say 'a fifth' than 'in a 3:2 ratio of frequencies'.

Apart from 3:2, what other intervals might be considered consonant? Although there are cultural variations, it's generally been the case that consonances are produced by pitch intervals in simple integer ratios: 4:3, 5:4, 6:5, etc. The physicist Helmholtz proposed that this was because consonance was produced when pitches had harmonics in common.

For example, consider two pitches of 100 Hz and 125 Hz. These are in ratio 5:4. The first pitch has harmonics at 200 Hz, 300 Hz, and so on; the second has harmonics at 250 Hz, 375 Hz, and so on. They both have a harmonic of 500 Hz, and are therefore consonant. But they are 'less consonant', perhaps, than the pitches 100 Hz and 150 Hz (in 3:2 ratio) because these pitches have lower harmonics in common. In any event, if we allow for harmonics up to the 7th as producing consonance, and we limit ourselves to whole-number frequencies, then Helmholz's analysis produces consonant intervals of 6:5, 5:4, and 3:2 (but not, interestingly enough, 4:3 -- a major fourth in modern nomenclature). If we include the 7th harmonic we get some very peculiar results, but the reasons for excluding the 7th seem no less arbitrary than our reasons for limiting ourselves to small integer ratios in the first place. In any event, we have to wonder about the usefulness of a system of note generation that will give us a minor third (6:5 ratio) but not a major fourth.

Be that as it may, we can find three or four intervals within an octave that most people think of as consonant, and a couple more which are constestable. All these intervals are 'bunched up' in the middle of the scale, and if we want more notes we will have to content ourselves with adding some less harmonious ones. As I said, that is no bad thing in modern music -- melodies consisting of only harmonious intervals sound too much like plainchant for contemporary ears. Ratios of 8:7, 9:8, 18:5, for example, are reasonable candidates for a modern scale.

Rather than picking consonant pitch ratios arbitrarily, what we might do -- and there is some merit to the scheme -- is to proceed as Pythagoras' followers did, successively increasing pitch in a ratio 3:2, and then reducing the pitch in octave steps to bring the note back into the starting octave. For example, if our first interval step is 3:2, then our next will be 3:2 above the original 3:2, or 9:4 overall. But a pitch that is 9/4 times the pitch of the base note is nore than an octave above the base note, so we must halve it to reduce it by an octave. This gives us a final pitch ratio of 1.125:1 (or 9:8). So we now have three notes, the base note, a note of 1.125 times the frequency, and one of 1.5 times the frequency. We can carry on increasing pitch by 3:2, and reducing by octaves, to get additional notes. This way we can readily generate a very natural-sounding 5-note scale, and slightly less natural-sounding 6-note and 7-note scales.

If we continue with the Pythogorean proces beyond seven notes, we find that we get something that is almost a complete 12-note scale within the octave. Irritatingly, the 12th note is not the octave, but is in a ratio 2.027:1 with the base note. If we want to base the scale on octaves -- and we do -- then the interval between the 12th note and the octave is just a little too small (the pythagorean comma). This means that a melody that crosses an octave boundary with a particular interval can sound badly out of tune, given what is colloquially known as a 'wolf note'.

We can minimise the problem by selecting a particular base note and tuning both upwards and downwards from that note, rather than simply tuning upwards. This spreads the 'comma' out more evenly, making it less apparent, at least in a particular key. The problem with this scheme is that we end up with notes that should be exactly a octave apart being not quite an octave apart. Another way of expressing the problem -- albeit an anachronistic one -- is that the notes G-sharp and A-flat are not the same (actually these notes are generally not the same, functionally speaking, but in modern turning practice they sound the same -- more on this point later).

Another problem with the Pythagorean tuning is that, apart from the notes in 3:2 ratio, few of the other intervals sound particularly consonant. Ideally, as discussed above, we would like to include the intervals 4:3, 5:4, 6:5, and maybe others we have already indentified. The Pythagorean scale can't generate the intervals 5:4 or 6:5. This wasn't a problem in ancient Greek culture, or even in medieval Europe, as these intervals were not widely used.

For all its problems, Pythogarean tuning and similar schemes remained popular until the renaissance. In fact, if you are a guitarist, and you tune your guitar using the common practice of tuning each string from the next lowest one sounded at the fifth fret, then you are using Pythagorean tuning. Up to a point, an instrument tuned to a pythagorean scale is transposable -- that is, we can play a melody in the key of C and then, say, the key of F, and the intervals will be the same. Transposability became particularly important in the baroque era, because composers increasingly employed key changes within a piece of music to generate musical interest.

'Just' tuning

We can avoid Pythagoras's wolf notes and lack of consonant thirds by the simple expedient of picking a scale where the notes are based on simple ratios which happen to sound consontant, without necessarily standing in a strict mathematical progression with respect to one another. There are many such schemes, of which a subset known as 'just tuning' or 'just intonation' is particular important. We can make an effective just 7-note scale using the ratios 9:8, 5:4, 4:3, 3:2, 5:3, and 15:8. Such a scale would sound reasonably similar to a modern equally-tempered major scale, but with a couple of jarring discrepancies. We can also make just 12-note scales, of which the Pythagorean turns out to be a particular example.

The problem with 'just' tuning systems is that they don't transpose very well. Suppose, for example, that I have tuned my instrument used the note C as the root note, and all the other notes derived from it. Then I play the note sequence C-D in the key of C. If I play the sequence D-E in the key of D, then in a transposable tuning system I would get the same pitch ratio. But using the 7-note scale I described earlier, D-C is a pitch ratio of 9:8, while E-D is 11:10. The difference is not enormous, but it is noticeable, and it varies from key to key and interval to interval. Just intonation schemes generally perform particularly badly when there is transposition to a chromatic key (e.g., from C to Eb). In addition, if you tune a 12-note scale (what we now call a chromatic scale), sharpened semitones are of dfferent pitch relationship to their nearest whole tones than flattened semitones. This means that if you play a piece in, say, the key of Eb, which calls for the note Ab, and then you play a piece in E, which calls for G#, you have to retune, because G# and Ab are not the same pitch. Notes that ought to sound the same when played at a particular scale position in different keys, are known as enharmonics. A tuning system that has enharmonics at the same pitches -- as they should be -- is called enharmonically equivalent. Pythogorean, just, and meantone systems (see below) all have enharmonic non-equivalences. Of course this isn't necessarily a bad thing -- music is typically composed to suit the tuning system of the instrument it was composed for, and the enharmonic non-equivalence may have been part of the flavour of the music. However, with frequent key modulations, enharmonic non-equivalence can be a problem. It appears that there were attempts to construct instruments with sufficient notes to be able to tune Ab as well as G#, Db as well as C#, and so on. But they would, presumably, have been fabulously difficult to manufacture, as well as horrible to play.

Just tuning isn't merely of historical importance; just as there are vestiges of pythogrean tuning in modern practice, many single-key instruments are justly tuned even today.

Meantone tuning

Another tuning possibility, once the pitch ratio 5:4 (which corresponds with what we now call a major third) becomes important, is to base our scale on perfect major thirds, rather than fifths, and distribute the other notes between them. So we could tune the interval C-E to be a perfect 5:4 ratio, and the interval E-G#. But if we increase again in a 5:4 ratio, we end up with something close to, but not exactly, an octave -- 1.95:1. So, as in pythagorean tuning, we end up with a 'comma' (this time called a syntonic comma). But we have an additional problem with this scheme that we don't have with pythagorean tuning: we don't have a way to generate intervals other than the major third from our base note. Continuing to increase pitch in 5:4 ratio beyond the octave simply produces nothing useful. A possible solution to this problem is to take the thirds generated by the 5:4 ratios, and simply interpolate pitches between them. So, for example, if we started with the note C, tuned E to a ratio 4:3 with respect to C, we would make the D exactly half way between C and E, giving it a pitch ratio of 9/8, which is the same as in the just intonation scheme described above. Not all the intervals work out so favourably, however.

Interpolation schemes of this sort are known as 'meantone' or 'mean tone' tuning, and were popular in the 15th century, surviving in some places right up until the end of the 19th century. The major thirds and fourths in the base key (i.e., the key tuned) are 'pure' (i.e., in exact integer ratios), and you can transpose to a key a fourth or a fifth higher with reasonable results. However, the fifths don't sound so marvellous on their own, and the practice developed -- and has continued until this day -- of not allowing an 'open' fifth to sound. Fifths would always be supported by a major or minor third, or something equally strong.

In addition, there are still 'wolf notes' and enharmonic non-equivalence, and some keys and transpositions are completely unuseable. In a tuning based on the note C as the tonic, the key of F# (or Gb, which is different from F#), for example, would be a complete no-no. Which is why we don't really see much renaissance or baroque music in this key. Of course, those of us with only modest technical skills might regard the baroque emphasis on a small number of key signatures as a good thing.

Tempering and well-tempering

In the baroque era, the increasing use of instruments with many fixed pitches (such as the pipe organ and harpsichord), which cannot practicably be tuned between pieces in a performance, combined with the development of music which made frequent use of key transposition, gave a new urgency to the tuning problem. One thing that all the tuning schemes so far have in common is that they are all based on pitch intervals in integer ratios. That is, they all have ratios that can be described as N:M, where N and M are whole numbers, and usually small ones at that. Tuning schemes of this sort are frequently referred to as regular tunings. But, however much we may wish it otherwise, it is a plain fact that we cannot get perfect transposability with a scale of more than four notes based on integer pitch ratios, so long as we want octaves to remain at a 2:1 ratio. It's a mathematical impossibility. Something has to give.

In general, baroque musicians compromised on the regularity of tuning -- that is, they deviated slightly from tunings based on whole integer ratios -- in the interests of better transposability. This deviation is called tempering. A system of tempering that was intended to get the best compromise between the 'purity' of 3rds, 4ths, and fifth, and the ease of transposition, was called 'well tempering'. All these tunings systems are irregular, that is, they deviate from pitch intervals which are simple integer ratios, at least in places.

At least one hundred well tempering systems are known to have been in use between the 17th and 20th century. The particular system described by Thomas Young in 1799, and then refined a number of times, is still in relatively common use among musicians who specialise in 17th-century keyboard music. Like all well-tempering systems, Young's specializes in tuning 'pure' (i.e. 2:1) octaves, and deriving pure thirds (5:4) and fifths (3:2) from some of these octaves, and from each other, in such a way that tonal purity is maintained in the popular keys, and there are no unuseable keys. The 'comma' is variously distributed among the different intervals of the scale, and is more pronounced in some keys than others.

It is a characteristic of all well-tempered tunings that, although all keys are playable (unlike in meantone or just tuning) each key has a slightly different character. Like the meantone systems that preceded them, many well-tempering schemes emphasize interval purity in the most popular keys, while allowing for some extra tonal colouration in the others. But a goal was that no key would be completely unusable.

In the baroque era there were no univerally-used terms for what we now call well tempering, meantone, equal tempering, and so on. It is entirely possible that what we now call well tempering was referred to by some baroque musicians as equal tempering, and vice versa. This means that we don't really know for sure what tuning Bach and his contemporaries would have used. By Bach's day meantone tuning systems, of which there were several, were the dominant form of tuning in Europe. But Bach probably did not use meantone, at least not for many of his works. The presence in some of his compositions of notes such as C## and Ebb strongly suggest that these works at least were not intended to be played on meantone-tuned instruments, where these notes would not even have existed.

Bach certainly knew of well-tempered tuning, but it isn't safe to assume that even a composition like The well-tempered clavier was intended to be played on a well-tempered instrument, as we now use that term, because of the problems in translation. The WTC consists of pieces in each of the major and minor keys available on a 12-note scale, and it was widely believed in the 19th century that Bach was demonstrating the capabilities of equal tempering, by demonstrating that all the keys sounded the same. However, most authorities these days think that Bach predominantly favoured well-tempered tuning, and that he composed music to exploit and demonstrate the harmonic differences that existed between different keys. Bach would probably have known of equal tempering (some writers have even claimed that he invented it) but, as I will explain in more detail below, it is unlikely that the technology existed in his day to tune a keyboard instrument accurately to equal tempering. There is some evidence that Bach used his own well-tempered tuning system, at least for some instruments.

And so, at last, to the equally tempered scale

An equally tempered scale is one in which the ratio of pitches between successive notes is the same, and where successive notes will always lead, eventually, to the next octave. Pythagorean tuning, as we have seen, uses identical ratios, but does not lead back to the octave. Well-tempering preserves the octaves, but at the expense of unequal note intervals.

Mathematically, there is only one equal tempering for a scale of a particular size. That is, with a twelve-note scale there is only one way to implement equal tempering. There is a different way to implement a five-note scale, and a 24-note scale. All equally tempered tunings have this in common: whatever the key, the interval between notes is always the same ratio. The ratio of the pitches C and D in the key of D is the same as for D and E in the key of D, and E and F# in the key of E, and so on.

It's not at all difficult to work out the (one and only) ratio of note pitches needed to create a 12-note equally-tempered scale. We know that if we apply the ratio 12 times, we must end up with an overall ratio of 2:1 -- an octave. So the pitch ratio of a single note is the 12th root of 2.0, or approximately 1.0595:1. In this tuning, the ratio of a major third is 1.26:1, which is audibly different from the just ratio of 4:3 (1.25:1). A fifth is 1.498:1, which is probably as close to 3:2 as tuning precision allows. A fourth is 1.335:1 -- again just slightly different from the just ratio of 4:3 (1.33:1).

In a sense, it is pure serendipity that the notes of a 12-note equally-tempered scale even approximate to the just or well-tempered equivalents. The surprising closeness of the equally-tempered fifth (1.498:1) to the pythagorean fifth (1.5:1) is a happy coincidence -- there is no mathematical relationship between the two.

Equal tempering has three huge advantages over any form of well tempering. First, the composer can modulate keys at will, because all keys have exactly the same characteristics. This makes composition less technical, in the sense of being less dependent on knowledge of the precise characteristics of a particular instrument. Second, it is easy to transpose, and that means that a chromatic keyboard instrument such as the piano can play with restricted-compass instruments that are usually played in specific keys. The ubiquity of the modern-style concert orchestra during the late classical and romantic eras required some kind of universal tuning system to make the large number of different instrumental sections manageable. Third, there are no jarringly unpleasant intervals in any key, and certainly no wolf notes.

Undoubtedly equally-tempered tuning would have been liberating for composers in the 19th and early 20th centuries who were looking to push the limits of what can be achieved with a 12-note chromatic scale. Freed from techical limitations, composers were able to concentrate on melodic chromatisicm to a greater extent than ever before. The rise of equally-tempered tuning took place side-by-side with the rise of the modern piano. The universality of the piano as a teaching instrument, and as a compositional tool, gave rise to a whole generation of composers and performers who had never worked in any tuning but 12-note equally-tempered. Novel and exciting orchestral works were produced prolifically during this period, by composers such as Debussy, Stravinsky, and Schoenberg.

The main problem with equally tempered tuning, however, is the same as its main advantage: all keys sound the same. Unless you are cursed with pefect pitch, a solo piano piece sounds exactly the same whatever key it is played in. A piece of music that modulates keys will benefit from the modulation, but the new key will still sound the same, so far as intervalic relationships are concerned, as the old one. With well-tempered tuning, a change of key from, say, C to E make a dramatic difference to the feel of the music. This key consistency is fine, of course, if you're playing music specifically composed during the early decades of the equally-tempered era, as this music will have been written to exploit what equal tempering has to offer, and may not even be playable in any other tuning. But an equally-tempered instrument is not ideal for playing music that was composed for meantone or well-tempered tuning. Not only will it not sound the same as it was intended to sound, but it will generally sound somewhat dull.

Contemporary composers have, from time to time, experimented with new tuning systems, but mostly these have been radically unfamiliar tunings, not subtle variations from what most people are familiar with. Such composers sometimes comment that we have done everything that can be done with a 12-note scale, and that the future development of music requires completely different scales -- 24-note scales, 56-note scales, or even no scale at all. But, arguably, the problem is that we have done everything that can be done as a 12-note equally-tempered scale, because it provides no key-to-key variation in musical flavour.

Closing remarks

Tuning is a much more complicated issue than most people, even most musicians, realize. Even within the confines of a 7-note or 12-note scale there are subtle variations on how notes can be tuned. It isn't true to say that one is 'right' and the others 'wrong', because there is no 'right' here. There are only different kinds of compromise. The larger the number of 'just' intervals we have in a particular key, the harder it will be for the instrument to play in different keys. The compromise made by equal tempering is that no interval apart from the octave is just, but at the same time no interval is a painfully long way off just, in any key.

It's interesting that skilled a capella vocal singers don't sing intervals in any particular tuning system. By the standards of a properly tuned piano, vocalists aren't even particularly accurate in pitch. So a particular singer may sing the interval C-E with a pitch ratio close to 5:4, then the interval F-A closer to the equally tempered ratio of 1.26:1. Moreover, a particular performer might well sing the same interval differently when pitch is descending, compared to when it is ascending. When singing in a group, vocalists seem to adjust their own tuning to a sort of group standard, that can vary during the performance. This all seems to suggest that the human ear has a certain tolerance for non-just intervals.

And finally: given that piano music has almost invariably been composed for equally-tempered tuning, and given that (by definition) all keys have the same interval relationships, why did so many composers of the classical era write solo piano music in keys like Db major and G# minor, which are awkward to read? A rational explanation is that these keys tend to lead to convenient finger positioning, and playing piano is all about getting your fingers into the right place. But I've heard some really way-out explanations as well. Other comments welcome.