First some definitions, to make history fit more easily into the framework of modern music.

A pitch ratio of 2/1, for reasons that will become apparent further on, is called an octave. This also applies to an interval that is approximately 2/1. Each octave is divided into 12 (not 8) equal, or approximately equal ratios called semitones. Each semitone is further divided into 100 equal ratios called cents. Cents are not used directly in music notation, but in determining or specifying pitch & interval accuracy.**

Two semitones form a tone, other intervals are named according to the span of natural tones. For example D & A span a total of five natural (white) keys, so this interval is called a fifth. A & D span a total of four white keys, hence this is called a fourth. A fifth actually spans eight semitones, a fourth spans six.

This strange, illogical naming has been foisted on us by history & makes slightly more sense if this is taken into account. From the pitch point of view a third has nothing to do with three, a fifth has nothing to do with five etc. It has all become part of musical jargon, mysterious to those not initiated.

History.

Pythagoras (who lived from 569 BC to 475 BC) is credited with establishing an arithmetical basis for music. During experiments with pipes & strings he discovered a precise correlation between length and pitch. Pipes produce different pitches when blown or struck. In both cases halving the length raises pitch by an octave.

He then found, using strings, that different notes played together sound pleasant (or harmonious) if the ratio between their pitches is a small whole number. The string's full length produces the fundamental pitch, halving it gives an octave, shortening it by a third gives a fifth, by a quarter gives a fourth, & by a fifth gives a third.

The Pythagorean scale, developed from these observations, is based on the perfect fifth, a pitch ratio of 3/2. This is then divided into semitones, to give a scale of 12 notes to the octave. A problem occurs, 12 intervals based in this way on the perfect fifth, do not exactly add up to a ratio of 2/1. What we now call an octave.

Starting at a particular note, say A, increasing the pitch by a perfect fifth 12 times in succession gives approximately an A seven octaves higher. It actually overshoots by 23.46 cents. This interval, nearly a quarter of a semitone, is called the Pythagorean Comma.

If the 12 notes obtained are successively multiplied or divided by 2, keeping within the 7 octave span, a 12 note scale is produced. This is the Pythagorean scale & works quite well for music in certain keys. The starting key & that a fifth above are fine. When playing in most other keys the comma causes increasing dissonance.

In attempts to combat this, a number of other tuning schemes have been devised over the years since. If the intervals used are small whole numbers, a requirement for consonance, then a comma remains. In Mean Tone for example, used during the renaissance, thirds are accurately tuned for better sound & power. This causes problems with other intervals & makes transposition difficult. Higher (wolf) fifths are particularly bad.

Two major developments were the 'pentatonic' & 'heptatonic' scales. These have been & are still used in various cultures. They use five or seven selected notes from the twelve note scale. Roughly equivalent to the black keys & white keys respectively on a modern keyboard.

The 'diatonic' scale ( Do, Re, Mi etc. ) is a heptatonic (7 tone) scale, used in Western music. The eighth note is approximately twice the pitch of the first, this is the origin of the word 'octave' for this ratio. between the eight notes are five large intervals & two smaller. The large intervals are called tones, the smaller, half the ratio, are termed semitones.

The earliest keyboards were based on this scale, having only one type of key, what we now call natural keys. It is likely that these were black, as wood was then easier to colour black than white. Music could be successfully played with this scale, our current stave notation is based on it. The comma still existed however, complexity of harmony is also limited, due to the missing tones from the Pythagorean scale.

The charming simplicity of early English music, madrigals etc. resulted from this limited scale. Eventually composers became more adventurous & were frustrated by these restrictions. Five additional keys were added between the whole tone steps, thus returning to the original Pythagorean scale. These extra (accidental) keys are set back, narrower, distinctly coloured & raised for playability.

The more complex music made possible by this extension re-emphasised the old problem with the comma. It was obviosly impractical to retune keyboard instruments for every key change. Due to bulk / cost considerations, multiple keyboards for varying keys were also not usually an option.

Over the years a number of alternative keyboard layouts have been tried. One technique is to add keys giving slightly different tunings, enabling more intervals to be exact. Such as paired black keys, one flat from the natural above, & the other sharp from the natural below. All suffer from added octave length & complexity.

Equal Temperament.

Eventually the equally tempered scale was devised, the one we use today. In this the comma is not eliminated, it is equally divided amongst the 12 semitones. Each fifth is made flat by almost 2 cents, so the full circle closes. Critics say every interval, other than each octave, is equally out of tune. This over simplification is true, but equal temperament has made three centuries of complex & beautiful music possible.

  ^{Just intonation.}  

Interest remains in 'just intonation', a type of scale where all intervals are pure, with no tempering or comma. Many attempts have been made to resolve the issues involved, with little success.

Years ago I worked out an alternative design, using large buttons rather than conventional keys, arranged in five rows. This has 22 keys per octave, is compact & has the advantage of a common chord fingering for a particular key. It is however difficult to imagine keyboard players taking to it. It reduces the number of dissonant intervals, but as with all other such schemes, some small commas remain.

Modern digital electronics has the potential to finally produce an instrument with no comma & no tempering. In which all intervals are exact, including the octaves. It need look no different to modern keyboards, & require no new playing technique. Playing together with current instruments may present problems however.

The principle uses the capability of modern computers to perform rapid calculations. Each note played is tuned to give exact harmonic intervals with those already playing. A possible problem, there is no exact standard tuning, every note played can vary in pitch to match those already sounding.

For example: If 'middle A' is defined as 440 c/s, a problem may occur if this note becomes, for example, an odd harmonic. It may need tuning away from 440 c/s. This will not be serious in staccato playing, but a legato style or overlapping sustained notes can lead to pitch drift. Each note played in succesion will be re-tuned to others already sounding, after many notes the base pitch may be some way from A 440.

This is another manifestation of the Pythagorean comma, & is easily demonstrated. Start with a low A, then whilst holding it, play the E above, a perfect fifth. Release the A, then whilst still holding E, play the B above, again a perfect fifth. Continue this legato run of perfect fifths, a total of 12 steps, you will now be playing an A seven octaves above the first. Whilst holding that A, play the A you started with.

It is obvious, from the arithmetic of music, that if every interval played is pure, the first A must now be 23.46 cents sharp to its original pitch. It may be said that this is an extreme case, not likely to be met in practice. With a system of perfect tuning, any deviation is undesirable.

The difficulty can be avoided by care in the writing & playing of legato pieces etc. This rather defeats the purpose of such an instrument, by limiting what can be played on it. A run of consecutive thirds, possibly the worst interval in the equally tempered scale, will soon result in an audible shift of base pitch. If the keyboard is played together with 'conventional' instruments it will be even more noticeable.

With really clever programming a computer can monitor this shift & constantly correct it, possibly by sliding the average pitch back to concert. This must be done so as to avoid the ear noticing it, but fast enough to handle rapid playing. A difficult compromise again, it is still just fiddling round a difficulty.

Properly implemented, it should be possible to play any piece of music, however complex, without the slightly impure tone & beats we have got used to. The purity of music from other cultures, but using our music.

Another approach is a more conventional instrument, tuned to the Pythagorean scale. Setting the key of this scale to suit the music, changing it whenever the score requires. This is an easy task for a computer based instrument, like most digital keyboards. It seems however a bit of an easy way out, not true just intonation.

For an acoustic keyboard instrument, even this limited approach requires considerable mechanical complexity. Lacking the elegance of most current instruments. It looks increasingly as though the entire concept is only suited to electronic instruments & the violin family.

Flutes can easily be made to a Pythagorean scale, but a different instrument is required for each key change, not an elegant solution. As for the piano, it is difficult enough to build in tuning stability with a fixed scale. A system of moving bridges or servo operated tuning pins seems to be storing up trouble. Electric string keyboards can have much lighter total tensile stress, reducing the load on key shifting mechanisms.

Shifting to a different set of strings will allow two different tunings with a practical instrument. Many pianos already have such a shifting mechanism, designed to strike either 3 or 2 strings (the una corda or 'soft' pedal). This can be modified to strike either of the outer strings, with the middle one left out from the triples. This will result in a quieter, different sound, probably with less sustain.

A further problem for just intonation with mechanical instruments is harmonic deviation. If a piano, for example, is tuned with pure intervals (for a specific key) between string fundamental pitches this is not the full answer. The harmonics are still out of tune, as with a tempered instrument. A compromise tuning is required, splitting the difference between harmonics.

If this is not done the result, with complex chords, will be multiple clashing beats, just as with the tempered scale. Balancing the harmonics may finish up with tuning similar to the way I do it. But without the versatility of equal temperament. It should be remembered that most struck or blown instruments exhibit this deviation.

Two strings, tuned accurately to an interval, still exhibit beats when plucked or struck. Even one string on its own shows beats. This also applies to struck plain bars or tubes. Blown pipes also show this effect, usually to a lesser degree.

So in starting a piece intended to support Just Intonation I seem to have finished up finding more problems. A system most suited to digital electronic keyboards & bowed instruments. The Theremin is also ideal for Just Intonation, as is the human voice. The situation is pretty much as it's always been.

My embryonic views on this matter are not cast in tablets of stone. Many have done considerable work in the sphere of Just Intonation. I will include links to sites providing useful information. This subject has engaged many minds for many centuries.

Incidentally, some of my work on stringed claviers touches on this subject. It is briefly mentioned in the section on Controlling Harmonic Deviation.

Michael Harrison is a Composer & Pianist. His work on just intonation spans many years & is best described in his own words. Click on his logo to view an interesting & informative site.

Kyle Gann is a Composer, Musicologist & Musician. This paragraph links to his informative site, a useful resource for musicians. Includes pages on historical tunings, Just Intonation & the history of pitch intervals.

To be continued.

Natural intonation, the scale of nature.

The 12 tone scale is a natural scale, Pythagoras no more invented it than Christopher Columbus invented America. In both cases it was a discovery, that had been made many times before. Pythagoras was the first man known to have set out the simple arithmetical nature of the subject.

To start, consider a simple regular shape, such as a straight rod, bar, tube or tensioned string. If made from suitable material it will respond to suitable stimulus, by exhibiting simple modes of vibration. When struck, a tone is produced, its pitch being inversely related to the length, divided by the speed of sound in the material.

If struck downwards, a wave of downwards movement travels down the length. At each end it is reflected back, as an upward movement. At one pitch, at which the length is half a wavelength, the waves add, maintaining vibration. A single, static wave is formed, alternately up & down.

This is termed a 'standing wave' for obvious reasons & only occurs at frequencies where one or more half waves fit exactly into the length. At other frequencies, any waves produced destructively cancel after reflection & don't last long. These transient frequencies are often termed 'strike tones'.

So such an object, common to most musical instruments, gives a lowest frequency, termed its pitch or fundamental frequency. Usually a number of other frequencies are produced, simple multiples of the fundamental. In most cases the fundamental is strongest, the multiples getting progressively less intense.

These multiples are called harmonics, & the range usually produced is termed a harmonic series. In modern parlance a tone at twice the fundamental is called the second harmonic, three times is logically the third, & so on. It was not always as logical as this, a few sad souls still stick to the old confusing terminology.

Of the harmonics the second & third are most important. Between them they create the twelve tone scale. Some of the other harmonics fit into this scale. Set a limit for the range of notes to match the span of human hearing. Limit it to allow harmonics to be heard. Seven octaves is enough for most purposes.

Take a pitch as a starting point, say 27.5 cycles per second, 4 octaves below our concert standard 'A'. Multiply it by 3 & divide the answer by 2. Do the same with the result a total of 12 times. Note each of the results, the final one is an 'A' 7 octaves above the first.

Each of the tone produced can be multiplied by 2, to give its second harmonic. It can also be divided by 2, to give the tone it is the second harmonic of. This can be done a number of times, stopping if the result goes outside the 7 octave range. At the finish the 12 tone scale is obtained, over 7 octaves.

Thus it can be seen that this scale exists in nature, Arguably it is the only musical scale that can exist, all others are subsets or elaborations. The diatonic scale, for instance, uses 7 out of the 12 tones. The Middle Eastern 24 note scale adds a note between each semitone. The resulting intervals are logically called quarter tones. They allow a greater degree of expression, often heard in authentic music of the region.

Comma & Natural Tuning.

The Pythagorean comma, 23.46 cents, see History.

The reason for the Pythagorean comma is that powers of 3 & powers of 2 do not produce the same numbers. Although at critical points they nearly coincide. So one can say that nature has built in errors.

There is an answer to this dilemma, let nature determine the scale. Making a decision that what is natural is not wrong may open the way to a pleasant sounding instrument. Surely this is what music needs?

Tuning meters & frequency counters will not give this kind of scale. Our ears can, an analogue oscilloscope can also provide very strong clues, to be any use its display needs to be meaningfully interpreted. 'Strobe' tuners can help to some extent. The best of these has a resolution of 1 cent, together with an accuracy of around 1 cent. Thus the potential error can be 2 cents, equal to the tempered comma.

Start with the modern tempered scale based on A 440. Set up a ring of fifths, transposed whilst tuning to stay close to this pitch. A region around the keyboard centre is now initially tuned. Tune octaves upwards & downwards from this area.

Tune higher notes to the second harmonic of the lower. Tune lower notes so their second harmonic zero beats with the higher. If excessive beating with the fundamental (harmonic deviation) shows, then compromise. Tune in between a dead beat fundamental & harmonic. Practice & a good ear will determine the best point.

To complete the task, check every fifth & twelfth. Fine adjust if required, checking the octaves again. Also double check using downwards fourths, these should sound similar to upwards fifths. Thirds, never very consonant intervals, must look after themselves with this method.

I mention the sound of fifths, fourths etc. This is deliberate, I never consciously count beats, contrary to the instructions in many tuning manuals. Instead I listen for the sound of a correctly tempered interval. This comes with experience, some have an ear for this naturally.

Sometimes a particular string, reed or tine etc. will not allow a pleasant compromise. It may be flawed, with structural irregularities, & require replacement. For instruments with complex resonator shapes, e.g. non-tubular bells, such deviation is part of their character. This should be allowed for in their playing.

Natural tuning seems at first glance to be a cross between Pythagorean & tempered. Actually it is closer to tempered, since it is built up from the tempered central region. Pythagorean simply 'dead beats' fifths, using the fundamental. Tuning progressively sharp at the upper end & flat at the lower is called stretch tuning. Natural tuning is slighttly different, starting from the tempered central octave and balancing harmonic differences.

Done properly, the result sounds harmoious, and in tune. Although a tuning meter or frequency counter may not agree. It is the method I use & has never produced any complaints. Some customers have commented that the tuning sounds mathematically accurate. I take this as a compliment, since many electro-mechanical instruments in the piano family have substantial harmonic deviation, due to their short resonators.

Being able to juggle conflicting requirements, to produce a good sounding instrument, is an art. The art of good tuners everywhere.

To be continued...

Harmonic Deviation.

What I call harmonic deviation is called 'inharmonicity' in America. In the context used here it is the error or deviation of harmonics from a true integer relationship with their fundamental pitch.

This deviation is common with many mechanical resonators but is unusual with electronically generated tones.

A simple mechanical resonator has one dimension greater than the other two, its length, this determines its fundamental pitch. For example a taut steel string may be 10 thousandths of an inch in diameter but 30 or more inches long

As described under 'Natural Scale' it responds to suitable provocation by exhibiting simple modes of vibration. A tone is produced, its pitch inversely proportional to the length, divided by the speed of transverse waye propogation in the string. The mechanism for this, and the effect of tension are described later.

The length of the resonator is half the wavelength of the fundamental frequency. Harmonics are usually also produced & sustained. These are pitches which are a simple multiple of the fundamental, i.e. where more than one half wavelength exactly fits in the length. The animation represents initial propagation of a 2nd harmonic wave. A more elaborate image will depict the actual movement of string deformation.

In practice harmonics are not as accurate as the simple theory predicts. Generally they are slightly sharp to the fundamental. In other words they are higher in pitch than they 'should' be. This out of tune effect is harmonic deviation, it can be quantified, usually in 'cents', for each significant (audible) harmonic.

A number of reasons are put forward for this deviation. I will cover each of them in turn, shorn of fancy mathematics. Complicated formulae are not needed for understanding or evaluating such causes.

The main (real?) reason is finite stiffness or rigidity of a practical string. Because of this a taut string, pushed at right angles to its length, does not bend abruptly at its termination points. Instead it describes a curve, the part closer to the termination moves less than the simple theory assumes.

This reduces the effective length of a string & is more significant with steel than with softer materials. The effect increases for thicker strings, everything else being equal. Harmonics, for a given amplitude, require a tighter bend, effectively shortening the string further. The higher the harmonic, the more pronounced is this effect.

Bass strings need to be thicker, to store enough vibrational energy to match the upper frequencies. Harmonic deviation is thus more pronounced, to an unacceptable degree. The solution adopted for steel strings is to add weight, giving more mass & thickness for a given stiffness. This is achieved by wrapping the vibrating length with a more ductile material such as mild steel or copper. An added benefit is a shorter string for a given pitch.

Equal temperament requires most intervals, other than octaves, to be tuned flat. This makes the effect of sharp harmonics more pronounced, especially noticeable in the mid-upper region. Take for example an interval of one octave plus a fifth, a frequency ratio of nearly 3 to 1. The upper note is about 2 cents flat, noticeable in this region. The 3rd harmonic of the lower note is sharp, say this is 2 cents.

The frequency difference is thus 4 cents, more noticeable, & actual harmonics are often more than 2 cents sharp. When the entire instrument is set up this way it will sound out of tune or dissonant. Despite the fact that a tuning meter will show it as correct. Stretch tuning is often adopted to improve the sound. With this technique strings above the middle section are tuned slightly sharp & those below are set flat. Thus the scale is 'stretched'.

This is usually done to a formula or table, determined experimentally for a given type of instrument. Effectively some or all the results of tempering are undone. The interval above (a twelfth) may be better, but all the octaves are out, the 2nd harmonic is not as sharp as the 3rd, a compromise is called for.

Even the 'accurate' octave suffers from harmonic deviation. This is less pronounced as it is not aggravated by flat tuning.

Some advanced strobe tuners have a set of 'stretch tuning' tables programmed in. Suitable for obtaining average results with a variety of instruments. Every instrument is slightly different , especially when pianos are considered. Strings are different even within one instrument. For this reason I do not favour the 'stretch tuning to a table' approach, using instead what I call natural tuning.

This method is covered above, some may be puzzled by its obvious similarity to stretch tuning. There is however a subtle difference, natural tuning is not done to a table or formula. Notes are tuned to one another, using their own harmonic series, employing judgement to compromise where needed. It is all compromises in any case, the trick is making it sound good, in the end this is what matters.

As already noted, harmonic deviation is common to many mechanical resonators, but is not usually exhibited by electrical resonant circuits. Therefore instruments using such technology do not require stretch tuning. Electronic instruments using binary dividers cannot, by definition, be stretched in any case. This also applies to those using binary multiplication (there are some).

More soon...

Controlling Harmonic Deviation.

In 2001 I started some initial design work on electric stringed claviers, in the process touching on this subject. It is possible to manipulate the deviation of specific harmonics, in either direction. In my case the original intention was to make a single string sound like two or three. It worked very well, showing that controlling harmonic deviation is possible.

In the past others have tried small masses, added to a string at a harmonic antinode. This can work, slowing wave propoagtion more for that harmonic than for the fundamental. An unwanted side effect is that small waves will reflect both ways from the discontinuity. Adding two other frequencies, with the weight at a node, these may have even more deviation.

Way back to the early days of claviers, a lot of care has been taken to eliminate string irregularity. A high standard of precision was attained quite early on, by the best companies, maintained & improved since. This is a skill that has been passed down over many generations.

An obvious way of improving flexibility may have undesirable side effects. Making a string from a number of twisted strands is an old idea, improving flexibility whilst maintaining tensile stregth. This will reduce harmonic deviation, but the string's plane of vibration will tend to slowly rotate. Resulting in pronounced amplitude modulation, also inter-strand friction will absorb energy, reducing sustain.

My work involves several factors, including bridge design, this is able to increase deviation more easily than reduce it. Back as far as the 19th century, experimenters have tried ideas such as clamping strings. the most promising technique involves control over wave propogation rates in strings. This works well to provide deliberate negative deviation.

When time permits, further tests will determine whether reliable exact cancellation is practical. One idea is to introduce controlled harmonic sharpening, swamping strings' variability & balance it with controlled flattening. We will see.

Piano Tuning, Harmony & Tone.

The concert piano-forte is a complex instrument, there is more to its tone & sustain than initially meets the eye. Most of the 88 hammers strike three strings together. The vibration of each of these strings reinforce one another, improving tonal quality, increasing volume & sustain. This is partly responsible for the sound we recognise as a piano.

For this to happen the three strings are tuned as precisely as possible to the same pitch (frequency), so they vibrate essentially as one. If the relative tuning is slightly off, a complex interaction takes place. Between the strings, bridge & soundboard. Strings vibrating at different speeds cause a twisting stress in the bridge.

This stress couples the strings, as does air displacement. After a few cycles of vibration they tend to pull into 'sync' with one another. This pitch shift causes the initial 'twang' common with 'honkey-tonk' or bar pianos. Another, more serious problem is the changed modes of vibration due to bridge coupling.

Less energy is transferred into the bridge-soundboard combination if the strings are in anti-phase. So they tend to settle into this mode, if only two strings are struck, they tend to vibrate in opposite directions. This causes cancellation of the fundamental pitch, leaving a weaker tone, with more second harmonic. With three strings, two may vibrate in phase, or all three with different phases.

Naturally this has a disturbing effect on the piano's tone. If one trio of strings is out in this way, others are probably in a similar state. The interaction between strings also dissipates energy, so sustain is often reduced. Such a piano will not be satisfying to play & will not do justice to the pianist's virtuosity.

There is more: A piano exhibits complex reactions & sympathetic resonances between many of its strings. Especially those that are harmonically related, this is particularly noticeable when the damper or sustain pedal is depressed. These interactions contribute to the piano's sonority & character.

If strings are out of tune they will respond less to the vibration of other strings. The piano will sound dull and less interesting. Any instrument with many strings thus requires more care, skill & experience in tuning.

There is another mechanism affecting strings vibration, regardless of tuning accuracy. Bridge pins cause a horizontal stress on strings zig-zaging between them. This makes movement in this direction easier. The vertical movement imparted by the hammers tends to change to vibration parallel to the soundboard.

It must be borne in mind that there is more to tuning than just getting the strings into accurate pitch. Each string must also be accurately 'set', this is part of a good tuner's art. If not properly done, a piano may not stay in tune even during one concert. A single Fortissimo passage may be enough to undo the tuner's efforts.

Another effect of strings being incorrectly set is detuning of the 'Duplex Scale'. Unbalanced tension causes these free sections to introduce dissonant overtones into the sound. The piano sound is less harmonious, but it may not be obvious why.

Detuning.

The result of substantial detuning between piano strings is quite different. Usually the instrument sounds atrocious & more or less unplayable. Deliberate careful detuning, (or a happy accident) can produce an interesting result, usually best with a small bright instrument.

A 'Honkey-Tonk' piano has a bright, thin sound, unsuitable for slow or serious playing. Its strings do not resonate together & 'help' one another. It can sound good with fast, bright, happy dance music etc. To get this effect one string in each tenor triad is set sharp or flat enough to defy the synchronising effect, producing a distinct 'beat'. If the strings are old, flat tuning is safer.

In some cases a stronger effect is obtained by detuning one more string, by an equal amount in the opposite direction. The centre string is kept at the base pitch. Hardening the hammers, to produce a more 'tinkly' sound, also helps. Such tricks may upset many 'purists' & are best not performed on a top class instrument.

It was said that a popular performer in this style of music put drawing pins (thumb tacks) in her piano's hammer tips. This shortens the life of strings, especially those tuned sharp. They are already above their designed tension. The shock stress may also accelerate hammer pivot wear.

I intended to write a section on tuning, then I found this site. I can only bow to a master:

Learning how to tune your own piano serves several purposes, developing an insight into its inner intricacies. Selecting this text link leads to an excellent site, an on line down-loadable book, by Mr. Chuan C. Chang. Teaching playing technique, theory & the elements of tuning etc.

This text link will take you straight to the 'Tuning' chapter.

Versions other this book in other languages:

French. This version includes a browser trap, depending on your browser, you may need to close it to escape.

When I have time, I will include more information on this unique man and his books: