Note: This page has been printed from the Musical Tuning and Temperaments web site which contains a 'Musical Calculator' diagram in the lower frame of the page. All references to 'the diagram' refer to this musical calculator.

Since the early 1900s, almost all Western keyboard instruments have been tuned to Equal Temperament, a system in which all of the notes of the chromatic scale are spaced equally between the tonic and its octave. This has led to a common misconception that such a tuning is 'right'. In fact, from a musical perspective EVERY note is in the wrong place and the great composers for these instruments such as Chopin, Liszt, Beethoven and Bach, whilst admiring the quality of modern pianos, would wonder why they were all out of tune!

The exact positioning of the notes within the scale is a complex subject with a long history, especially where the tuning of fixed pitch instruments such organs and keyboards is concerned. This site aims, to provide an introduction to the topic with the help of the interactive diagram at the bottom of this page.

To start at the very beginning, it will be instructive to question the whole concept of notes and scales. Nature produces sounds at every pitch, such as the wind blowing through the trees or the human voice, but musical instruments and almost all modern music is based on sets of discrete notes. In looking at how many notes there should be and how they should be spaced, the driving force turns out to be the ratio between notes. Simple ratios, 2:1, 3:2 etc are pleasing to the ear, complex ratios 78/34 are not! The simplest of the ratios 2:1 (or a doubling of the pitch) produces an octave and it is one of the few almost universally agreed facts that the octave and its tonic must always be in the ratio of 2:1. Since the octave has its own octave at double its frequency, the spectrum of notes can be divided into octave-wide scales which will all have identical properties except for the doubling of pitches.

Music made up simply of octaves would be rather tedious, and so the next task is to divide the scale into a set of discrete notes, and this is where the fun starts!  Once again the ratios between notes come into play and the next simplest ratio, 3:2 gives us a tonic and its 5th which is another 'pleasing' interval and must appear in our new scale. If we make two more seemingly sensible assumption, that all of the notes in the scale should be evenly spaced out and that we need no more than 20 notes, then something very interesting happens. Some simple mathematics will show that there are only two possibilities for the number of notes in a scale, 12 and 17. No other number of notes produces a scale with a 5th in it. Western music has settled on a 12 note scale, both because its 5th is closer to the 'natural' ratio and music based on a 12 note scale is easier to learn and play. However, for anyone interested in a musicological diversion, a move to a 17 note scale (with enharmonics such as Ab and G# seen as separate notes) was seen as the next natural progression from atonal music in the early 20th century and had quite a following.

A 12 note chromatic scale may seem an attractive route to follow, but, and its a big 'but', the 5th does not lie exactly at the same pitch as the 9th note of this evenly spaced scale, even though it is so close that our ears cannot tell the difference in a line of melody. Other simple ratios such as the major 3rd (5:4) are so far out that our ears can detect a distinct difference. This evenly spaced chromatic scale is therefore only an approximation.

To illustrate this problem another way, there are two ways to find the major 2nd of the scale, based on intervals with simple ratios. The first is to go up a 5th and the back down a 4th. The second is to go up a major 6th and back down a 5th. The problem is that these two ways arrive at notes of slightly different pitches. Though this seems odd at first, both notes are right in their own way because it is the interval that is important, not the exact pitch of the note. A good musician will naturally flatten the major 2nd when descending from a 6th to make the interval of a 5th correct. This is why music teachers often want particular notes to be slightly sharpened or flattened. Once again, the conclusion to draw is that there is no 'right answer' when deciding where every note of the scale should sit.

If we accept that it is intervals that we should concentrate on then musicians will have to bend the actual pitch of some notes to make the music sound right. This slight bending of notes is possible with the voice, on string instruments and on most woodwind and brass instruments. The problem comes with fixed pitch instruments such as pipe organs and keyboard instruments. These instruments have to be tuned to one, and only one pitch per note and it is the choice of these pitches that defines each 'Musical Temperament'.

Many different tunings and temperaments have been used through the ages and the chapters that follow will examine the important ones. Before we start, however, it is useful to define the difference between a 'tuning' and a 'temperament'. In a tuning, the notes are tuned using a particular method, based on the ratios of pairs of notes. The notes are in the right place, in theory, and tunings are normally good for playing in the tuning key and for playing the type of music they were intended for (medieval, classical etc.). However, they usually have significant problems either with particular notes in the scale, or with modulations to other keys. A temperament, on the other hand, starts with a tuning and then 'tempers' it by shifting some of the notes slightly from their natural pitch to improve problem intervals or problem keys. The most radical of all of the temperaments is 'Equal Temperament' which shifts every note from its natural position, but creates a set of scales which are all equally good (or bad depending on your point of view) and which allows modulation freely between any keys.

The Musical Calculator at the bottom of the page will be used to illustrate the features of various musical tunings and temperaments. The calculator may be controlled directly using the Control Panel on the left, but clicking on 'special' text will set it up in a particular way. For instance, click here to show Equal Temperament, relative to itself. The calculator should now be displaying what is called Equal Temperament, showing a set of major scales, with each note displaying its note name. Whenever you come across an underlined section in the text, click on it to illustrate the current point.

If you clicked on the special text above, the calculator will be showing a series of scales centered on the home key of C Major. The scales are laid out like a flattened circle of fifths with the 'near' keys next to the home key and 'far' keys further away. Clicking on any of the key names in the left-hand column will make that the new home key. Try clicking on F#, the top-most key currently displayed. You should see this become the home key, with the keys then ranging from C Major through to B# Major. As in a standard circle of fifths, the diagram will not go further than B#, so clicking C# Major will display its enharmonic key, Db Major. (Practice moving around between the Major keys.)

The relative minor key is shown in the right-hand column. Clicking one of these keys will make it the home key, and display the minor scale, rather than the major. Shifting between major and minor is also possible via the radio buttons in the Control Panel, also offers a third option of displaying the full chromatic scale.

A tuner of an organ or keyboard has to start somewhere, and this 'musical calculator' assumes the key of C as the starting point. Even if the home key is then changed, the tuning will still be relative to the tuning key of C. This simply mirrors the reality that a keyboard cannot be re-tuned on-the-fly and therefore regardless of the key you are playing in, the keyboard will remain tuned to C.

Up to now, the musical calculator has simply shown scales and the relationship between them. Clicking on Cents/Tonic in the Control Panel will show the pitch in cents for each note. (If you are not familiar with the concept of 'cents' read the chapter Ratios and Cents before continuing with this chapter) Since the musical calculator is currently showing Equal Temperament, each note is evenly spaced, 100 cents apart, and the diagram shows that this is true for every key (the big advantage of Equal Temperament).

Every other temperament or tuning leads to a situation where the pitch of intervals is different in different keys. Natural Tuning shows this very well. The note F, the 4th in our tuning key of C, has a pitch of 498 cents, relative to the tonic, but the Bb in the key of F, its equivalent 4th, has a pitch of 520 cents relative to its tonic, a difference which is very audible! This is an unfortunate consequence of Natural Tuning which is explored in more detail in its own chapter, but it is clear to see that many intervals are different in the different keys. What takes a bit of working out is how different each interval is. To make this easier to see, there is a useful option Cents/Tuning Key which shows the interval in cents relative to the same interval in the tuning key of C. In our previous example, it is now easy to see that the F-Bb in the key of F is 21 cents sharp, relative to the C-E.

This difference between intervals in the different keys is going to be crucial to much of the discussion of temperaments and tunings, because it will define how useful a particular tuning or temperament will be. Large differences will render some notes (sometimes whole keys) unusable, which may or may not be important depending on the style of music. To make this point as clear as possible, each of the notes in the diagram is colour coded, according to how far out it is relative to the same interval in the tuning key. Sharp notes go from a dark red, for relatively benign differences, up to 'white hot' for unplayable notes and flat notes go from a relatively okay light-blue through to black, representing notes which should be banished to the outer darkness! If the note is green, there is no difference in the interval and it will sound exactly the same as the equivalent interval in the tuning key

One further useful facility, hovering over a note will display a popup showing all of the detail for that note.

Lastly, a note about enharmonics (if you'll excuse the pun). Looking at note names, the 4th in the key of Db is a Gb. On a keyboard it is obviously the same key as the F#, an enharmonic. However, the way in which pitches are derived for the different tunings and temperaments will lead to a situation where the pitch of these two notes will normally be different. Since the diagram simulates the keys on a keyboard, only one of the enharmonics can be used, and the Gb is actually a F#. The reason for this is that the order in which the keys are tuned goes up and down in 5ths from C, (C,G,D,A,E,B,F#,C#) and (C,F,Bb,Eb,Ab), producing the chromatic scale:

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

All pitches are those of the notes in this chromatic scale, even if the theoretical note name would be an associated enharmonic.

Pythagorean Tuning is probably the earliest organised form of tuning, credited to the Greek philosopher and mathematician Pythagoras, though no writings survive from that time to substantiate this. There is documentary evidence for use of Pythagorean Tuning in the 9th century and it was the predominant method of tuning organs and keyboard instruments through to the 15th century.

The method tunes using the interval of the 5th, formed by the simplest ratio 3/2 (702 cents). The 5th is an easy interval to use since it produces very pronounced 'beats' when slightly out-of-tune, which become slower as the note approaches the perfect ratio. Using this method it is possible to tune a succession of 5ths C-G-D-A etc. and, by moving down an octave at appropriate points, it is fairly simple to tune each of the notes of the scale.

The finished effect can be seen to have each 5th at 702 cents with one rather significant exception. Unfortunately, starting at C and moving up 12 perfect 5ths, we reach B# at 8424 cents (702x12) which should be the enharmonic of C, 7 octaves higher at 8400 cents (1200x8) but clearly it is 24 cents sharp. This gap is known as the Pythagorean Comma and shows up in our circle of 5ths where one of the intervals G# to Eb, has to be 24 cents narrower. the so-called 'wolf interval'.

Looked at another way, the Pythagorean Eb forms a pure 5th in Eb-Bb but is 24 cents flat of the note D# we would desire for the pure 5th G#-D#. Likewise G# is perfectly in tune for the 5th C#-G# but is 24 cents too sharp for a pure 5th with Eb (Ab-Eb). Thus, on a 12 note Pythagorean keyboard tuned with a chain of 5ths from Eb to G#, the 5ths Ab-Eb and G#-D# are not available, although the notes Eb(D#) and A(G#) are both very useful in other 14th century sonorities. Fortunately, Ab-Eb and G#-D# doesn't occur often in pieces from this era so the problem is mostly academic when playing music for which the tuning was intended.

For the purist, the Pythagorean Comma is 23.46 cents, based on a 5th of 701.95 cents but almost all of the literature uses 24 cents, based on a rounded up value of 702 cents for the 5th. If you place the mouse over the Eb you will see that it is actually 23.46 cents flat.

Even though Pythagorean Tuning contains wolf intervals, it may be seen that all of the keys from Bb to E behave identically allowing free modulation between them, which is one test of a 'good' tuning; though switching to the minor, modulation looks less attractive!

Looking at Pythagorean Tuning relative to Natural Tuning shows another very interesting effect. The Major 3rd and the Major 6th are significantly sharp, relative to the natural ratios of 5/4 and 5/3. Though this is a problem for modern music (Baroque onwards), where consonant major 3rd are expected, this was a positive advantage for medieval music, where tense or active 3rds asking to be resolved, were the order of the day. It is interesting to compare Pythagorean Tuning with today's Equal Temperament and note that the differences are not so great. The conclusion to be drawn from this is that, by accident, Equal Temperament is good base for playing medieval music

One last useful element to be drawn from the 'musical calculator' concerns the size of semitones. Looking at the pitches of Pythagorean Tuning, it will be seen that the scale is made up of a set of tones, each 204 cents wide and two 90 cent semitones to complete the 1200 cent scale. This is fine except that the tone is much larger than two semitones, so looking at the full chromatic scale, it may be seen that there have to be two types of semitone, one 90 cents wide and one 114 cents wide.

The ability to modulate to near keys, the perfect 4ths and 5ths and the sharpish major 3rds were all factors which made Pythagorean Tuning a good choice for medieval music. It was only in the 15th century as composers began to want more consonant 3rds that its use waned, in favour of Meantone Tuning.

Natural tuning, sometimes called Just Tuning or Ptolemaic Tuning is something of an oddity in that it has some attributes which would commend it as the best possible tuning but its drawbacks are so great that it is never used in practice. It has academic significance, however, as a tuning against which other tunings should be compared. If we take Pythagorean Tuning with its perfect 5ths and detune the D-A so that it is 22 cents flat, we get a remarkable effect, Natural Tuning. Looking at the tuning key of C, there are perfect ratios for the minor 3rd (316), major 3rd (386), 4th (498), 5th (702), minor 6th (814) and major 6th (884). The table in the chapter on Ratios and Cents will confirm these ratios and pitches. In fact, every simple ratio is concordant and so this looks like a perfect contender for 'Tuning of the Millenium'.

Unfortunately, the first problem with Natural Tuning is obvious from the colours in the chart. Modulating to any key except the tuning key and its relative minor produces unacceptable intervals. In the major keys, there are 3rds which are 41 cents sharp (pretty well 4ths) and the minor keys are just as bad.

A second problem, which may be even more significant is that there are 2 different tone intervals. The interval between the 4th and 5th is 204 cents, but the interval between the 5th and the 6th is only 182 cents. Examining the chromatic scale shows that there are also 3 different sizes for semitones. This make Natural Tuning useless in practice, since as a melody progresses through a set of intervals, it will quickly reach a point where the notes do not correspond to the original notes of the scale, due to going up one size of tone and back down another. (It would make Schoenberg appear positively tuneful!)

In conclusion, Natural Tuning is an academic oddity, but it is very useful as a test of other tunings. Because it is perfectly concordant in its tuning key in the notes which modern music requires to be concordant, it will be a very useful test of other tunings and temperaments to see how much they differ from Natural Tuning. For this reason, there is a button on the Control Panel below which shows the selected tuning relative to Natural Tuning.

Pythagorean Tuning produces an almost perfect set of fifths. The diagram below shows all of these 5ths at 702 cents with the exception of one, which has to be at 678 cents to close the circle of fifths. Looking at this relative to the Just Intonation this wolf note is 24 cents flat, a gap known as the Pythagorean comma. The other key feature of Pythagorean Tuning, is the sharpness of the 3rds. Again, relative to the natural ratio of 5/4, the Pythagorean 3rds are 22 cents sharp. This 22 cent difference is known as the syntonic comma (or the comma of Didymus)

The syntonic comma was seen as beneficial in medieval music where 3rds were not meant to be consonant. However, the baroque style demanded 3rds in their natural ratio, so the syntonic comma had to disappear. In the intervening centuries, the two basic methods have been to detune all of the 5ths equally, creating regular temperaments or to share the comma amongst a few of them, creating the irregular temperaments with 5ths of two different sizes.

The regular temperaments are usually specified by the fraction of the syntonic comma that is used to reduce each fifth. Thus, Meantone 1/4 comma reduces each 5th by 5.5 cents (22/4). It may be seen in the diagram that this creates a set of perfect 3rds, in the near keys, but renders the far keys useless. It may also be seen that wolf note 5th present in Pythagorean Tuning at 24 cents too small is now worse at 36 cents too large.

Is it possible to get the benefits of the perfect thirds without the drawback of the wolf 5th? The answer is yes, and the method is to flatten some of the 5th, but not all. Temperaments which use this solution ar known as 'irregular' since they lead to two different sizes of 5ths. One example, Kirnberger III uses the 1/4 comma of the previous example, but only the 5.5 cent flattening to 4 of the 5ths. The diagram shows that the wolf has disappeared, but at the expense of the 3rds which are still relatively consonant in the near keys, but drift back to the syntonic comma in the far keys.

As a last example, if the syntonic comma is spread evenly across all of the 5ths, we have Equal Temperament. The 5ths are now all 2 cents flat, indistinguishable from the natural 5th to the human ear. However, the 3rds are all 14 cents sharp; an acquired taste, but given that this has been the norm for all keyboard instruments since the early 20th century, probably one which we have now acquired.

In conclusion, the Pythagorean comma is an unwanted, but natural outcome of musical geometry. Its partner, the syntonic comma, started as a 'feature' but quickly became a 'bug' (as the computer programmers would have it) and the history of tunings is very much about the reduction of the syntonic comma to acceptable levels, with the removal of the Pythagorean comma being a useful by-product, if possible,

A defining feature of Renaissance music was the move away from the active 3rds of Pythagorean Tuning to consonant 3rds at their natural pitch ratio of 5/4 relative to the tonic or 386 cents. To achieve this Meantone tuning was introduced in the early 1400s and remained the primary method of tuning for the next 250 years. The basic idea was to reduce the syntonic comma, the difference between the natural 3rd and the Pythagorean 3rd, by detuning the 5ths, aiming to make the 3rds as consonant as possible without the 5ths becoming dissonant.

The most obvious choice, known as 1/4 comma Meantone reduced each of the 5ths by one-quarter of the syntonic comma (5.4 cents) which creates a set of perfect 3rds. Meantone, because every 5th is reduced by the same amount and therefore the 3rd is always made up of two equal tones, as can be seen if the pitches are shown relative to the tonic.

1/4 comma Meantone, sometimes called Aaron's Meantone or just plain Meantone has perfect 3rds in the near keys and seen relative to itself allows free modulation to near keys which all have identical properties. One drawback, however is that the wolf, which was -24 cents in Pythagorean Tuning has worsened to 35 cents in 1/4 comma Meantone.

A further drawback is that the human ear is more sensitive to poorly tuned 5ths than it is to 3rds and making the 3rds perfect, at the expense of a 5.5 cent reduction in the interval of the 5th is not the best compromise. The alternative is to use different fractions of the syntonic comma and the following table, along with the diagram, shows their properties:

A number of interesting facets become apparent. Although 1/4 comma has the best 3rds, 1/6 comma is arguably the best compromise. The wolf note can only be reduced at the expense of poorer 3rds. However, 1/8 comma Meantone is considered 'well-tempered' since the wolf is less than 10 cents, though the 3rds are too active for Renaissance and Baroque music. Something special happens with 1/11 comma Meantone. The wolf disappears completely and this tuning turns out to be the same as our modern day Equal Temperament, for reasons which should be obvious when seen relative to itself.

By the late 1600s, composers were beginning to make much greater use of transposition between keys and the existence of wolf notes in tunings such as 1/4 comma Meantone were becoming unacceptable. However, the need remained for consonant 3rds and 5ths a seemingly insoluable dilemma. The only answer was to compromise and the compromise chosen was to flatten only some of the 5ths rather than all of them. If we take an early example French Ordinaire it may be seen that this creates two different intervals for the 5ths, with some 5 cents flatter than the natural pitch and some 5 cents sharper. It has reduced the wolf to 14 cents, from 36 and kept a significant number of the 3rds at their perfect ratios.

Such temperaments are not meantones, since a further effect is to create different size tones, an example being the key of Bb with its major 2nd at 204 cents and its major 3rd at 397 cents (only 193 cents higher). Instead, with their attempt to resolve all of the above needs, these temperaments are called 'well-temperaments'

Two further examples, Kirnberger III and Werckmeister spread the syntonic comma (22 cents) and the Pythagorean comma (24 cents) respectively, across 4 intervals (each choosing a different set of intervals) removing the wolf altogether at the expense of some sharper 3rds.

Lastly, in this set of examples, Vallotti/Young spreads the Pythagorean comma across 6 intervals (4 cents) to remove the wolf, creating a narrower range of 3rds in the process.

The is one further, very important difference with the well-temperaments which has not revealed itself so far. If 1/4 comma Meantone and Werckmeister are compared not with natural tuning but relative to themselves, a very interesting difference appears. The meantone tunings are identical in each of the playable keys but the well-temperaments are different! This means that modulating to different keys will not only have the effect on the listener which we recognise today, but there will be a 'colour' difference as well with different interval sizes in different keys. To look at this a different way, a piece of music may be composed for a particular key because of the way that key sounds, relative to others even with the same tuning.

This takes the discussion neatly to J.S. Bach and the Well-Tempered Clavier. This set of keyboard pieces is deliberately written in each of the 24 major and minor keys. Because music was Bach's medium, rather that words, he neglected (or didn't see the need to) write down two very important facts. The first is that he doesn't tell us which of well-temperaments he used, though the smart money appears to be on Werckmeister. The second and much more fascinating fact, which remains the subject of academic debate to the present day, is whether he wrote the 24 preludes and fugues as a celebration of the fact that these keys were available on one 'clavier' or wether he was celebrating the subtle differences in colour created by the new well-temperaments.

Equal Temperament gradually took over from the various 'well-temperaments' at the end of the 19th century and has been the 'standard' tuning system for keyboard instruments ever since. Each semi-tone is exactly 100 cents wide, producing a tuning system which is identical in all keys, allowing free modulation to any key.

However, the history of equal temperament goes back much further than this. It was accepted as the standard tuning for the lute by the mid 1500s, as a compromise solution, but really the only way to tune a fretted string instrument. There is a written reference by Abbott Girolamo Roselli in 1588 and therefore it is appropriate to ask why it was not adopted for keyboard instruments until much later.

Part of the answer is that it is more difficult to tune keyboard instruments to Equal Temperament than to Pythagorean Tuning or Meantone tunings, but a more convincing answer can be seen if Equal Temperament is viewed relative to Natural Tuning. The 5ths and 4ths are good, but both ,major and minor 3rds are significantly different to the natural ratios, which would have been unacceptable in Baroque and Classical periods. A rather more contentious difference can be seen by comparing Equal Temperament with a well-temperament. Looking at the tunings relative to themselves, Equal Temperament is identical in all keys but Werckmeister has different intervals in different keys, 'colouring' the keys. This move towards homogeneity has not always been viewed as an advantage! (Dumbing down 19th century style?)

One last thought. It is a strange coincidence that after a journey through a plethora of meantone tunings and well-temperaments, we have arrived in the 21st century at a tuning, in Equal Temperament, which is relatively close to the Pythagorean Tuning of Medieval times. Looked at relative to each other, the 5ths are extremely close and the 3rds are both sharper than their natural ratio and relatively close at only 8 cents apart, so modern Equal Temperament is better choice for playing Medieval music than all of the tunings and temperaments in between.

An understanding of the basic idea of 'cents' is absolutely necessary, since its is used throughout this, and other, descriptions of musical tuning and temperament. However, for those who are somewhat shy of mathematics, I have presented this chapter in two parts: an initial description which tells you all you need to know to read the rest of the chapters and then a more mathematically rigorous description for those interested.

Imagine a ruler which could be used to measure intervals. It would make sense that this 'ruler' was one octave in length so that we could use it to measure all of the intervals within an octave. The ruler will need some markings on it, in the same way as we have centimeters or inches on a real ruler. Since there are 12 notes in a chromatic scale, lets go from zero, at the tonic to 1200 at the octave. In that way, we will have a note roughly corresponding to each 100 marks (which is why they are called cents).

If we look at Equal Temperament each semitone falls exactly on a 100 cent mark, which is what defines Equal Temperament.

The really important thing about 'cents', as opposed to frequencies, ratios or any other way of measuring pitch is that we can simply add and subtract them. If we consider the chromatic scale of C in the diagram, the 5th is an interval of 7 semitones or 700 cents. If we go up a further major second (200 cents), then we arrive at a 6th which is the 9th note of the chromatic scale and has a pitch of 900 cents (700+200). Come down a perfect 4th (500 cents) and we get to a major 3rd at 400 cents (900-500).

In the Introduction, it was asserted that Equal Temperament is an approximation and, in musical terms, the 5th is really to be found at a pitch ratio of 3/2 relative to the tonic (or 1.5 times the pitch if you prefer). Using the mathematics to be found in the second half of this chapter, it turns out that this is 702 cents. Given that we can safely add and subtract cents, this means that the 5th, on a keyboard tuned using Equal Temperament, is 2 cents flat. As a rule of thumb, we cannot detect differences of less than 5 cents, so it not just "close enough for jazz", its perfectly okay for classical music as well!   If we now consider Natural Tuning, which is based on naturally occurring ratios and so should be the 'most agreeable' tuning, and concentrate on the tuning key of C, the chromatic scale shows that none of the notes lie on the 100 cent marks. Unfortunately, this Natural Tuning has many other problems which make it unusable in practice. (See the chapter on Natural Tuning.) In fact, if you randomly switch between the various tunings and temperaments, you will find that the only thing that remains constant is that the tonic is always zero, and the octave is always 1200. This underlines the point made in the introduction that the octave is always fixed at twice the pitch of the tonic or 1200 cents higher, using our 'interval ruler'.

The difference or interval between two notes is always specified in terms of a ratio of their frequencies, rather than simply subtracting the higher frequency from the lower. The reason for this is that the interval of a 5th between the A at 440Hz and the E at 660Hz is the same, in musical terms as the 5th between the A at 880Hz and the E at 1320Hz. In both cases, the ration is 3/2 even though the differences in frequency are 220Hz and 440Hz respectively.

It is, of course, mathematically possible to just work with ratios. However, working with ratios has a number of problems. The first is that adding and subtracting ratios is a 'bit of a pain in the neck'. If you don't believe me, try the fairly simple task of working out the size of the major second (as a ratio) formed by going up a major 6th (5/3) and then descending a 5th (3/2) and then working out the difference between that and the equivalent major second formed by going up a 5th (3/2) and down a 4th (4/3). The answer is 1/72.

A second problem is seeing easily which is the greater of two ratios. It is fairly obvious that 3/2 is a higher pitch that 4/3 but what about 6561/4096 and 729/512, which appear in Pythagorean Tuning. (One is an F# and one a G#!)

Since ratios are the 'currency' of intervals, the way to simplify matters is to use logarithms. The key fact about logarithms is that multiplication and division of numbers is the same as addition and subtraction of their logarithms. So, if we work with the logarithms of the ratios we will be able to simply subtract the two numbers to find the interval. Since the range we are working in is an octave, and the octave is twice the tonic, base2 logarithms are the answer because this will give us numbers between 0, log2(1/1) and 1, log2(2/1). As a further refinement, multiplying the logarithm by 1200 will allow us to work in integers most of the time, with the tonic at 0 and the octave at 1200. In a 12 note chromatic scale, this makes each semitone interval about 100, hence the name 'cents'

Put rather more mathematically, we define an interval in 'cents' as:

Interval = log2(Note Pitch/Tonic Pitch) x 1200

For some of the common intervals, we get the following for the key of C:

This agrees with our musical calculator for Natural Tuning,

To demonstrate the ease with which we can work with cents, let us return to the problem outlined earlier. The major second formed by going up a major 6th and then descending a 5th is 884-702=182. The equivalent major second formed by going up a 5th and down a 4th is 702-498=204. Therefore the difference is 204-182=22 cents, an audible difference

The following sites provide more information on this fascinating topic and the author would like to acknowledge many of them as sources for the material in this site:

• Margo Schulter Pythagorean Tuning and Medieval Polyphony (comprehensive and very readable).
• Arthur Benade, Fundamentals of Musical Acoustics (great book)
• Kyle Gann An Introduction to Historical Tunings
• Bradley Lehman Keyboard temperament calculator/analyzer
• Pierre Lewis Understanding Temperaments
• Huygens-Fokker Foundation Tuning & temperament bibliography
• The Tuning and Temperament lists at Yahoo, - dmoz open directory project - Alta Vista

About the author: I am a software developer by profession, and also an amateur cellist, living in Manchester, UK. I remember being told by my cello teacher that C# and Db were different notes and since I had no frets, I should think and play intervals, not notes. However, this only scratches the surface and it was only when I delved more deeply that I began to appreciate just what a fascinating topic this is. This website is a result of that voyage of discovery. If you have any comments on the content, corrections or ideas for improvement please email them to: Chris Tyler, comments on this website

http://www.ppexpressivo.co.uk
Last revised: 23rd Feb 2003

For those wondering about the website name. It comes from a joke, often told by violists to get their own back:

How do you get a cellist to play forte? Mark the cello part "pp expressivo".