Harmonics and just temperament

=Musical note intervals=

This is background material (for the most part), rather than how to instructions for the program, though I will explain how things are done in the program, when this is relevant.

See also [harmonics_and_just_temperament.htm Harmonics and just temperament]

Absolute beginners
This section is included since some users of FTS may be completely new to the subject of musical intervals. Anyone can enjoy FTS and have fun with it, and one doesn't need to be a musician in any sense to use it.

A tune has two components, pitch and rhythm. Scales are used to describe the way a tune goes up and down in pitch.

A scale is relative. If you sing a familiar tune, then you can sing it starting from any note, and it will sound like the same tune. Similarly, you can sing a major scale starting from any note and it will sound like a major scale.

When you hear music described as being in a particular key, this refers to the note the scale starts from. So for instance a C major scale starts at C.

Suppose you sing a C major scale, up as far as a G. Then suppose you decide to sing a new major scale starting from the G you have just sung. Your new key would be G major - still a major scale, but with all the notes at new pitches.

Nowadays the pitch for concert pitch C has been standardised - but in origin, the idea of a key is relative too. If you sing music that starts in C major, and then it moves to G major, then it doesn't matter what exact pitch you used for your C, so long as the two keys are related to each other in the same way. A few centuries ago indeed, C was often played much flatter than today, indeed, it was often what we now call a B or lower. Historically authentic performances of Baroque and early music often use lower pitches for all the keys. There was no fixed standard of pitch in Baroque times, and some other musicians of the time used a high pitch, higher even than today. See towards the end of the on-line article Everyone in Tune, whatever that means, by Anne Johnson (NYT, March 14th, 1999).

Since scales are all relative, and so are the keys, what matters is the interval between two notes. This depends on the ratio of frequencies.

When one note is double the frequency of another, it sounds like the same note, only higher. It is said to be an higher. Perception of notes an octave apart as the same note is universal to all cultures.

The numbers that you see in the box below the drop list describe the ratio of the frequencies to the first note of the scale, either as pure ratios, or in another notation called cents which is also a way of describing the relative pitch of two notes. It's a convention to show whole numbers in scales as a ratio too, for instance, the octave is shown as 2/1. The first note of a scale is shown as 1/1. (You can choose whether or not to show the ratios as 1/1, 2/1 etc from ).

Nowadays, there is an accepted international standard pitch for all the notes, known as concert pitch. For instance, in concert pitch, the a above middle c is 440 cycles per second. At least, that's what it is supposed to be. However, there is a tendency for the pitches of instruments to continually increase with time - musicians like to be able to play sharp. This happened before the standard was set, and is continuing. Many instruments are now tuned to a=442, and even pianos for accompaniment of instrumentalists in competitions may be tuned to a=442, with others urging against the adoption of such high pitches.

Some musicians have absolute pitch, which means that if you sing a note, they can hear just by listening whether it is a concert pitch C, or slightly higher in pitch, or lower. But this is fairly rare. Musicians with absolute pitch have some perception of relative pitch as well (possibly weaker than for those without absolute pitch) - otherwise they wouldn't be able to recognise a tune as being the same when transposed into another key. You can change the absolute pitch of a fractal tune in FTS via the window.

This introduction describes music as it is usually practiced in my country of origin - UK - and generally western music. The same applies to other cultures, but the emphasis may differ - for instance the idea of keys and of key change being important is a particular feature of "Western music". Other cultures may keep within a single key - to an untrained Western ear sometimes such music may at first hearing sound monotonous because of that (e.g. Indian music for instance especially, because of the drones) and maybe one doesn't fully appreciate the significance of melody, subtle rhythmic variation, and lyricism which play a far more central role. The other way round, Western music may not be fully appreciated at first if one is used to these steady pitch systems because the significance of key shifts and the chord progressions may not be realised on first hearing.

Perhaps a brief introduction to the idea of hertz may help too. just means cycles per second. The standard tuning for the a above middle c is 440 cycles per second, or 440 Hertz. The higher the pitch, the greater the frequency, and so, the more cycles per second. The numbers double for every new octave.

That hopefully gets you up to speed for the next section.

Newbie notes
Most of this material will be new even for professional western musicians. A professional will most likely have heard of the overtone series. Anything else in the way of tuning theory will probably have been covered sketchily, if at all, unless one is a specialist in a relevant area (e.g. historical performance, possibly). This is understandable, as tuning theory isn't relevant for most contemporary western music - everything is played in the same underlying scale, with the exception of a few authentic historical performances. However, there is a fair amount of interest in it now among some classically trained musicians, who are stimulated by the new unexplored terrritory of these various tunings.

For those new to the idea of ratios and cents describing a scale, this introduces the terminology: [cents_and_ratios.htm cents and ratios]

Then you can try out these javascript applets included with this help: [ratios_with_factors.htm ], and [mean_tone_in_cents.htm ]

If new to ideas of the overtone series, just temperament scales, and how some of them are constructed from the overtone series, this section introduces some basic ideas

[harmonics_and_just_temperament.htm Harmonics and just temperament].

See also [harmonic_series_notes_sound_well_together.htm Why two notes of the harmonic series sound good together].

The partials (component notes of a timbre) of a string instrument are very loud. In a cello concerto, the partials will at times be louder than many instruments of the orchestra, though they are usually not heard as separate notes unless one listens out for them with a keen ear.

In this clip [help_files/cello_c2_partials.mid cello partials], try listening to hear how the note played on the pan pipes continues into the cello note that follows it (like a kind of continuing resonance after the note stops sounding). You can also try singing the note, and in fact it a nice way to get into playing just intonation is to play a sustained drone note on, say, a cello (or on the cello voice of your synth or soundcard) and then sing the partials in turn, or sing tunes going up and down the harmonic series.

[testmymidiplayer.htm ]

Here is a drone cello note to sing along with.

[help_files/cello_drone_c2.mid Cello drone on C2] (lowest note on the cello).

The just intonation major third may sound flat to a classically trained western musician when first heard. On the other hand, to a classically trained Indian musician, the major third of classical Western music may sound extremely sharp on first hearing. Perhaps after singing along with the cello drone you may understand why - Indian music uses drones a lot.

The seventh harmonic may sound very flat to a classically trained western musician on first hearing. It too is used in many types of music (not in Indian music). For instance, it is sometimes used in Blues / Jazz.

The idea of the prime limit of a ratio is a very helpful one in this field.

The seventh harmonic is a seven limit note. This means that it is divisible by 7, rather than using multiples 2, 3 and 5 as in the numbers used for the five limit just intonation scale. The eleventh harmonic, and ratios between numbers divisible by 11 are called 11 limit, and ones that use 13 are called 13 limit. However, ratios such as 9/8 involving 9 are called 3 limit because you can construct it using 3 and 2 only: (3*3)/(2*2*2). This has musical meaning too, as you can reach 9/8 using two 3/2s as: 3/2 followed by 3/2 then transpose down an octave 1/2. So what matters in this definition is the largest prime divisor of the denominator and denumerator.

For more on this see [#three_limit_etc Three limit, five limit etc].

The seventh harmonic is used in the seven limit dominant seventh. This gives a wonderfully consonant chord, for those who have the taste for it.

There are two commonly used five limit versions of this note - 9/5 or 16/9, which give two just intonation five limit dominant sevenths.

Here is a clip of the three dominant sevenths to compare: (each time resolving to the just intonation major third triad 1/1 5/4 3/2).

[help_files/7o4_dom_seventh.mid 7/4 dominant seventh]

[help_files/16o9_dom_seventh.mid 16/9 dominant seventh]

[help_files/9o5_dom_seventh.mid 9/5 dominant seventh].

The 9/5 one is perhaps less decisive than the other two. To my ear, the 16/9 dominant seventh sounds the most decisive of the three. Many find the 7/4 dominant seventh the most harmonious and mellow of the three. What do you think?

Here is the [help_files/12tet_dom_seventh.mid twelve tone equal temperament dominant seventh] by way of comparison, resolving to the twelve tone major chord in twelve equal. This is what we are used to hearing all the time.

A few commonly used major thirds are: pure ratio 5/4 (as used in those clips), equal temperament 400 cents, Pythagorean 81/64, 19-tone major third 378.947 cents, and 31 tone major third 387.097 cents.

The pure fifth is 3/2, and the twelve tone equal temperament one is 700 cents, which is a tiny bit flat (by about 2 cents). This choice of the ratio for the fifth is generally accepted, as the 3/2 is a particularly consonant and clear ratio and is very close to 700 cents, and there are no other nearby pure ratios using numbers anything like as small as this.

Though scales can be described using numbers, and the subject can get quite mathematical, the primary motivation is to make notes that sound good together. If it sounds good, it is okay as a scale, and that is all there is to it really. So the thing to do is to tune up some of these scales and try them out and see what you make of them.

Tones, semitones, and n-et
An octave can be divided into any number of equal parts as a way of fitting pitches into a system. The modern piano tuning relies on a division into twelve equal parts. However other systems are available. Microtonal guitar fretting is often based on 19, or 31 equally spaced notes per octave, with various other ones also favoured such as 22. The system with 72 equally spaced notes is highly regarded by many. So are the systems with 53 notes, and 55, and various other numbers.

The major and minor scales both are made up of steps of two sizes, whole tones and semitones. The major scale goes T T S T T T S where T = whole tone and S = semitone.

The is a larger step than the. However, ; there are a number of ways of playing them depending on the scale. So, the semitone, despite its name, needn't always be exactly half the size of a whole tone. In historical tuning systems there are many different sizes for these, and in fact, an interval may come in several different sizes in the same scale.

However in the system that uses twelve equally spaced notes to an octave, there are exactly twelve semitones to an octave, all the semitones are the same size, and a semitone is exactly half of a whole tone.

One needs to be careful to distinguish " ", and " " here.

The standard terminology is to speak of the standard (piano tuner's) scale as being a. That means twelve distinct pitches. It's another meaning of the word " " to mean a distinct pitch rather than a particular interval size. As whole tones, the twelve tone scale consists of.

Just to confuse things, one often abbreviates "whole tone" as "tone" when the context is clear. But, if one keeps in mind the two uses of the word it becomes clear enough what is intended.

Another useful term. One often speaks of. This stands for "19 tone equal temperament", which means, 19 equally spaced notes per octave. Another term used by some is 19-edo meaning "19 equal divisions of the octave" - because sometimes one may want to study equal divisions of some other interval instead of the octave, say, 3/1. A newer abbreviation now used by many is, and that's what I'll use in this help as it is a little shorter, "19 equal temperament" - the word "tone" is redundant.

To give a few examples of whole tones and semtones: in 19-et, the whole tone consists of 3 steps, and the semitone of 2, so the semitone is two thirds of a tone. In 31-et, the whole tone consists of 5 steps and the semitone of 3. In 17-et, the whole tone consists of 3 steps and the semitone of 1, so in this system, the semitone is particularly small (and rather elegant) at only a third of a whole tone. In fact, 17-et has the smallest semitone of any major scale in any n-et.

Since the major and minor scales are so prevalent, intervals are often defined in terms of them. So for instance, a major third is the interval from the first to the third note of the major scale. The minor third is the interval from the first to third notes of the minor scale. A fifth is the interval from the first to the fifth notes, and is the same in the major and minor scales.

Just as there are many ways of tuning the tones and semitones, there are many ways of tuning the major or minor thirds, and indeed, some variation in opinion about which tunings are best - for instance one may like ones major thirds to be sharper even than for 12-et, in which case 17-et major thirds may be attractive, or the Pythagorean major thirds. These are very "bright". Or one may like a gentler more mellow tuning for them, such as one has in 31-et. The just intontation major third at 5/4 is the most mellow of all.

There's less discussion about the fifths as most agree that a fifth at about 702 cents (3/2) sounds best in harmonic timbres. However one can choose to use sharper or flatter fifths and think of these as "unstable", and no longer treat the fifth as a point of rest in ones tuning, so that is one possibility. Or one may use a timbre that works well with wide or narrow fifths.

Even fifths a quarter of a semitone sharp or flat or more can sound convincing in a suitable timbre, or when used by an ingenious composer. Try listening to Jacky Ligon's Ten Thousand Things on his cd Galunlati for a wondeful piece of music with singers singing fifths a quarter of a tone sharp with the accompaniment of non melodic instruments..

The circle of fifths
In some unaccompanied ethnic / folk music one might just keep to the notes of a single scale, perhaps of a few notes only. In Indian ragas there is a drone accompaniment, and the musicians again keep to a single scale without modulation. However in other types of music, one might wish to modulate. What that usually amounts to is that one wants to be able to start a new scale using the same pattern as the scale one is already in, and one wants to be able to start it from any note of the scale one is already in. For instance, having sung a particular tune, one may want to go to another note of the scale, and sing the tune again starting from that note instead, and one wants it to sound the same as before.

The interval of a fifth is the next simplest one after the octave, and most scales have this as one of the intervals, and most tunes have this as one of the notes in the tune (the Soh in Sohl Fa notation). So, if one wishes to modulate, one is likely to want to be able to go up by a pure fifth from any note reached so far.

An interval is pure if you can play both of its notes together without beats. Whether one hears beats depends on the timbre as well as the tuning of the notes. The rest of this section applies to harmonic timbres such as voice, strings, indeed most instruments of the modern orchestra.

For instance, it's possible to play a C and a G, the first and fifth notes of the C major scale, to sound perfectly in tune. This happens when the frequency of the G is exactly one and a half times the frequency of the C. (For more on this, see [harmonics_and_just_temperament.htm Harmonics and just temperament] ).

This interval is called a fifth because it is played using the first and fifth notes of the C major scale.

You can go through all the notes of the twelve tone scale by going upwards a fifth at a time, in the sequence C G D A E B F# C# G# D# A# F C.

Since the frequency multiplies by one and a half each time, the frequency of the last C in this sequence is the same as for the first one multiplied by 1.5 taken to the twelfth power. This works out at close to times the original.

That is a little seven octaves, since seven octaves would be exactly times the original frequency.

So we end up with two different C's. What is a piano tuner to do, if he or she has to tune all the C's on the piano to be in tune with each other, and all the fifths as well? Or, to keep things simple, let's say, harpsichord tuner, because in actual fact, the piano timbre isn't quite a harmonic one (because of the extremely high tension in the strings), but has detuned octaves, and the tuning of a piano is a very complex matter as a result. The harpsichord however is a good approximation to a harmonic timbre.

Here is the circle of fifths as a tune smithy file:

[help_files/pure_circle_of_fifths.ts pure circle of fifths.ts] (the notes have all been transposed down into the same octave).

[tune_smithy_links.htm Tune smithy links options]

Here it is as a midi file:

[help_files/pure_circle_of_fifths.mid pure circle of fifths.mid]

[reset_tuning_details.htm ]

Notice how the last C in the sequence is out of tune with the first one.

So the answer to the harpsichord tuner's question is that it can't be done. Some compromise is needed.

One can keep pure fifths between most of the notes, but leave one [wolf_fifth.htm wolf fifth] which is out of tune. The result is a tuning which sounds good in some keys, but not in others. A tuning with all except one of the fifths completely pure is called a Pythagorean tuning.

One might prefer to make all the fifths the same, and all a little bit flat. That's equal temperament, which is the one piano tuners usually use nowadays. Actually the tuning itself is older - it has been used since the middle ages for lutes because it is fairly easy to tune the lute to this tuning by placing the frets accordingly. Since the maths for placing the frets is fairly modest (one needs to be able to calculate square roots and cube roots, but logs are not necessary), there is some speculation about whether it could have been known even before the middle ages, but no hard evidence that it was.

All keys sound the same in this system.

In other systems in which the fifths vary in size, between pure, and somewhat flat, distributing the flatness of the wolf fifth over several keys, and these are known as well tempered scales.

Bach wrote his Well Tempered Clavier with pieces in all the major and minor keys in a well tempered scale. Bach was showing off the different character of the various keys, because of the variation in the tuning as one moves from one to another. Some popular books say that he wrote it for the equal temperament system, but this is incorrect. Confusingly, well temperament was called equal temperament in his time, because you could play equally well in all keys. However, in his day, what we now know as equal temperament was only used for lutes, and never used for harpsichords and keyboard instruments. There is some discussion about which well temperament he had in mind, but general agreement that he meant what we now know as well temperament rather than equal temperament.

If new to historical tunings, you may want to start with An introduction to historical tunings by Kyle Gann.

Another partial solution is to just keep going, and add a new C which is sharper than the one that began the scale. The Arabic Pythagorean scale does just that. and is constructed by continuing the circle for another five notes, and adding them in as new notes. You can modulate a fair amount using only notes of this scale.

You can make the Arabic Pythagorean scale from pure fifths in FTS using a window. Though it is already in the drop list of scales, it may be worth going through this to give insight into how it is constructed (and possibly suggest ideas for making other scales).

What you do is go up by fifths for seventeen notes, and as you go, reduce all the notes found into the first octave by dividing them by suitable powers of two. The Arabic Pythagorean scale then starts at the second note of this new scale.

Here is how it's done:

Show a Select. Enter as the scale (for a major fifth). Change the to octaves, and click the button, then. Set the number to the right of the button to, for the number of notes to reduce, and click the button. Then click the arrow to cycle the scale round by one note. Click, to copy the you have just made into the main window, ready for use.

You make the twelve tone Pythagorean scale in the same way - this time you need to go up by twelve notes. and start at the second last note of the new scale.

So follow the same procedure, choose as the number of notes for the button, and at the end, click the arrow to cycle the scale round one note to the right. The Pythagoraean scale twelve tone scale consists of the notes 15 16 18 19 21 22 24 25 26 28 29 31 32 of the Arabic Pythagorean scale (also 1 2 4 5 7 8 10 11 12 14 15 17 18).

In all these systems so far, the major thirds tend to be sharp - the major chords and arpeggios sound very bright. In fact, if you play a major chord in, say, the twelve tone equal temperament, and then tune the major third of the chord flatter by about 14 cents or so, then you will find it gets much fuller, richer and warmer, more mellow.

So, because of that, there are other systems in which many of the major thirds are pure, and fifths are flatter even than for equal temperament. In fact, going back to Bach's time, church organs were often tuned in a system called quarter comma meantone, which is such a system. This system is particularly popular for church organs because of the rich full chords.

The need for a tuning system for the notes mainly applies to keyboard instruments, and fretted instruments. Singers, and players of other instruments naturally adjust intervals they play depending on the context, so that they fit in with the harmony of the piece, and can use pure intervals in that way whenever they wish, varying the pitches of notes depending on the context of the scale or chord in which they occur.

Special microtonal instruments have been constructed - microtonal guitars, keyboards with split keys (this is a very early development with some instruments constructed in the middle ages, some pictures here: Denzil Wraight - Italian Keyboard Instruments - 19 tone, and split sharps), and instruments with a hexagonal or square lattice layout.

For modern electronic lattice type keyboards, see for example, the Microzone (hexagonal layout), and the Zboard (square layout, guitar type fingering) from StarrLabs. Another interesting idea is the Notebender keyboard by John Allen. All the keys can be moved longtitudinally, so that you raise or lower the pitch of any individual key by sliding it away / towards you. This page Keyboard and tactile interfaces describes other interesting electronic keyboards. Hugh Le Caine's electronic sackbut of 1948, also used horizontal pressure to change pitch. For acoustic and links to many more instruments, see John Starrett's list of Microtonal Instruments, and for both acoustic and electronic instruments - the Huygens-Fokker foundationlist of microtonal instruments

In Pythagorean temperament, each fifth multiplies by 3/2, so after a few fifths you can get rather large numbers like 243/128. The numbers are all powers of three or powers of two. The major third is 81/64, which is rather sharp, but okay to modern ears, and the minor third is 32/27.

The just temperament scale favours simpler ratios like 15/8. The major third is 5/4 and the minor third in this system is 6/5. The major third in this system, when in isolation, can sound flat to modern ears, because we are so used to the equal temperament, in which it is rather sharp. It is however beautifully in tune once one gets used to it. It gives particularly sonorous major chords.

The term just temperament is also used more generally for systems favouring small ratios, or for ratios in general, in preference to equal or well or mean-tone temperament.

In the quarter-comma mean-tone temperament, the fifths are flatter even than for the equal temperament scale, in order to make the major thirds in tune. In this system, the wolf fifth is sharp rather than flat. It dates back to a time when musicians were used to the just temperament major third, and the wolf fifth seemed a small price to pay to be able to have pure thirds. There are other mean-tone temperaments such as sixth-comma, which are in between quarter-comma and equal temperament.

See [wolf_fifth.htm wolf fifth] again.

We have already seen two ratios for the major third - 5/4 for the just temperament one, and 81/64 for the Pythagorean one.

Here they are as a Tune Smithy file: [help_files/just_then_pyth_then_just_major_thirds.ts just_then_pyth_then_just_major_thirds.ts]

If a string quartet tunes to perfect fifths, then the strings of the 'cello are

C' G' D A

and of the violin

G d a e'

The e'of the violin is then a Pythagorean major third above the C' of the cello raised by two octaves.

The ratio between the Pythagorean and the Just temperament major third is called the syntonic comma 81/80. This is quite a large amount (21.5 cents), so if a cellist finds the third harmonic on the A string and compares it with the fifth harmonic on the C' string with the strings well tuned in perfect fifths, it may well be apparent. Similarly a violinist will probably hear it when comparing the fifth harmonic on the G string with the third harmonic on the e' string. Sometimes string instrument players get involved in researches into tuning systems as a result of noticing this "discrepancy".

When you get to smaller intervals like a tone or a semitone, many different numbers are put forward as suitable ratios for the pure interval by theorists. The reason for all this variety is that you can get to the same note by various paths.

For instance, you may wish to use minor or major thirds as well as fifths. It takes four minor thirds (diminished seventh), or three major thirds to go up an octave. To reach a pure octave, the minor thirds have to be flat by about 16 cents, and the major thirds sharp by about 14 cents. The fifths for the circle of fifths have to be flat by about 2 cents.

You can also get to the octave by using six major whole tones (each needs to be flat by about 3 cents).

Here are some example Tune Smithy files:

[help_files/pure_circle_of_major_thirds.ts pure_circle_of_major_thirds.ts], [help_files/pure_circle_of_minor_thirds.ts pure_circle_of_minor_thirds.ts] , [help_files/pure_circle_of_bluesy_minor_thirds.ts pure_circle_of_bluesy_minor_thirds.ts] , [help_files/pure_circle_of_major_whole_tones.ts pure_circle_of_major_whole_tones.ts].

Each returns to the 1/1 at the end, which of course sounds out of tune after going round the circle.

The ratios in the Pythagorean systems are all expressible as multiples of two or three. For instance, 9/8 is (33)/(222), and 81/64 is (99)/(8v8) or (3333)/(222222).

In the just temperament scale, the ratios are all multiples of two, three or five. The reason is that they are all constructed using fifths and major thirds, which use the ratios 3/2 and 5/4, and these intervals can only introduce extra multiples of three or five, (and multiples of two to shift notes into the same octave)

See [harmonics_and_just_temperament.htm Harmonics and just temperament].

Some of the tunings are particularly suited to particular instruments, at least by historical association or tendencies deriving from the instrument design.

The voices are good for tunings, as singers, if left to their own devices, often tend to sing in just intonation intervals (perhaps less true for "Western music" with so much twelve equal music as a model, but even then, it's found that a capella choirs slide into just intonation in long sustained chords). The is suitable for the temperament for historical reasons, and because they sound good on these instruments - the harpsichord has prominent fifth partials and the church organ has long sustained notes that will show up any beating between notes. The is good for the, as those are the notes played on a natrual trumpet. The sounds well on, as string players tune their open strings in perfect fifthss (usually). In actual practice, string players play intervals in various ways depending on context, and may deliberately avoid open strings if they need some particular nuance of pitch. The scales are suitable for an voices, again for historical reasons, and because as fixed pitch instruments, these are the very types of instruments the well tempered scales were designed to help. The are good for the scale - the Shakuhachi is another Japanese instrument and the two together are also a very traditional Japanese combination of instruments. The scales can be played with - as an approximation to the instruments of a gamelan orchestra using the standard Midi voices for General Midi. The are suitable for the voice, as the Sitar is one of the Indian instruments used for playing them.

Three limit, five limit etc.
A scale is called three limit if you can express all its ratios as multiples of two and three. It is five limit if you need five as well. More generally, the prime limit of a scale is the largest prime number you need to express all its ratios.

For the bluesy minor thirds, you need the ratio 7/6, so this needs a seven limit scale.

In the more general situation, you first have to factorise the ratio into its prime factors - prime = numbers with no further divisors.

So e.g. 9/5 = 3*3/5.

Then to say that this is 5-limit means that the largest number you see in the factorisation is a 5.

22/21 = (11*2)/(7*3) so it is 11-limit, and so on.

It ties in with the idea of a lattice.

To lattice out a scale, take for example this commonly used 5 limit just intonation twelve tone scale:

Factorise it:

where ^ is the exponential notation 2^5 = 2*2*2*2*2 (5 times)

Ignore the factors of 2:

Now put the ratios on the same line if they have 3 to the same power, where 1/3 and 3 are counted as different powers:

You can make a hexagonal or square keyboard to play this.

Then the major triad is

and wherever you play that particular pattern on the keyboard you get a major triad.

The minor triad is

If a scale is 5 limit it has a two dimensional lattice. If it is seven limit, then the lattice gets three dimensional, with factors 3 5 and 7; and an 11-limit lattice has four dimensions (four space dimensions, not time dim).

Many scales have been developed with ratios involving numbers such as 7 or higher. I have included one of these from an article by David Canright as the 7-limit twelve-tone scale in the box. It includes the blues notes 7/4 and 7/6.

You can also find his 13 limit scale from

See his on-line articles, such as: A Tour Up The Harmonic Series, and On Piano Retuning.

Some links
See Just intonation explained by Kyle Gann.

Also look at the articles on just intonation on the web by David Canright. The ones especially relevant here are: A Tour Up The Harmonic Series, On Piano Retuning , Pentatonics I Have Known , and Superparticular Pentatonics.

I used his articles as the source for the 7-limit twelve-tone scale, and for some scales used in the example compositions with the program. He has many more, including some new ones made, or found in computer searches.

This material is discussed extensively on the tuning list at yahoogroups, and there you will find a great deal of active research in progress, much of it unpublished. See also the tuning-math list for in depth mathematical discussion, and the makemicromusic list for the practicalities of actually making and writing music based on microtonal scales.

See also Jacky Ligon's The Microtonal Activist E-zine, the Huygens site, and John Starrets microtonal music page.

There are many other sites, but that will get you started as nearly all will be linked to from one or other of those.

For a very extensive bibliography, see Huygens site and look under Literature | Bibliography (700 K at present) (maintained by Manuel Op de Coul).

Some of the entries in the preset Scales drop list
See also [#A_few_of_the_scales_in_the_drop_list A few of the scales in the drop list].

Werckmeister III scale (1681): (famous historical well tempered scale from the time of Bach)

1 256/243 192.180 cents 32/27 390.225 cents 4/3 1024/729 696.090 cents 128/81 888.270 cents 16/9 1092.180 cents 2

Vallotti & Young scale (Vallotti version) (famous historical well tempered scale from Mozart's time)

1 94.135 cents 196.090 cents 298.045 cents 392.180 cents 501.955 cents 592.180 cents 698.045 cents 796.090 cents 894.135 cents 1000.000 cents 1090.225 cents 2

Sixth comma mean-tone may be a better tuning than Vallotti / Young to use for baroque music in the period between Bach and Mozart.

Quarter comma mean-tone is suitable for the period before Bach, and also for organ music until quite late. It has pure major thirds in most positions. If you go up from any note in major thirds, since three major thirds make an octave, two of them are 5/4s, and one is a 32/25. So four out of the twelve major thirds are 32/25. It has one wolf fifth from G sharp to E flat not intended to be played at all in music of the time (especially since the corresponding triad also has a 32/25 major third in it).

You can make other mean-tone scales using

Pythagorean twelve tone is especially suitable for playing Gothic music.

For some more famous historical temperaments, see Understanding Temperaments.

Slendro scale: Gender wayang from Pliatan, South Bali (Slendro), 1/1=305.5 Hz

1 235.419 cents 453.560 cents 704.786 cents 927.453 cents 2/1

Modern Pelog designed by Dan Schmidt and used by Berkeley Gamelan

1 11/10 6/5 7/5 3/2 8/5 9/5 2/1

Pelog scale: Gamelan Saih pitu from Ksatria, Den Pasar (South Bali). 1/1=312.5 Hz

1 153 cents 315 cents 552 cents 706 cents 848 cents 1058 cents 2/1

Seven tone tuning from Thailand.

The Thai modes are for approximately seven equal scales, but not exact seven equal. There are several Thai scales in the SCALA archives (thailand2.scl to thailand5.scl are all seven note ones).

The one used here is Khong mon (bronze percussion vessels) tuning, Gemeentemuseum Den Haag 1/1=465, which isn't at all regular, but is a nice scale to play in with the Thai modes, which is why I chose it - i.e. for musical reasons. No idea if this scale is actually used with those modes in practice.

Arabic 17-tone Pythagorean mode, Safi al-Din :

256/243 65536/59049 8192/6561 1024/729 262144/177147 32768/19683 4096/2187 1048576/531441

If you select above the box, you can see the intervals as the ratios from the previous note.

For instance, the Arabic 17-tone scale, if you choose to show the ratio from the previous note, turns out to be made up of intervals of two sizes, 256/243, and 531441/524288, which you can compare with the Pythagorean diatonic with ratios 9/8 and 256/243. The ratios shown in red above are also in the Pythagorean 12-tone scale (all of the 12 tone notes are there except for 729/512 and 243/128). The widest intervals of this Arabic scale are the same size as the smallest ones of the Pythagorean diatonic one, and two wides plus one small are the same size as the wide notes of the Pythagorean diatonic. The intervals are in the order W, W, S, W, W, S, W, W, W, S, W, W, S, W, W, W, S (W for wide, S for small), comparing with W, W, S, W, W, W, S for the Pythagorean diatonic.

Bohlen Pierce scale

1 27/25 25/21 9/7 7/5 75/49 5/3 9/5 49/25 15/7 7/3 63/25 25/9 3

This one is rather unusual as it repeats at an octave plus a fifth, instead of at the octave.

For details about the Bohlen Pierce scale:

The Bohlen Pierce site

For the theory for the 22 tone scale (which is designed to favour septimal intervals)

Paul Erlich's 22 tone equal temperamemt scale

For a summary of the 31 and 19 tone scales, and a fair number of other scales:



For an introduction to notations for the 31 tone system, see John Allen's Notation for microtonal scales, part 1.

You can find many more scales in the scales archive for the freeware SCALA program by Manuel Op de Coul. See [More_scales.htm#Lists_of_Scales Lists of Scales] to find out the easiest way to access these in FTS.

[some_of_the_scales_and_modes.htm Scales, modes and intonation]

[More_scales.htm#Maths_info Some info about the numbers]

Almost equal tone systems
The Javanese Slendro scales have five approximately equal divisions in each octave. Some of the scales anyway - it is quite a variable tuning.

They don't correspond to anything in Western music, and have their own unique flavour which has fascinated Western composers, Debussy being an early example. The Pelog scale is an unequal seven tone one. The originals of these scales are also rather unusual because they often have detuned octaves - getting flatter with increasing pitch. Try this one: Gamelan kodok ngorek (1/1=270 Hz)

1 227.965 cents 449.275 cents 697.675 cents 952.259 cents 1196.79 cents

steps

1/1 227.965 221.31 248.4 254.584 244.531

Try listening to [1.082b/slendro_with_detuned_octaves.ts slendro_with_detuned_octaves.ts], and [1.082b/slendro_without_detuned_octaves.ts slendro_without_detuned_octaves.ts].

You can listen to gamelan music on-line at the American Gamelan Institute.

Also at Bali and Beyond, and for an overview of the gamelan,

Music from Thailand has approximately equally spaced notes, and this tuning is also found in Mozambique.

For the Mozambique tuning, see Chopi Scale (pdf file), from the World Scale Depository maintained by Kraig Grady. The scale was recorded by Hugh Traceyin the 1940s.

The tuning is from a Xylophone tuned by Venancio Mbembe of the Chopi people of Mozambique. You can hear him playing here: TIMBILA TE VENANCIO and read about him here, and his instrument the timbila xylophone: Venancio Mbande timbila musician, composer, Mozambique / South Africa For some pictures of Chopi Land, and xylophones there, see the Recording Trip to Chopiland (site no longer live, but available on wayback machine). There's another page by Andrew Tracey about the music of Chopiland here

For videos and cds by him:

Amazon artists - Venancio Mbande

Venancio Mbande

There are several Thai scales in the SCALA archives (thailand2.scl to thailand5.scl are all seven note ones).

The exact equal temperaments 2-et, 3-et, 4-et and 6-et can all be found as subdivisions of the equal temperament twelve tone scale:

4-et: Diminished seventh chord such as C D# F# A C, four equal divisions of a minor third, much used in Western music. 6-et: whole tone scale, six equal divisions of a whole tone, such as C D E F# G# A# C. 2-et: tritone, C F# C. 4-et: three equal divisions each of a major third as C E G# C.

So, these are exactly equal in 12-et, and give approximately equal divisions in other tunings of the twelve tone scale.

To make n-et scales in FTS, see [More_scales.htm#equal_tone Equal tone systems] on the Scales page.

Microtonal music in MIDI
[#intro Intro] - [#pb_range Pitch bend range] - [#15_chann Pitch bends and the limitatin to 15 (or 16) channels] - [#sit_run_out_of_out_chann Situations where one can run out of channels] - [#tracks Tracks and GM synths] - [#same_num_chann Playing the notes for a part on the same numbered channel ]- [#lingering_resonances Lingering resonances after a note switches off]

Intro
MIDI is a system designed around the Western equal temperament scale. However, it also has an option to individually adjust the pitch of each note, which makes it useful for microtonal music too. Normally this option is used in combination with a pitch bend wheel for pitch glissandi and the like - however by using "instant" pitch bends you can also retune individual notes and so get microtonal music in Midi. In this way you can get any pitch you like within the entire range of midi notes.

Actually Midi also has a standard called the MTS tuning standard which is far better than pitch bends for microtonal music, indeed pretty close to ideal - but unfortunately it has been rarely implemented until recently, with manufacturers opting for custom methods of retuning notes instead. Using instant pitch bends is a work around. It is suitable for most present day midi equipment, including GM synths and sound cards which may indeed not have any retuning capabilities at all. If your synth does support MTS, then you can use this with FTS. To find out more see [#mts_tuning_programs MTS tuning programs] and the tip of the day [tips.htm#tip_72 Single note retuning]

Pitch bend range
FTS sets the pitch bend range to its standard value of +- 2 semitones (which it may be at already). This should work for any MIDI play back device which responds to a pitch bend wheel. Some may only respond to the most significant part of the data FTS sends - these will only play the desired pitch accuratel;y to approximately the nearest cent - reasonably good even so. They may have other limitations pn the pitch resolution however. The data FTS sends is exact to well beyond normal human sensitivity to pitch - but the Midi Specification is very flexible in its requirements for the devices that play the data and has no requirements for pitch resolution. Most of it consists of suggestions and recommendations rather than requirements. An FM synth may be better than a wave table synth this respect. If you are intereseted to follow this up see the FAQ [#what_pitch_accuracy_to_expect What level of pitch accuracy can I expect to find typically?].

Some few devices (soft synths anyway) also respond to pitch bends if the range is set appropriately, but don't respond to the pitch bend range message that FTS sends - although it is included in the standard, again devices aren't required to observe it. You may need to set the pitch bend range yourself, and somtimes the range may be preset to zero, meaning that everything will be played in twelve equal.

You can test to see if FTS was able to retune notes as desired on the desired device using. See the [testmymidiplayer.htm Test my midi player] page for more information.

Pitch bends and the limitation to 15 (or 16) channels
MIDI pitch bends are applied to the entire channel. This means that you need a new channel for each distinct pitch bend in play.There are 15 channels available on GM synths - these have channel 10 set to non melodic percussion so that it can't play pitch bends. So usually you can have a maximum of 15 pitch bends in play simultaneously. (16 channels for a non GM synth).

To hear this limitation in operation, choose the Indian Shruti scale, choose for the window, then using which sustains notes played by the mouse (or the space bar sustain) sound all the notes in the scale. Just move the mouse from left to right across the scale, then back again.

When you reach the highest note, you will hear that some of the lower notes have stopped sounding. When you get back to the bottom note, you will hear some higher notes have stopped sounding.

However this is rather an unlikely situation in practice, - the ragas are based on only a few notes normally. Normally they use various selections from the twenty two shrutis.

Similarly, the thirty one note equal temperament scale has this same limitation - indeed even the nineteen tone one - you can't play all the notes at once. However again, selections of notes are used, maybe of seven, or perhaps twelve notes at a time, so it is going to be rather rare to need more than fifteen of these notes at once in a single chord.

Indeed, there should be enough channels to play all notes simultaneously in any of the modes listed in the SCALA mode names archive to date. Joe Mandelbaum's fourteen out of nineteen for 19-tone equal temperament is the mode with most notes so far.

So, we see, in many situations, there are plenty of channels for pitch bends available. However in some situations one can run out of them, and some of the notes in play will get switched off.

Situations where one can run out of channels
One is unlikely to run out of channels in a single part for the reason mentioned. : a single selected in the window may need to be played on several midi out channels at once. See [basic_concepts.htm#Parts ] for details about this.

You can run out of channels more easily if you vary effects such as modulation, or portamento from one to another. This is because in MIDI, these effects are set for channels as a whole. This means, a new channel needs to be allocated for each combination of controller settings needed, and then for each combination of controllers, channels also need to be allocated for all the pitch bends in play.

Lets take an example - suppose you have four voices, each using different effects. This may happen quite easily since Pan is an effect - the position of the voice left or right. You may also want to apply tremolo or modulation (vibrato) to one part and not to another. Suppose now that you want each of the voices to play a four note chord and the chord requires four separate pitch bends for each voice.

Then you need 16 channels, which is more than the number available, and so can expect that one of the notes will get cut off early. This situation could easily arise when using FTS to re-tune a complex score. To do this deliberately, one could play four parts simultaneously, with each part having its own effect or pan position, and each part playing all four notes of a a four note tempered or just intonation chord (indeed, maybe they all play the same one - that won't help the situation here).

One of the notes would need to be switched off. FTS would switch off the first note played of all the ones in this chord - because it switches off any notes that are still sounding in the channel before it applies the pitch bend.

In FTS, when a performer is playing along with the fractal tune, preference is given to the notes played by the user, from the MIDI or PC keyboard or the mouse - and notes in the fractal tune are switched off if necessary.

Tracks and GM synths
This situation gets a little easier however, if all the instruments you use have the same effects, and if you also use a GM synth to play your score. This is often done because this is a standard that guarantees that your violin part in the score is always sounded on a vioiln voice, etc).

A GM synth is a multi-timbral device - it can play on all the channels at once with separte pitch bends and voices for each. The guarantees only one voice per channel, and if one kept to this, one would run out of channels easily when there are multiple voices in play, as you would also need new channels for each voice..

However I've checked up on this, and in practice use a concept of a independent from channels. Lower end of the range devices have. Then you can normally play up to this number of voices simultaneously - and they can all be played in the same channel if desired (you may be limited to fewer tracks if it implements some voices using two simultaneous tracks - this is often done for the honky tonk piano).

Then if the total number of voices in play in all the channels exceeds the number of tracks, the GM soft synth or sound card will make a choice of one of the voices as the "least important" one sounding at the time and leave it out.

See Phil Rees's article MIDI channels, voices, timbres and Modes (look under Multi-Mode)

The standard setting in FTS is for a soundcard or GM synth modern enough to implement this notion of tracks, and to plays multiple voices per channel.

However many synths can't do this. For those, click on (this means one that doesn't use tracks internally) or. See [midi_relaying.htm#synths_etc Setting FTS appropriately for synths or sound cards] for more about this

Another situation in which one might run out of channels is if one uses a non octave scale - then one may well often need more than 16 pitch bends in play simultaneously.

Playing the notes for a part on the same numbered channel
You might sometimes wish to play the notes for each part in the same numbered channel: part 1 in channel 1, part 2 in channel 2, and so on. One situation in which this is useful is for a save to Midi of the fractal tune. This lets one look at the midi file as a score, using any notation software that can read a midi file and automatically convert it to a score.

To do this, unselect

As before, FTS will switch off any notes still sounding in the channel before applying the new pitch bends.

Lingering resonances after a note switches off
Some instruments such as the Glockenspiel or Koto can continue to resonate long after the end of the note, and the pitch bend will change the pitch of this resonance. With these instruments, you will probably want an "All sound off" before applying a new pitch bend to the same channel. Alternatively, you can switch the resonance off by setting the expression controller to 0 momentarily if your synth doesn't recognise the All Sound Off. See s.

However you may hear tiny click like effects with this setting as the sound is abruptly cut off. For an instrument like the flute which has a rather quiet resonance after the end of the note, the click effect is may be more noticeable than the pitch bent resonances. You can unselect it, from

The here refers to the sinewave pitch variation vibrato that FTS can apply to notes as an alternative to modulation.

FTs now uses very few pitch bends in most situations, as it relays notes on channels that already have the correct pitch bend assigned to them wherever possible. So this setting is only significant if one has, e.g. a non octave scale, or is in some other situation that requries many pitch bends such as the fibonacci tonescape.

You can also try switching off the channel changing and also remove the all sound offs, for an interesting effect - notes all played in the same channel with no all sound offs before the pitch bends - you can get a wierd and interesting ghostly effect with a voice like the Glockenspiel playing in the Slendro or Pelog scales with pitch bent resonances!