Overview - Seeds etc - User guide - Musical note intervals - Scales - Midi in - Other pages

Musical note intervals

Some basic concepts - Equal temperament - The_circle_of_fifths - Using ratios to describe a scale - Equal tone systems - Some info about the numbers - Harmonics and just temperament - Summary of how to use scales in the program - Special notations - Fractal Tunes - Microtonal music in MIDI

Some basic concepts

This section is included since some users of FTS may be completely new to the subject of musical intervals. Hopefully it will give enough information so that the rest of this page will make some sense!

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

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

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 decide to sing a new major scale starting from the G. Then your new key would be G major.

The idea of a key is also relative - if you sing music that starts in C major, then moves to G major, 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. In fact a few centuries ago the standard pitch for C was much flatter, indeed, it was what we now call a B. Historically authentic performances of Baroque and early music often use lower pitches for all the keys.

Since scales are all relative, what matters is the ratio of the frequency of a note to the frequency of the note that starts the scale. The numbers that you see in the Intervals box below the Scales 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.

Some musicians have absolute pitch, which means that if you sing a note, they can hear whether it is a concert pitch C, or higher in pitch, or lower. But this is fairly rare. Musicians with absolute pitch still hear the notes in a tune or chord as relating to each other in musical intervals, as well as hearing the absolute pitch of the notes.

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

An octave can be divided into any number of equal parts as a way of fitting the pitches of scales into a system. One particularly prevalent system in use today has twelve equally spaced notes, so that each one is exactly a twelth of an octave. These twelve notes are then further divided into 100 subdivisions called cents. Any musical interval can then be expressed in cents. One can show subdivisions of a cent using a decimal point.

The major scale has two steps, large ones called tones, and small ones called semitones. Tones and semitones are not fixed in size; there are a number of ways of playing them. However in the system that uses twelve equally spaced notes to an octave, there are exactly twelve semitones to an octave. So a cent is a hundredth of a semitone in this system.

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.

I hope that is enough information to get you up to speed.

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Equal temperament

This is the standard tuning for modern pianos. It has twelve semitones in an octave, all of equal size.

The major, or diatonic scale (as it is sometimes called) runs 0, 2, 4, 5, 7, 9, 11, 12 in the equal temperament scale.

The numbers here are degrees, which start at 0 for the first note of the octave. You can read 4 as four twelfths of the octave away from the start note. Then 0 is a zeroth of an octave from the start note, or in other words, the start note itself.

The Note numbers shown below the Arpeggios or modes drop list are degrees.

Another way to put it is that 0, 2, 4, 5, 7, 9, 11, 12 are the numbers you have to add to any MIDI note to get a major scale starting at that note.

For instance, middle C is MIDI note 60, and the MIDI note numbers are in equal temperament semitones, so if you start there, and add 0, 2, 4, 5, 7, 9, 11, and 12 , the C major scale runs 60, 62, 64, 65, 67. 69. 71 and 72 .

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The circle of fifths

In some unaccompanied ethnic / folk music one might just keep to the notes of a simple scale, perhaps of a few notes only. But 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 to the same pattern, starting from any note of the scale one is playing in.

The interval of a fifth is the next simplest one after the octave, 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.

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 ).

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 is the same as for the first one multiplied by 1.5 taken to the twelfth power. This works out at close to 129.75 times the original.

That is a little over seven octaves, since seven octaves would be exactly 128 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?

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

pure circle of fifths.ts (the notes have all been transposed down into the same octave).

Tune Smithy will open all links in the same instance by default, unless you are in the middle of editing a fractal tune.

If you want to open it in a new process, hold the Shift key down while clicking on the hyperlink.

Here it is as a midi file:

pure circle of fifths.mid

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

The answer 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 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.

Here are some tune smithy files you can use to experiment with chords round the circle of fifths in any of the twelve tone tunings:

pythagorean circle of fifths.ts - sustains each note in the circle of fifths to overlap with the next one.

pythagorean_circle_of_major_chords.ts - plays major chords starting at each note in the circle of fifths

pythagorean_circle_of_dominant_sevenths.ts - plays dominant sevenths starting at each note in the circle of fifths

You can change to any of the other twelve tone scales from the drop list. Also you can follow the same patterns to explore any other 2, 3 or 4 note chords round the circle of fifths. To do this, change the Note numbers for the Arpeggios or modes to any other chord with the same number of notes. Retain the 7 at the end as that is what moves it up by a fifth to the next one in the circle.

All are saved with Channels | Ranges all set to the first octave above middle C, and When outside range set to wrap around . So all the notes are transposed into this octave - chanage the top and bottom notes of the range for channel 1 if you want to spread them over a larger range.

You will want to have Tune | Options | Show sustain as ties unticked if you wish to look at the chords in the Tune window while playing them

(Reason: the Tune window ties a sustained note no matter how short the note is that it ties to, and these chords have been constructed using notes of zero length, so it will show notes of one chord as sustained into the start of the next chord, when they are in fact switched off in the middle of the cluster of notes starting the next chord).

One might prefer to make all the fifths the same, and all a little bit flat. That is the equal temperament, which is the one piano tuners usually use nowadays.

It's advantage is that you can play in any of the keys, and they all sound the same.

Then there are 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. Confusingly, well temperament was called equal temperament in his time, because you could play equally well in all keys. See 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 with some of the modes using only notes of the scale.

You can make the Arabic Pythagorean scale from the pure fifth using a New Scale Window .

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

Here is how it's done:

Tick File | Scales Options | Add Reduce buttons to New Scale Windows . Enter 1 3/2 as the scale (for a major fifth). Show a New Scale window, expand to 16 octaves, and click the Expand button. Click Select All . Set the number of notes to reduce into the first octave to 17 and click the Reduce button. Then click the left arrow to cycle the scale round by one note. Click Apply , to copy it into the Scale and Intervals boxes in the main window, then if you want to show it as blue dots in the New Scale window, click Get Scale played .

To make the twelve tone Pythagorean scale, 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 12 as the number of notes for the Reduce button, and then click the right 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.

There are other systems in which many of the major thirds are pure, and fifths are flatter even than for equal temperament.

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.

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, between quarter-comma and equal temperament.

See wolf fifth

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: just_then_pyth_then_just_major_thirds.ts

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 in the choice of ratios is that you can get to the same note of the twelve-tone scale by various paths. For instance, you may be able to use minor or major thirds as well as fifths. It takes four minor thirds (diminished seventh), three major thirds (tritone) to get to the note an octave above. For these 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:

pure_circle_of_major_thirds.ts , pure_circle_of_minor_thirds.ts , pure_circle_of_bluesy_minor_thirds.ts , pure_circle_of_major_whole_tones.ts .

The ratios in the Pythagorean systems are all exprssible as multiples of two or three. For instance, 9/8 is (3×3)/(2×2×2), and 81/64 is (9×9)/(8v8) or (3×3×3×3)/(2×2×2×2×2×2). It is called a three limit scale, because it only goes up to 3.

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 of two to shift notes into the same octave)

See Harmonics and just temperament .

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

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 Scales box. It includes the blues notes 7/4 and 7/6.

You can also find his 13 limit scale from Scales box | More Scales | Canright's 12-tone scales from On Piano retuning.

You can try out all these scales with the program, and play in them using the p.c. keyboard, using the New Scale... window. To start with, you may want to choose a temperament authentic for the voice. The choir voices are good for the just temperament, as choirs tend to sing pure thirds to bring out the sonority of chords.

The harpsichord is suitable for the mean-tone temperament. The trumpet is good for the harmonic series, as those are the notes played on a natrual trumpet. The Pythagorean temperament sounds well on stringed instruments, as they tune their open strings in perfect fifths, though in actual practice, string players play other intervals in various ways depending on context. The well-tempered scales are suitable for an organ. The Koto and Shakuhachi are good for the Japanese Koto scale. The Slendro and Pelog scales can be played with chromatic percussion, strings, flute, and ensemble.

Have a look in the Groves Dictionary of music if you want to see lots of details about all this. For instance, the entries on INTERVAL, JUST TEMPERAMENT, MEAN TEMPERAMENT, and PYTHAGOREAN TUNING.

For an on-line source, 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.

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Using ratios to describe a scale

The ratios you most often encounter in scales are closely linked to the harmonic series, see Harmonics and just temperament .

Here are some scales in cents, ratios, or mixed notation. They are all on the Scales drop down 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

Pygmie scale:

1 8/7 21/16 3/2 7/4 2/1

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

Gamelan Saih pitu from Ksatria, Den Pasar (South Bali). 1/1=312.5 Hz (this is the Pelog scale used on the program's drop down list)

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

A folk scale from Rajasthan, India

1 9/8 5/4 4/3 3/2 15/8 2/1

Japanese pentatonic koto scale

1 9/8 6/5 3/2 8/5 2/1

Xylophone from West Africa

1 152 cents 287 cents 533 cents 724 cents 890 cents 1039 cents 2/1

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

1 256/243 65536/59049 9/8 32/27 8192/6561 81/64 4/3 1024/729 262144/177147 3/2 128/81 32768/19683 27/16 16/9 4096/2187 1048576/531441 2/1

If you tick As steps above the Intervals 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 are in the Pythagorean 12-tone scale (all of them 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.

Try the well tempered scales with the note height pattern 0 4 7 12 7 to play major chords, going round the cycle of fifths, or 0 3 7 12 7 for minor chords. Add a sustain of say 3 notes. One of the Organ voices would be a good choice, to hear the chords.

You can find many more in the scales archive for the freeware SCALA program by Manuel Op de Coul.

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

You can paste the lines with the notes of the scale into the Intervals box. It doesn't matter that the notes are on separate lines - the program will put them all on a single line for you. You need to add a 1 to begin the scale if it's left out, as it is for all the scales in the Scala archive.

You can set a maximum size for the quotients for all the ratios generated by the program in the File | Number options | Max ratios quotient box.

Tick Show cents or small ratios to show pure ratios if the numerator and denominator are small, or have only small factors, otherwise cents.

Tick Factorize ratios to factorize numerator and denominator for pure ratios. This will show ratios like this:

2^4/(3*5)

* here means ×. The reason for using * is that it is easier to type it from your p.c. keyboard than ×. To type the character ×. you would probably need to cut and paste it from Start | Program Files | Accessories | System Tools | Character Map , or to type Alt+0215 on the numeric keypad.

The 2^4 means 2 to the power of 4, i.e. 2×2×2×2 (2 multiplied by itself four times). That's an especially useful notation for scales like the Arabic Pythagorean with numbers like:

2^8/3^5

Use the Max factor box to set a maximum factor for the ratios, beyond which numbers are shown as cents for Show cents or small ratios . For more details: Max factor expo

If you want to see all the numbers as cents, tick Show cents from the Number options window.

To see them all as decimals, untick Show cents or small ratios and Show cents , and set the Max ratios quotient and Max quotient for n(..) values to 1.

You can also use Herz to describe a scale using the absolute frequency of the notes. You do it like this:

440 Hz 495 Hz 550 Hz 660 Hz 733.3333 Hz 880 Hz

for the just intonation pentatonic scale starting at a=440 ( A4 ). Tick File | Number options | Show herz to show a scale in this form.

Some info about the numbers

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Equal tone systems

Since twelve is divisible by two, three, four and six, musicians can play other equal divisions of the octave using notes from the equal temperament twelve tone scale.

The diminished seventh chord, such as C D# F# A C has four equal divisions of a minor third, and is much used in Western music.

The tritone with three equal divisions C E G# C, and the whole tone scale, with six equal divisions of a whole tone, such as C D E F# G# A# C are also used. The one with two equal divisions, is C F# C.

The Javanese Slendro scales have five approximately equal divisions in each octave.

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

Try listening to slendro_with_detuned_octaves.ts , and slendro_without_detuned_octaves.ts .

To listen to original Balinese and Javanese music Bali and Beyond , and for an overview of the gamelan, Gamelan virtual tour (Chico's music heritage network) .

Music theorists have also introduced scales of 19 and 31 notes among others. See MICROTONAL in Groves. The reason for these particular numbers is that they give a way to divide the octave in equal spacings that preserves many almost pure intervals.

You can specify any of these systems in the Intervals box, using the same n(..) notation. For instance, for 19 note micro-tonal, use

1 n(1/19)

This is a notation I've invented to give an easy way of typing equal temperament scales into the Intervals box.

You can make scales from notes selected from within one of these equal tempered scales.

The easiest way to do this is to use the New scale button.

To see the scales in the n(..) notation, untick Show cents or small ratios.

Choose a scale from the drop down list. If you want to make one of the equal temperament microtonal scales not in the list, such as 41 notes type a number into the Microtonal notes box.

Click to select/unselect notes from the scale. When it's ready, click Apply to start using it for the melody.

Using the n(..) notation, the major chord of the twelve tone scale in the equal temperament scale is:

1 n(4/12) n(7/12) 2

You can read n(4/12) as "four twelfths of the octave".

This notation starts at 0 for the first note of the scale.

1 is n(0)

These are the same numbers as for the Arpeggios or modes box except that you have to put each one inside the n(.../12) expression.

In this notation, you can also write the major chord of the twelve-tone scale as

1 n(1/3) n(7/12) 2

since note 4 of the 12 tone scale counting from 0 is the same as note 1 of the three tone scale (tritone).

Try making the major chord by selecting notes from the equal temperament 12 tone scale, and you will find that this is what you get.

You can write any other scale the same way.

The diatonic scale for the 31 note system is

notes 0 5 10 13 18 23 28 31 (for the octave)

or in the n(..) notation, with 1 for n(0):

1 n(5/31) n(10/31) n(13/31) n(18/31) n(23/31) n(28/31) 2

For the 19-tone system , it's

notes 0 3 6 8 11 14 17 19

or in the n(..) notation:

1 n(3/19) n(6/19) n(8/19) n(11/19) n(14/19) n(17/19) 2

See Microtonal scales (Microtonal synthesis home page)

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Some info about the numbers

Here is the expression for n(..), which is used by the program:

n(x)= 2^x

In fact, you could type in this expression instead, for instance, 2^(1/12) for the equal temperament scale.

Here ^ stands for exponentiation (alternatives are ** and |).

You can type in any arithmetical expression, and it will calculate the result.

The expression for each number has to be typed without any spaces, so that it is easy to tell where one number ends, and the other begins. Expressions can use brackets.

Use * for multiplication and / for division.

You can use upper or lower case. N(x) is the same as n(x).

As well as this special N() symbol, it recognises E() for the natural exponential and L() for log. So you could also use E((1/12)*L2) instead of N(x) if you like. It also recognises S() for sine, C() for cos, T() for tan, P for pi, and R() for square root.

Also it recognises ' or ` (char. next to 1 on standard Windows keyboard) as a shorthand for cent. The ` applies to the number following it. So `100 means 100 cents from the base note of the scale, or one semitone. `x is short for 2^(x/1200).

It also recognises M(..) as an alternative to N(..) - this is for compatability with an earlier version of the program.

Recognises a comma ' , ' as alternative to a decimal point ' . '. This is because it is fairly easy to type 0,8 instead of 0.8 - the comma and full stop keys are next to each other on the standard keyboard. They also look similar when the font size is small. Otherwise, the parser would read the first as 0 , which could cause some puzzling moments, for instance if it is the sustain, so that the notes are set to length 0, and you no longer hear anything.

When ever the program needs to show an interval, for the Fractal Tune as Intervals score, or for one of the boxes, the program converts it to a fraction if possible (unless you have Show cents ticked). It looks for either a ratio, or the n(..) notation using ratios, that gets within 1e-12 of the exact value needed. It looks for whichever of those has the smallest quotient. If that doesn't succeed, it shows it as a decimal. The algorithm used to find the ratios is very fast (using continued fractions - see Hardy and Wright, "Theory of Numbers" for the details).

So any fraction it finds will be extremely close to the exact value. It will exactly equal it if the ratio is originally given as a small exact fraction with small quotient and denominator. In other cases, where the original is given in cents, or in the n(..) notation, or after repeating some interval such as 11/10 many times, it could find a close approximation that happens to be a fraction.

For the 19 tone microtonal scale, it would show the ratio 419914/404871 because this just so happens to be within 1e-12 of n(1/19).

To avoid that type of case as far as possible, it also checks that the denominator and quotient can be factorised entirely into small numbers less than 50. That's because scales usually use the first few harmonics, and a harmonic even as high as 13 is unusual.

It goes up to 50 as a generous over estimate for most situations - it would be an unusual scale indeed that used the 29th harmonic. However some of the superparticular scales in the SCALA archive go this high, and even higher.

Result of factorising: 419914/404871= 2*11*19087/(3*19*7103) . Since 19087 exceeds the max factor of 50, it shows it as cents.

To see the factors of the pure ratios, tick Factorize ratios

You can change the highest harmonic to look for using the Max factor box ( Tune .... then Options ).

Set this to 1 to look for any ratio, whatever it's factors.

If set high, or to 1, you may find that the program pauses a while when testing or factorizing large numbers. It's factorizing algorithm isn't particularly fast. If it get's held up for a while, hold down the escape key to break out from factorizing.

If you have Show cents or small ratios ticked, it looks for the closest approximation as a small ratio (default - up to a million for the quotient) and in cents. It shows cents if there is a value in cents that achieves the resolution of 1e-12, or if there is no pure ratio within the resolution that can be factorised entirely into small numbers less than 30. In other words, it uses the mixed cents / ratios notation most usually used for musical scales.

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Harmonics and just temperament

Play a low note on a stringed instrument. Then lightly touch the middle of the string with one hand while continuing to bow (or pluck) with the other, and you will get the note an octave above. Touch a third of the way across, and you get the note an octave and a fifth above. A quarter of the way across, and you get to two octaves.

In the next octave you get the major third, fifth again, and then the seventh harmonic is a note which isn't used in our system, so sounds out of tune to us (ratio 7/4). It is used in jazz. Then you get another octave again, this time three octaves above.

The notes get gradually closer together as you get higher.

harmonic_series_first_five_octaves.ts

Tune Smithy will open all links in the same instance by default, unless you are in the middle of editing a fractal tune.

If you want to open it in a new process, hold the Shift key down while clicking on the hyperlink.

In fact, all these notes are present in the original low note, and if you have keen hearing, you might be able to hear some of them, as its "overtones". All you are doing by touching the string lightly is to select out some of the overtones by damping other ones so that they stop sounding. (Some instruments, notably the piano, and the church bell, have extra non harmonic partials, which are out of tune with the fundamental and contribute to the characteristic piano, and bell sounds).

If you are a string player yourself, try sounding an overtone, then play the original note, and see if you can hear the overtone still sounding.

Since they all belong to the same note, if you play two notes of the harmonic series, particularly the ones lower in it, they will sound good together. Intervals between notes of the harmonic series sound particularly pure.

If new to this subject, you may find it confusing at first that ordinals such as "fifth" are used in two ways.

Fifth, used on its own, refers to the fifth note of the major scale, and in this case, the major scale starting from the fundamental.

When one wishes to talk about harmonics, one says third harmonic, fifth harmonic, ....

The other thing one has to get used to is that notes that are an octave apart sound as if they are the "same note", only higher. So the third, sixth and twelfth harmonics are all the "same note"; they are all fifths. They are the octave and a fifth, two octaves and a fifth, and three octaves and a fifth above the first harmonic.

The fifth is the same in both minor and major scales, but the third differs, so one also talks about the major or minor third. If one says "third" on its own, this is understood to be the major third unless the context is clearly about minor thirds.

Confusingly (if you are new to this subject), the fifth harmonic is the third note of the major scale, and the third harmonic is the fifth note of the major scale.

To play a pure major chord, as it is most usually described, enter:

1 5/4 3/2 2

into the Intervals box.

Here 3/2 is the third harmonic in the second octave. The 5/4 is the fifth harmonic, in the third octave. It's divided by four this time, because you have to drop it down by two octaves to get the note back into the same octave as the 3/2.

If you multiply a ratio by two, or divide it by two, you get the "same note".

You need to include the octave in your scales - the program takes your scale, and repeats it, so you need to include the octave to tell it where to begin the next copy of the scale.

If you were to type:

1 3/2

then the scale would go up in pure fifths - the circle of fifths in fact.

It's 1 5/4 for the circle of major thirds, and 1 7/6 for the circle of bluesy minor thirds.

Starting from C, the notes of the harmonic series are:

 C  C  G  C  E  G  -  C  D  E  -  G  -  -  B  C
 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16

in_tune_harmonic_series.ts (played on 'cello)

The ones shown with dashes are out of tune in the ordinary twelve-tone scales.

 

We've already got all the white notes except for F and A.

Here it is again, this time showing only the new notes each time, all transposed down into the same octave. The original harmonic series notes are shown in brackets.

in_tune_harmonic_series_new_notes.ts (harmonic series: 1 3 5 9 15 2).

Here it is with all the notes. Tune drops back to the fundamental after each of the "out of tune" notes. They aren't really out of tune, notes of the harmonic series always sound good together, but they are certainly more adventurous.

all_harmonic_series_new_notes.ts (harmonic series: 1 3 5 7 1 9 11 1 13 1 15 2).

We can now express all the new notes as ratios. For instance, the B is 15 times the fundamental. You need to drop it down a few octaves to get it into the same octave as the fundamental, which you do by dividing by two as many times as needed. The result is 15/8.

The other notes are D = 9/8, E = 10/8 (= 5/4), and G = 3/2.

Here are the notes of the scale so far in ascending order:

1 9/8 5/4 3/2 15/8 2

As a tune smithy file

diatonic_notes_from_harmonic_series.ts (played on violin this time).

We can find the missing F because it's a fifth below C, and the A because it's a third above the F (alternatively, as a fifth below E).

Using 3/2 for the fifth, and 5/4 for the major third, the F is 2/3, or moving it up by an octave, 4/3. The A is then a third above that, so it's ratio is 4/3 times 5/4, which is 5/3.

As a tune smithy file

diatonic_notes_from_harmonic_series_with_F_and_A.ts

So the just temperament diatonic (i.e. major) scale is

1 9/8 5/4 4/3 3/2 5/3 15/8 2

diatonic_notes.ts

We can now fill in the black notes using major thirds. We have to multiply by 5/4 to go up by a major third, and divide by it, or equivalently, multiply by 4/5, to go down a major third.

Going down a major third from the octave C gives 2 times (4/5), so 8/5, which is our A flat / G sharp. Doing the same from the G at 3/2 gives 3/2 times 4/5, which is 6/5, our E flat / D sharp. We can go down from F at 4/3 to get D flat / C sharp as 16/15. Going down a major third from the D at 9/8 gives the B flat / A sharp as 9/10, or after moving it up an octave into range, 9/5.

Notice, that if you went up a major third from E, you would get to G sharp as 5/4 times 5/4, or 25/16 instead of 8/5, which is how a musician sensitive to just intonation major thirds might play it in the scale of E major from this E.

If playing in pure just intonation, you could be forced to play a minute shift in pitch when changing from a G sharp to an A flat.

This is what one sounds like: enharmonic_shift_for_G_sharp.ts (plays diatonic scale in C, then goes down to the A flat in A flat major, then goes up to the G sharp in E major from the E of the just intonation C major scale, then shifts back to the first tuning for the A flat ready to go back up to the original C in A flat major).

We now have all the notes except the F sharp (or G flat). It's a major third above D at 9/8, so 9/8 times 5/4, which gives 45/32, the most complicated ratio in the just temperament twelve tone scale.

Try just_temperament_twelve_tone_constructed_from_diatonic_scale.ts

The notes played are

1 9/8 5/4 4/3 3/2 5/3 15/8 2 8/5 3/2 6/5 4/3 16/15 9/8 9/10 9/8 45/32 1. Try showing the Tune... window and watch as the notes are played.

 

So the complete twelve tone scale is

 C    C#    D    D#   E    F     F#    G    G#   A    A#    B   C
 1  16/15  9/8  6/5  5/4  4/3  45/32  3/2  8/5  5/3  9/5  15/8  2 

This scale is great for music with lots of major thirds and that doesn't stray too far from the basic key.

You can try playing a major third in the just , Pythagorean, and the equal temperament scales to see the difference. You can use the New Scale... option, as described on the Seeds_etc page.

It's a transposing keyboard, so set the pitch of the lowest note appropriately, and choose a voice such as Church Organ that brings out the harmonies especially clearly.

Try a major third such as C to E. If the lowest note is set to middle C at Midi note 60, frequency 261.63 Hertz, then you should hear 16 beats per second for the Pythagorean major third, and 10 beats per second for the equal temperament third.

Try dropping down a couple of octaves and the numbers will be the easer to count 4.0879, and 2.5956 beats per second. (41, and 26 beats in ten seconds).

Now try the just temperament third, and you shouldn't hear any beats at all. (However sometimes you may find that a single note played on your midi synthesizer may have slow variations in volume, maybe depending on the voice, which can confuse things).

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Summary of how to use scales in the program

F or the Western equal temperament scales, or other twelve tone systems, the easiest way to specify a scale, or chord, is to choose one of the twelve tone systems in the Scale box, and enter the note numbers in the Arpeggios or modes box, giving the lowest note of the scale or chord as 0.

For instance, try 0, 4, 7, 12 for a major chord. The scale doesn't have to ascend from each note to the next, so 0, 7, 4, 12 is also okay.

The scale should begin with 0, and end with 12 if you want it to repeat identically in each octave.

If the twelve-tone system you want isn't listed, you need to paste the numbers that define it into the Intervals box.

Enter ratios without any spaces before or after the division sign like this:

1 5/4 3/2 2

You can also use cents.

Another notation you can use is n(1/...) for equal temperament.

For instance, it's:

1 n(1/12)

for the 12 tone scale.

You can use:

1 n(4/12) n(7/12) 2

to select notes within the equally tempered scale, in this case, the fourth and seventh notes of the chromatic scale, for a major chord, with 2 for the octave to complete the chord.

Use the New scale... button to select notes from scales by clicking on dots above a keyboard for the ones you want to select / unselect.

Having set up a scale, you can choose a mode, or arpeggio from it to play, using the Arpeggios… button.

See New arpeggios , New scales

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Special notations

Use

2

(no zero before it)

for the harmonic series, which otherwise you'd have to write as

1 2 3 4 5 6 7 8 9 .... (as many notes as you need for the tune)

This is a special notation devised for the program. When the first number you enter for a scale definition is anything other than 1, the program interprets it as a kind of generalised harmonic series.

The number 2 is the frequency of the first harmonic, as a multiple of the frequency of the fundamental.

The harmonic series is then found by repeatedly adding the frequency difference between the fundamental and the first harmonic.

For instance, the frequency for A is 440, so the harmonic series starting at A is

440, 2*440, 3*440, 4*440, 5*440, ... adding 440 each time.

Negative numbers for the seed will give an upside down harmonic series for these scales:

1 1/2 1/3 1/4 1/5 1/6 ....

In the same notation, 1+1/5 would be short for a first "harmonic" of 1+1/5, i.e. a generalised harmonic series of

1, 1+1/5, 1+2/5, 1+3/5, ...

or starting at 440,

440, 440+(440/5), 440+(2*440/5), 440+(3*440/5),...

Since 6/5 is the ratio of the sixth to the fifth harmonic, this gives the harmonic series from the fifth harmonic upwards.

The upside down series will be

1 1/(1+1/5), 1/(1+2/5),....

Any ratios of the form (n+1)/n will give successive members of the harmonic series from the nth harmonic upwards.

Other ratios like 5/3 or 11/9 also pick out members of the harmonic series, with gaps. For instance 5/3 picks out every other harmonic from the 3rd upwards.

 

Another notation is

#g 1/2

for the geometric series n(0), n(1), n(1+1/2), n(1+1/2+1/4) etc,

i.e. 0 cents 1200 cents 1800 cents 2100 cents 2250 cents 2325 cents 2362.5 cents 2381.25 cents,...

each interval half the previous one

It would be

#g 1/3 for n(0), n(1), n(1+1/3), n(1+1/3+1/9) etc, and so on, each interval a third of the previous one.

The number after the #g is the common ratio for the series.

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Fractal tunes

You can enter your own tunes, and the fractal music will change to respond.

You enter the numbers for the positions of the notes in the scale:

How these are interpreted in terms of pitch depends on what scale and mode you have selected to play them: 0 for the lowest note, 1 for the next, and so on.

The program makes a fractal tune out of the numbers, by adding a copy of the pattern of note heights to each note.

For instance, suppose your pattern is 0 1 2 0.

You can enter 0 1 2 0 into the note heights box, and set the level to 0 , to hear the pattern on its own.

Then set the number of Levels to 1.

It will add the complete pattern 0, 1, 2 and 0 to each of its numbers, giving:

0 1 2 0 (add 0 )

1 2 3 1 (add 1 )

2 3 4 2 (add 2 )

0 1 2 0 (add 0 )

or, all on a single line,

[0 1 2 0] [1 2 3 1] [2 3 4 2] [0 1 2 0]

Now set the Levels to 2.

This time it adds the complete level 1 pattern to each of its numbers, to get

[0 1 2 0] [1 2 3 1] [2 3 4 2] [0 1 2 0] (add 0 )

[1 2 3 1] [2 3 4 2] [3 4 5 3] [1 2 3 1] (add 1 )

[2 3 4 2] [3 4 5 3] [4 5 6 4] [2 3 4 2] (add 2 )

[0 1 2 0] [1 2 3 1] [2 3 4 2] [0 1 2 0] (add 0 )

or, all on a single line

([0 1 2 0] [1 2 3 1] [2 3 4 2] [0 1 2 0]) ([1 2 3 1] [2 3 4 2] [3 4 5 3] [1 2 3 1]) ([2 3 4 2] [3 4 5 3] [4 5 6 4] [2 3 4 2]) ([0 1 2 0] [1 2 3 1] [2 3 4 2] [0 1 2 0])

and leaving out the brackets:

0 1 2 0 1 2 3 1 2 3 4 2 0 1 2 0 1 2 3 1 2 3 4 2 3 4 5 3 1 2 3 1 2 3 4 2 3 4 5 3 4 5 6 4 2 3 4 2 0 1 2 0 1 2 3 1 2 3 4 2 0 1 2 0

It iterates this process for as many levels as you ask it to do. For a more visual idea of what a fractal tune is, click on Tune... while one of the tunes is playing.

To see the numbers, click Options... choose Score from the drop down list, and select Note heights from the Notes as drop down list.

If the numbers are too small to read, change the width in seconds, or height in octaves for the score, or resize the window.

string_quintet.ts is arranged for a string quintet, with the first violin playing every note, the second violin playing the note that begins each pattern at the second level, the viola playing the first notes at the third level, and so on down to the contrabass.

Listening to that, and trying it with various patterns of note heights, could give a good idea of how the fractal tune works.

Try showing one channel as note heights and the rest as dots. To do this, select dots from Tune | Options | Notes as , and note heights for Tune | Options | Channel to follow, notes as . Then vary the Chann. to follow , and notice how all parts are playing the same pattern of note heights, at varying speeds.

marimba_with_string_quartet.ts is the same as string_quintet.ts , with a marimba for the top line, and changes to the volumes of the notes in the tune, and the durations of the notes in the top line.

You might also like to listen to major_and_minor_scales_fractal.ts , for a more complex pattern. The recorder is playing it very fast at the top, then glockenspiel, wood block, and church organ follow at increasingly slower speeds. You will have to wait some time for the church organ to change a note - more than two and a half minutes. The pattern is modulating back and forth through major and minor keys all the time and is meant to suggest some natural sounds, like the wind, as it can be on occasion.

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Microtonal music in MIDI

Although MIDI is a system designed around the Western equal temperament scale, it also has an option to individually adjust the pitch of each note. You can use that to get any scale you like.

To do this with the program, select one of the scales, or specify your own scale using the methods described earlier on this page.

The program will then adjust all the pitches accordingly.

When adjusting pitches, FTS assumes that the pitch bend range for MIDI is set to its default value of +- 2 semitones. However it sends the necessary MIDI messages to change the pitch bend range to +- 2 semitones on all channels it plays on, so it should also work for any MIDI play back device which implements a coarse pitch bend range controller.

You can test for the pitch bend range by typing the word test into the Musical seed box, and clicking Play . If everything is okay, you should hear repeated notes all at the same pitch.

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 (since the notes can't be played on the non melodic percussion channel 10), so a maximum of 15 pitch bends in play simultaneously.

To see the limitation of 15 channels in operation, choose the Indian Shruti scale, choose Select all for the New Arpeggio window, then using Shift + mouse move to sound all the notes in the scale, move the mouse from left to right across the scale, then back again. When you reach the highest note, some of the lower notes will stop sounding. When you get back to the bottom note, some higher notes will stop sounding.

However in practice, none of the Shruti modes need as many notes as this.

When playing from MIDI in, or the p.c. keyboard enabled for sound, the Channels in play for melodic voices are kept free for the fractal tune if possible (except those playing rests or non melodic percussion). However, if all the other channels are used, notes are played in the channels in play too.

When there are no other notes being played, there should be enough channels to play all notes simultaneously for all 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.

When playing along with the fractal tune, it is easier to run out of channels for pitch bends, because of the way FTS organises the channels.

There are enough channels to have, say, seven distinct notes of the scale played simultaneously from MIDI in, or the p.c. keyboard (e.g., all the notes in any of the 12 tone diatonic scales) and at the same time, eight voice channels in play for the fractal tune, (these can need up to eight extra channels for the pitch bends when the fractal tune is actually playing, so that successive notes for same voice channel and differing pitch bend can be played on distinct channels, but these can be shared with the p.c. keyboard or MIDI in).

So in many situations, there are plenty of channels for pitch bends to play the tune, and play along simultaneously

So there are plenty of channels for the pitch bends in most situations.

Situations where one can run out of channels

You can run out of channels if you vary effects such as modulation, portamento from one voice channel ( def. of voice channel ) to another. This is because in MIDI, these effects are set for channels as a whole, so that the channels the notes are actually played on need to be set to the correct values for both the effects, and the pitch bends..

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

When playing along with the fractal tune, preference is given to the notes played from the MIDI / p.c. keyboard enabled for sound / mouse movement, and notes in the fractal tune are switched off if necessary.

Playing all the notes for a voice channel on the same channel

You might sometimes wish to play the notes for a voice channel all in the same channel, for instance to save as a Midi file, in order to look at the result as a score. Another reason might be to use MIDI mapper to assign separate output devices according to the channels. To do this, untick

Channels | Effects | Ok to change channels for pitch bends

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

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. So you will probably want an "All sound off" before applying a new pitch bend to the same channel.

However this can also cause tiny click like effects 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 untick it if you wish, from

Channels | Effects | All sound off before same channel pitch bends (except vibrato)

You can also try switching both off for an interesting effect - you can get a wierd ghostly effect with a voice like the Glockenspiel playing in the Slendro or Pelog scales from the pitch bent resonances!

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