Scale Notations

Notation for n-et
Example: for 19-et, use:

This is a notation I've invented to give an easy way of typing equal temperament scales into the various scales boxes. One can read it as degree 4 of 19-et - so it's the note you make by using 4 in the for 19-et.

It is short for 2 4/19, or as one would type it in the scales boxes: 2^(4/19), four nineteenths of an octave. This is the interval that is fairly close to 7/6; one of the special points of interest of 19-et.

Another example: is the tritone (half an octave). The closest low numbered ratio is 7/5.

To show the scales in this notation, select.

Using this notation, the major chord of the twelve tone scale in the equal temperament scale is:

Here, is four twelfths of the octave, which is the same as a third of an octave - three twelve equal major thirds span the octave. So another way of writing it is as:

So you can cancel out common factors of the top and bottom numbers in this notation, just like normal ratios.

1/1 is also

2/1 is also

You can also make n-et scales using To do this, you type the number of steps you want for the octave into the box (blank originally). Then select them in the usual way by clicking on the blue dots to make your new scale.

The diatonic scale for the 31-tone system (microtonal synthesis web site) is

Arpeggio: 0 5 10 13 18 23 28 31

or in the special

For the 19-tone system (microtonal synthesis web site), it's

notes 0 3 6 8 11 14 17 19

or in the n(..) notation:

, 31 note system

scale notation
Use this window to choose what notation to use to show the scales.

You can use any of these notations to enter values - and it doesn't matter which one you are currently using to display the scale. For instance if you show the scale in cents and ratios, you can still paste a scale in using the hertz notation, so long as you include the Hz symbol after each entry like this:

You can also do this at any time, whether the current notation shown is hertz or cents, or ratios or whatever. The in the shows that it is in hertz. More generally, just copy whatever notation is used to show the scale in any of the notations to enter values in that notation, at any time.

It is probably best to show the 1/1 if you want to work with hertz, so that you can see its frequency in the main window. With other notations it's probably best not to show it. When the first value of a scale is in hertz, it is taken as being the frequency for the 1/1 whether or not you choose to show the 1/1. This lets one change the pitch of the 1/1 just by prefixing a new hertz value before the first entry of the scale.

For more about these various notations see [#Calculator Calculator] and [More_scales.htm#Special_notations Special notations].

- shows a close approximation to the interval as a small ratio if possible, and otherwise, shows it in cents. It shows cents if there is no pure ratio within the resolution that can be factorised entirely into small numbers less than 30 and with the quotient less than a million. All these paramaters can be configured from the window. Anyway, to summarise in brief, this option uses the mixed cents / ratios notation most usually used for musical scales, and normally one can just use it as it is - only needs to be configured in rather special situations.

- this shows suitable values in a special n-et notation devised for the program. = 18th degree of 31-et. Configured using. Again it will show them as cents or small ratios if those notations are more suitable and only uses the special n-et notation for values that are extremely close to integral number of steps of n-et for some n.

- Shows all values in cents (100 cents to a semitone, and 12 semitones to an octave).

- Show the pitch in Hertz. Since this is absolute pitch, then as you change the pitch for the 1/1 from the window, the hertz values shown for the scale will change..

- shows all the numbers in decimal format. This means that you use, e.g. 1.25 for 5/4. This notation is rarely used for scales - cents notation or ratios, are more commonly used as they show the position of the note in the octave (cents) or its relation to other notes (ratios). However, pure decimal notation is sometimes useful, especially for quantities that can't be expressed as ratios. To take one example, the golden ratio is of musical interest because it is as far as you can get from any small numbered ratios, and the golden ratio is usually given in decimal notation. The golden ratio is used in two ways in scale construction - sometimes a scale may be defined using a golden ratio related number of cents, or sometimes as a golden ratio in pure decimal notation and here I'm talking about the pure decimal notation version of it..

The reason decimal notation is seldom used is that e.g. 1, 1.1, 1.2 and 1.3 in decimal notation gives us the notes 1/1 11/10 6/5 13/10 which are unequally spaced in the octave - to stack musical intervals, one needs to multiply ratios or values in decimal notation rather than add them. As a mathematician would say, we hear logarithmically (i.e. with addition in the place of multiplication). You can read more about this in the section [cents_and_ratios.htm ].

To see the factors of the pure ratios, tick

Normally one uses 1200 cents per octave, i.e. 100 cents per equal temperament semitone. However when working with other numbers of tones per octave, it can be useful to use other numbers per octave, e.g. 1700 centisteps per octave for 17-et or 3100 centisteps for 31-et. etc.

This field lets one vary the number of cents per octave in order to make a notation especially suitable for work with such scales.

So for example the major third in cents is 386.314 cents. It's close to the tenth degree of 31-et and in fact in 31-et cents the major third is 997.978 (just a bit short of 1000 = 10*100)..

So FTS shows that as 997.978 cents=31. Alternatively you can write it as 31=997.978 - to show it that way select.

The = is a kind of shorthand notation for "equal temperament"... :-).

You have to do it all as one word 31=997.978 so that it is clear where one entry ends and the next begins - because otherwise the 31 there would be read as 31/1. Similarly if you use the words, you need to enter it as cents31= all in one word.

Then for example, 3/2 is i.e. 13 percent of the way between 1800 cents31= and 1900 cents31=, i.e. between the 18th and 19th degrees of 31-et.

The box to the right of it will let one set the number of centisteps for a non octave interval as well, such as 3/2, 3/1 etc.

Here's an example

9(3/2)=495.306 9(3/2)=900.0 9(3/2)=1538.56

That is the major chord 5/4 3/2 2/1 in Wendy Carlos's alpha scale of 9 equal divisions of 3/2. As you see, you put the interval of repetition in brackets after the number that shows the number of 100 cent units - here there are 9*100 "cents" to a 3/2 so the 3/2 is shown as 9(3/2)=900.0, and the 5/4 as 9(3/2)=495.306 - i.e. 495.306 centisteps using nine equal divisions of 3/2. As you see, in this notation, the major third 495.306 is close to 500.0 so close to the fifth step of this scale system.

I got the idea of this notation from a post by Margo Schulter to the tuning list suggesting the iota as a tuning measurement - this is a particular case of centisteps where one has 1700 centisteps (iotas) to an octave. The name centisteps as a generic term for this type of notation was also suggested by Margo Schulter.

This let's one choose to call any of the notations plain cents. The advantage of doing that is that you can just use the word cents when entering values, or use shortcuts such as the SCALA convention.

You can still use the notation 12= to enter 12-et based values. For instance if you have set the notation to Wendy Carlos's alpha, you could enter the major chord in mixed notation as: using the 12= value for the 3/2 as 701.955 cents.

. - in particular case of 12, this can also be useful for finding cents values to use for synth tuning tables.

If one changes the number to say 13-et, you will see this:

n(0/13) zn(3/13)-10.05 cents zn(8/13)-36.5 cents n(13/13)

The z converts the n-et notation value to cents so that you can add or subtract the difference to / from it. It's done this way as it is then easy to see which scale degree it is a difference from. If all teh values were in cents one couldn't easily see that.

You can read zn(3/13)-10.05 cents as

Here one could choose to use1300 for the box too, then you see:

0 cents13= 300-10.89 cents13= 800-39.55 cents13= 1300 cents13=

- so you see the cents diffs in the thirteen equal relative cents notation.

The word cents is omitted. All values in cents are shown with decimal point: 152.0 for 152 cents. This is useful when entering scales by hand, and makes the scales easy to read too. It is the convention used for entering scales into the SCALA program by hand, so SCALA users will be particularly used to it. The other way a decimal point could be understood is as 1.5 = 3/2. However, in scale definitions, one usually uses either cents, or ratios, and the pure decimal notation is seldom used, so the SCALA convention is very convenient ..

With this option, the relative cents notation is shown as

,, etc. You don't need to include the decimal point actually, as the = symbol itself is a sufficient clue that it is in relative cents notation.

You have to do it all as one token, i.e. no spaces between the 31=part and the number that follows (otherwise FTS will think the part and the part are two separate notes in the scale).

You use it in conjunction with the option (although that also uses decimal points, the Hz shows that the values are actually hertz rather than cents).

Use of ratios with this option  :

Any formula that includes the symbol is read as a ratio, whether it includes a decimal point or not.

You can use the pure decimal notation with this convention, if you type as.

If you have the SCALA convention selected, and want to use the symbol to express some fraction of a value in cents:. then add the word cents:, or use or to show that it is in cents:.

Alternative cents notations
This is a notation for those who are reasonably comfortable with the idea of number systems with varied radixes.

In twelve tone equal, the usual convention of 100 cents to a semitone works well, with 1200 cents to an octave. However in, say 5-et, one can't immediately see how close or far away a number is from the exact division into 5 notes.

To introduce this notation, let's see the main window Slendro scale in base 5 notation:

You can use instead of to show that it is to base 5.

In base 5, one counts consecutively like this:

1, 2, 3, 4, 10, 11, 12, 13, 14, 20, 21, 22, 23, 24, 30,... 44, 100, 101, ...

= (to base 10)

1, 2, 3, 4, 5,  6,  7,  8,  9, 10, 11, 12, 13, 14, 15,... 24,  25, 26, ...

100 = 25, and 1000 = 125, and 5-et is

So is in the vicinity of, so we see this is an approximately five equal scale, as most Slendro scales are:

The one that's furthest away is at 341.3 cents5, which is 3.2 cents5 away from 400 cents5, or a little under a semitone.

In fact, selecting we see

Even in base 12, it can be interesting to use base 12 notation, with a=10, and b=11, so that you still have a semitone of 100 cents, but you use a00 instead of 1000, b00 for 1100, and 1000 for 1200.

This is just intonation

In base 17 one counts:

The base can go up to 36, which needs all the digits and all the letters of the alphabet.

Here is quarter comma meantone to base 31. v=31, u =30, t = 29,... a = 10

That's not so easy to read, however, is very close to 200, and is even closer to 500. The is just one less than which is the "10" of this base. is just above - so just a little over = as the is the fourteenth digit in this system.

I don't suppose one will use this notation regularly, but it can be useful on occasion.

You can set the octave to some other value than 2/1, e.g .3/1 for Bohlen Pierce scales.

When one has Scala convention, decimal point = cents, then you use to mean.

Ratios options
You can change the highest factor to look for using the box.

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.

. Idea of the tolerance one is that if you enter a value like 1 cents, then it will find the closest pure ratio to the values in cents.

Ex.

Gamelan from South Bali (Slendro)

1/1 235.419 cents 453.56 cents 704.786 cents 927.453 cents 2/1

Set the tolerance to 6 cents and you find that to within 6 cents it is

1/1 8/7 13/10 3/2 12/7 2/1

Set the tolerance to 1 cent and you get

1/1 55/48 13/10 918/611 41/24 2/1

(so the 13/10 is pretty close!)

You can also set a list of prime factors to search for.

Calculator
Type in any expression into the field and you can show it as cents, ratios etc, and see what the result is - this is a straightforward text based calculator, but with special extra features useful for scales work.

Choice of ways of entering your expression:

- 5/4 means the just intonation major third, and can also be written as 1.25.

- all numbers are in cents, e.g. 400 = twelve equal major third. Now 1.25 will mean 1.25 cents.

- you can then use any of the accepted methods of entering values in the main window and other scale boxes. See [#number_options scale notation] for these. You can enter cents values by entering the number followed by the word cents, or you can also as a shortcut use ' or ` for cents e.g. '386 for 386 cents. Or if you use the SCALA convention (which is now the preset in FTS), just make sure you include a decimal point 386.0 and your value will be understood as cents. However you can also use the lattice notation, the alternative cents notation, the n-et notation, or hertz, or any of the notations that FTS understands for scales.

You see the result evaluated in the field. This field changes instantly as you change the expression. The drop list has choice of various ways to show the result - as cents, ratios or decimal, decimal only, or in the same way as the main window scale.

Expressions used in the calculator can have spaces: 5 / 3 etc.

However, normally, in other places in FTS, particuarly ones where you have several entries such as the scale fields, the expression for each entry has to be typed without any spaces, to make it easy for the program to find out where one number ends, and the other begins (otherwise it will try and run everthing together into a single expression).

These formulae can be typed into all the places where you can enter scales for FTS, and in fact, most of the fields will accept the same expressions, even ones where it isn't particularly much needed such as the number of parts in play etc - if you wanted to you could enter the number of parts as sqrt(25)*cos(0) and it would still be understood. Obviously if the number expected is an integer, it gets converted to an integer first, so if you enter PI for the number of parts, then it will be read as 3.

Use * for multiplication and / for division. Use ^ for exponentiation, so 2^3 means 2*2*2 = 8. You can have as many brackets as you like nested to as many levels as you like.

That covers most of what one needs. .

One notation is useful for making scales based on various numbers of equally sized steps- for n-et notation (same as the 4//19 type [#equal_tone Notation for n-et], which you can use in its place). The way it works is that is the fourth scale degree of 19-et. You can read it as a fractional octave notation, so means four nineteenths of the way through the octave - that's to say four nineteenths of the percieved pitch distance between 1/1 and 2/1. So it isn't a pure ratio type notation - in fact, the nearest ratio is 7/6, in this case. Cents is a fractional octave type notation too, dividing the octave into 1200 parts, so you could write for instance, 386 cents as .. You can use decimal points in this notation so e.g. or whatever is valid.

It's case insensitive: is the same as. Also, is the same as, for compatibility with very early versions of the program.

You can't normally mix this notation, or any other notation with cents.

To show how this works let's take an example. is 240 cents, but as a pure decimal is 1.149. Syppose you want the note 2.3 cents above n(1/5).

Ones first thought might be to write however, that will be read as - with the decimal value used for the n(1/5). So there you need some way to indicate that you want to use its cents value.

For mixed expressions like this, use the letter to convert a value to cents so that you can add it to the cents value, like this:

- which will make a note 2.3 cents above n(1/5), i.e..

Other symbols:

= natural exponential = natural log.

You can leave out the bracket if the symbol is followed by a number immediately, e.g. is short for ). Then, by way of example, by the rules of logarithms and exponents, will give the same value as, which is 8.

You can use all the usual trig functions like, ,,,etc, also you can use , etc for Bessel functions - these are used lfor making waveforms in FTS in the option to synthesize a chord in the New Scale and New Arpeggio windows. You can also use the step function =. So for instance K(1, 1.5, 2) is 1, and pi * K(1, 1.5, 2) will be the saem as pi. That's not particularly useful in the calculator but can be very useful for making some waveforms and you can use it in the calculator if you want to investigate how the waveform values get calculated.

The trig functions expect angles in radians - this is a measure based on the amount of the circumference of a circle swept out by the angle. So the full circle sweeps out two pi radians, pi is180 degrees, and pi/2 is 90 degrees. So for instance, sin(pi/2) in radians is the same as sin(90) in degrees = 1.

Most of these can be abbreviated to just the first letter. So abbreviations recognised nclude: P (for PI), G (for PHI), R (for SQRT), L for LOG, E for EXP, C for COS, S for SIN, T for TAN, and also (running out of initial letters at this point) H for COSH, I for SINH, U for ACOS, and V for ASIN.

You can use orfor e because the natural logarithm of 1 is e.

One might wonder what connections these notations have with scales. As it happens there is a lot of interest in golden ratio scales, partly because the golden ratio is in a certain sense "As far away as one can get from a pure ratio", and because of the similarity properties it has, and because it just makes very nice scales sometimes. There's also interest in using pi to define some scales. Square roots are often used as they correspond to dividing an interval exactly in two, e.g. a meantone is a square root (the square root of 5/4). I don't know of any use of cos, sine, or e to construct scales, but one never knows... They are there anyway if anyone needs them.

The calculator recogniseds a comma ' ' as a way of listing values. So if you do a list such as 5/4, 6/5 then you can use the options there to show all the numbers in teh list multiplied together (in this case, 3/2), divided (here, 25/24), added, subtracted etc. You could do these just as well using brackets, e.g. (5/4) * (6/5), and this option to add or multiply etc a list of values has the same effect as that, but saves you a bit of typing.

The comma is just ignored in scales, treated as a way of separating the values, like spaces. So for instance, 1/1, 9/8, 5/4, ... can be used if you prefer to enter your scales like that.

In single entry fields, the comma is treated as an ' ' alternative to a decimal point ' '. This is because it is fairly easy to type instead of - 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. If it wasn't done this way, the parser would read as, 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.

Recognises (apostrophe character) as a shorthand for cent. The applies to the number following it . So 100 means 100 cents. Techy note;: x is short for 2 x/1200. You can use the char (prob. to the left of 1 on your keyboard) as an alternative to.

You only need the single apostrophe before the number.

Ratios are calculated exactly in the Calculator.. It can use 64 bit numbers for both denominator and denumerator, which means that it can handle numbers exactly up to 2^63-1, which is large enough for all but a few very specialised tuning needs. Beyond that point, it will keep the ratio as a double precision floating point format number - which means that - it keeps a record of the first fourteen or fifteen decimal places of each number, and along with that, in a separate field (the mantissa) it stores another number for the position of the decimal point.

When you set it to show prime factorisations of a number, it uses a look up table of the first 1000 primes up to 7919. That is fine for nearly all except very specialised tuning needs as most scale constructors use low number primes - but a few do use very large primes indeed. It will be able to completely factorise any number up to 62710561 (7919^2). For larger numbers, it shows all prime factors up to 7919 but may miss out higher ones.

There's a very impressive on-line factorising java applet here if you want to check the factorisation of larger numbers - http://www.alpertron.com.ar/ECM.HTM - wait for the applet to load. Then enter any number into the top line into the applet. E.g. something like 1234710293478019875098170289750198275089127 or whatever - it can be a really long number. It will factorise it almost instantly.

Earlier versions of FTS used the double precision format only in the rest of the program, and relied on converting those to ratios as needed (ratios could normaly be recognised and distinguished from cents and n-et values if they were very close to a double precision number and if they had no prime factors greater than 50)

However now most of the program can recognise 64 bit ratios just like the calculator. At present a few parts still do this conversion of double precision numbers to ratios as needed (for instance for display of chord ratios using the y notation) - gradually the whole program should get converted to the new format.

Ratio Options
You can increase the maximum quotient from.

(work in progress)

Special notations
[#h_notation Harmonic series] - [#z_notation Hertz] - [#g_notation Geometric sum] [#notaition_harmonic_sum Harmonic sum ]- [#notaition_warp Warp] - [#notaition_formula_etc Formula Product and Sum] - [#more_details_for_harmonic_series_notation More details for the harmonic series notation].

To experiment with these to see how they work, try showing a window. Then you can see how the pattern of the blue dots changes as you change the notation. For the harmonic series ones you will probably want to show a much wider span in octaves - use the field for this.

You can use these notations in the scale definition area - e.g. if you ener h 2/1 there then you make a harmonic series scale.

The sum, product and warp notations make a scale with many terms in it (as many as you like). You can enter the scale into the scale definition field, however, with these large scales, you may want to enter it in the scale description, which you can do using e.g.

With the

Harmonic series notation
The thing about the harmonic series is that you can't describe it as a repeating series in cents notation or ratios, so this particularly needs a special notation. As you go up the scale, the notes get closer and closer together and so there is no repeat of the whole thing. It continues without a repeat, on and on, and eventually gets beyond the range of human hearing.

You can make a harmonic series using:

The idea is that you enter just the one term of the series after 1/1 assumed as the previous term, here the 2/1, and this gets expanded into an entire harmonic series.

(as many notes as you need for the tune)

Or in hertz, suppose that one started at 440 Hz:

So you just keep adding 220 Hz.

Treating 220 Hz as the 1/1, it works like this:

What then about notes below the 1/1, played on keyboard, or PC keyboard or as part of the fractal tune? The harmomic series starts at 1/1 so doesn't go below 1/1.

The answer is that FTS plays the subharmonic series:

You can enter other numbers here in place of the 2/1

In the same notation, makes a harmonic series

i.e.

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

If you are used to thinking in terms of hertz, this one, starting at 220 Hz, is

So the way it works is that you keep adding 44 Hz, the difference in hertz between the 1/1 and the 6/5.

In fact, it will be the harmonic series on the note at 44 Hz, (i.e. 220/5) but missing out the first four terms.

If you enter a number less than 1, say, in place of the , this just reverses the scale, so that it goes down in pitch as it goes to the right.

There are other options you can use too with this notation. See [#more_details_for_harmonic_series_notation More details for the harmonic series notation].

Hertz notation
You can enter a scale in hertz, using the Hz symbol, like this:

One should select before using this notation, so that you can see and set the pitch for the 1/1 as well as the other notes.

Also, to see all scales in this notation, use.

When you enter the scales, you can mix all the notations, e.g. mixed hertz and cents and ratios, e.g.

440 hz 5/4 3/2 2/1

That will make a major scale starting at 440 Hz, and so this method gives one an alternative way to change the pitch of the 1/1 without first showing the window.

If all the values are in hertz, you can just put a at the beginning:

(Septimal minor chord in hertz, pitch for 1/1 = 300 hz)

You need to leave a space after the z.

In the same way,

= n-et notation. n(2/7) for 2nd degree of 7-et etc.

= cents notation

= go back to normal mixed cents / ratios / hertz / decimal notation

You can change the notation in the middle of a scale like this:

which again makes the septimal minor chord.:

Geometric sum notation
or

or in the description field: or

for the geometric sum (what mathematicians call a series), ...

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

for etc, and so on, each interval a third of the previous one.

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

Just version of the Geometric sum notation
for thesum...

i.e. 1/1 2/1 5/2 11/4 23/8 47/16 95/32,...

This is the mathematician's use of the word "harmonic series" - not in general the same as the musicians hamonic series as you can see.

However, if you do

that does make the musicians harmonic series.

Warp
Example:

[2.4/mobile%20in%20sine%20warped%20seven%20equal.ts mobile in sine warped seven equal]

If you leave out the interval, you make it with 2/1 as the interval to divide up. If you leave out the number of notes as well, then you will make a twelve tone scale.

To make equal divisions without any warp using the same instruction, you go:

(or in the scale descritpion field, )

The first number after the scale making instruction is the amount of warp, which is 0. The second is the number of divisions,and the third is the interval to divide.

Enter that as the description of your scale (just that and nothing more) and the scale will change to show:

which is eleven equal divisions of 3/2.

Now add a bit of warp

If all you want are equal divisions without warp then another way to do it is to enter

into the equal steps field in the New Scale window - that's a percentage sign between the number of divisions and the 3/2.

Formula, Product and Sum notation
Example:

for the scale defined by that formula:

i.e. 4/3 3/2 8/5 5/3 12/7 7/4 16/9 2/1

That one is interesting because all its steps are square / (square -1)

4/3 9/8 16/15 25/24 36/35 49/48 64/63 9/8

The notation is

makes the same scale, but makes it as

4/3, 4/3 * 9/8, 4/3 * 9//8 * 16/15, 4/3 * 9/8 * 16/15 * 25/24, ...

also makes the same scale, this time as:

1+1/3, 1+1/3 + 1/6, 1 + 1/3 + 1/6 + 1/10, 1 + 1/3 + 1/6 + 1/10 + 1/15.

That's because the sum of the reciprocals of the triangle numbers is the same as the product of ratios of form square / (square - 1) and both yield 2 - 2/n. See Gene Ward Smith's post to the tuning list at Yahoogroups for 20th August 2004 subject "Pedagogical question".

Normally the product starts with 1/1 - and the sum starts from 0. But you can start them anywhere, so e.g.

which makes

5/4 19/12 7/4 37/20 23/12 55/28 2/1

the scale for [a_little_light_jazz.ts A little light jazz]

and you use the same start notation to set the first value for the product.

More details for the harmonic series notation
Similarly, would give the harmonic series from the 10th harmonic upwards and the corresponding subharmonic series.

Other ratios like 5/3 or 11/9 also pick out members of the harmonic series, with gaps.

For instance 5/3 (= 1+2/3) picks out every other harmonic from the 3rd upwards.

->

and makes

You can also use two or more terms:

will play harmonics 3, 4 and 7, then after that will repeat the spacing between them