# Scale Notations

## Notation for n-et

Example: for 19-et, use:

1/1 4//19

You can alternatively use n(4/19) for 4//19.

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 Arpeggio 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: 1//2 is the tritone (half an octave). The closest low numbered ratio is 7/5.

To show the scales in this notation, select File | scale notation & Number Opts | Cents, Et // notation, or small ratios .

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

1/1 4//12 7//12 2/1

Here,4//12 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:

1/1 1//3 7//12 2/1

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

1/1 is also 0//1 or 0//12 etc

2/1 is also 1//1 or 12//12 etc

You can also make n-et scales using Bs | Scale... To do this, you type the number of steps you want for the octave into the Equal steps 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

1 5//31 10//31 13//31 18//31 23//31 28//31 2

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:

1 3//19 6//19 8//19 11//19 14//19 17//19 2

## scale notation

File | 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:

440 Hz 513.333 Hz 660 Hz 880 Hz

You can also do this at any time, whether the current notation shown is hertz or cents, or ratios or whatever. The Hz in the 440 Hz 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].

Cents or small ratios - 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 Ratio Options | Max ratios quotient 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.

n-et cents or small ratios - this shows suitable values in a special n-et notation devised for the program. n(18/31) = 18th degree of 31-et. Configured using Ratio options | Max quotient for n(5/7) etc. notation . 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.

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

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

Pure decimal - 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 Cents and Ratios].

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

cents per octave 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 Use SCALA convention decimal point for cents .

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 User Inputi.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.

Set this notation as the one to call "cents" 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: Blue Text using the 12= value for the 3/2 as 701.955 cents.

Show cents as diffs from [ 12 ] et . - 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 (3rd degree of 13-et as cents - 10.05) cents

Here one could choose to use1300 for the cents per octave 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.

Use SCALA convention decimal point for cents 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

User Input, User Input, 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 User Inputpart and the 1813 part are two separate notes in the scale).

You can use it in conjunction with the Show Hertz 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 1.5 as 1.5/1 .

If you have the SCALA convention selected, and want to use the symbol / to express some fraction of a value in cents:. 1200.0/7 then add the word cents: 1200.0/7 cents , or use ' or ` to show that it is in cents: '1200.0/7 .

### 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:

0 cents5 44.3 cents5 142.1 cents5 243.2 cents5 341.3 cents5 1000 cents5

You can use cents5 instead of cents 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 0 100 200 300 400 1000

So 44.3 cents5 is in the vicinity of 100 cents5 , so we see this is an approximately five equal scale, as most Slendro scales are:

```0 cents5 44.3 cents5 142.1 cents5 243.2 cents5 341.3 cents5 1000 cents5
```
```0 cents5 100 cents5  200 cents5   300 cents5   400 cents5   1000 cents5
```

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 Show cents as diffs from 5-et we see

1/1 100-0.22 cents5 200-2.34 cents5 300-1.3 cents5 400-3.2 cents5 2/1

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

0 centsc a9.b centsc 1b0.9 centsc 2b3.7 centsc 3a9.b centsc 4b9.2 centsc 5a7.1 centsc 6b6.4 centsc 7b0.9 centsc 8a7.1 centsc 9b6.4 centsc ab0.9 centsc 1000 centsc

In base 17 one counts:

1 2 3 ... 9 a b c d e f g 10 11 ... 19 1a 1b 1c 1d 1e 1f 1g 20.. 90 ... a0... b0... ... g0 g1 g2 ...g9 ga gb gc ge gf gg 100 etc.

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

0 centsv 1tr.u centsv 4ul.9 centsv 80e.i centsv 9ub.h centsv d04.r centsv eu1.q centsv huq.4 centsv jtn.4 centsv mug.d centsv q09.m centsv ru6.m centsv 1000 centsv

That's not so easy to read, however, 1tr.u is very close to 200, and 4ul.9 is even closer to 500. The u is just one less than v which is the "10" of this base. d04.r is just above d00 - so just a little over User Input= as the d 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 5"44.3 to mean 44.3 cents5 .

## Ratios options

File | scale notation & Number Opts | Ratios options

You can change the highest factor to look for using the Max factor 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.

Tolerance in cents for pure ratios . 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

File | scale notation | Calculator

Type in any expression into the Values 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:

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

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

Same as main window scale - 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 Results 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- n 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 n(4/19) is the fourth scale degree of 19-et. You can read it as a fractional octave notation, so n(4/19) 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 n(386/1200).. You can use decimal points in this notation so e.g. n(4.1314 / 19) or whatever is valid.

It's case insensitive: N() is the same as n() . Also, M() is the same as N() , 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. n(1/5) 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 n(1/5)+2.3 cents however, that will be read as (1.49 + 2.3) cents - 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 z to convert a value to cents so that you can add it to the cents value, like this:

zn(1/5)+2.3 cents - which will make a note 2.3 cents above n(1/5), i.e. 242.3 cents .

Other symbols:

Exp() = natural exponential
Log()
= natural log.

You can leave out the bracket if the symbol is followed by a number immediately, e.g. Log2 is short for log(2) ). Then, by way of example, by the rules of logarithms and exponents, 2^3 will give the same value as Exp(3*Log2), which is 8.

You can use all the usual trig functions like Sin, Cos,Acos, Sinh, Cosh etc, also you can use J0, J1 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 K(a.x,c) = 1 for a < x <b, otherwise 0. 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 Exp1 or E1 for 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 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. If it wasn't done this way, the parser would read 0,8 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.

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 File | scale notation | Ratio options .

(work in progress)

## Special notations

To experiment with these to see how they work, try showing a Bs | New Scale... 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 Width in Octaves 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). With these notations, what you enter gets expanded into a complete scale with many entries, when the scales field is next refreshed - e.g. when you show Scale and Arpeggio as text (Ctrl + 104) or exit and start a new session or go to another task or More / Less.

### Description field - Make Scale

It is sometimes more convenient to put the scale definition into the scale description, when you have a scale with many terms and a short way of making it. You can do this using MAKE SCALE followed by the scale to make. E.g.

Make Scale geometric 1/2.

The scale field will fill up with this scale with all its terms while the description is left as it is so you can see what you used ot make the scale, and can easily edit it to make other simlar scales.

This works in the main window description field, or the Scale and Arpeggio as Text window - or the New Scale window - there you can enter it in the Select from field or the New Scale field.

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:

h 2/1

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.

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

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

```220 Hz   440 Hz   660 Hz   880 Hz ...
```

So you just keep adding 220 Hz.

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

```220/220 440/220 660/220  880/220 ...
1/1     2/1     3/1      4/1   ...
```

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:

. .. 1/9 1/8 1/7 1/6 1/5 1/4 1/3 1/2 1/1 2/1 3/1 4/1 5/1 6/1 7/1 8/1 9/1 ...

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

In the same notation, h 6/5 makes a harmonic series

1/1, 6/5 7/5 8/5 9/5 .,..

i.e.

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

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

220 Hz 220+(220/5) Hz 220+(2*220/5) Hz 220+(3*220/5) Hz,...

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 h 1/2 , in place of the h 2/1 , 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:

220 Hz 440 Hz 660 Hz 880 Hz

One should select Show 1/1 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 File | scale notation | Hertz .

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 Pitch window.

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

z 300 350 450 600 (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 = n-et notation. n(2/7) for 2nd degree of 7-et etc.

c = cents notation

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

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

z 300 350 r 3/2 c 1200

which again makes the septimal minor chord.:

### Geometric sum notation

#g 1/2 or

geometric 1/2

or in the description field: Make Scale g 1/2 or Make Scale geometric 1/2

for the geometric sum (what mathematicians call a series) n(0), n(1), n(1+1/2), n(1+1/2+1/4) , ...

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.

### Just version of the Geometric sum notation

#j 1/2

or

JustGeometric 1/2

for thesum (series) 1, 1 + 1/2, 1 + 1/2 + 1/4. ...

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

harmonic 1/1

that does make the musicians harmonic series.

### Warp

Example:

warp 37.52 7 ! Seven equal with 37.52 cents sine warp.

142.094 306.278 498.006 701.994 893.722 1057.906 2/1

Divides 2/1 into 7 equal divisions, then adds a sine wave warp so that pitches get streteched away from the centre of the scale - maximum warp is by 37.52 cents. This is the scale for [2.4/mobile%20in%20sine%20warped%20seven%20equal.ts mobile in sine warped seven equal]

The format is

warp <cents warp> <number of equal steps> <interval to divide up>

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:

WARP 0 11 3/2

(or in the scale descritpion field, MAKE SCALE WARP 0 11 3/2)

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:

63.814 127.628 191.442 255.256 319.07 382.885 446.699 510.513 574.327 638.141 3/2

which is eleven equal divisions of 3/2.

Now add a bit of warp

MAKE SCALE WARP 12.172 11 3/2

57.233 116.556 179.394 246.057 315.641 386.314 455.898 522.561 585.399 644.722 3/2

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

11 % 3/2

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:

formula 8 to 2/1 as 2-2/(x +1)

#f as shortcut for formula

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

formula <number of notes> to <note to stop at or append if you don't readh it> as <formula>

The variable in the formula has to be called x, and gets filled in with the numbers 1, 2, 3, etc to make the scale.

The word as can be omitted - it is just there to make it easier to read.

The to part can also be omitted in which case it just makes the specified number of notes.

Whatever value you put for the to interval gets added at the end - here it added the 2/1 at the end.

If the formula yields a 1/1 at the start or several such they just get ignored as one doesn't normally want repeated 1/1s. That happens with this example in fact, as the first term is 1. So it is an eight note scale including the 1/1. If we tried say 3/6/x we get

The formula may yield negative or zero values, e.g. -5/4. As those wouldn't sound notes whatever the choice for the 1/1, they just get skipped from the list and not shown at all. This feature is primarily designed for constructing scales so we aren't interested in negative frequencies (perhaps they might have meaning in some other context).

The sum and product are similar:

product 8 to 2/1 as x^2/(x^2-1)

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), ...

sum 8 to 2/1 as 2/(x(x+1))

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.

sum 8 to 2/1 start 1/4 as 2/(x(x+1))

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, h 11/10 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.

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

->

1/1 5/3 7/3 3/1 11/3 13/3 5/1 17/3 19/3 7/1 23/3 25/3 9/1, ...

and h 11/9 makes

1/1 11/9 13/9 5/3 17/9 19/9 7/3 23/9 25/9 3/1 ......

You can also use two or more terms:

h 3 4 7

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

```1   3   4   7   9  10  13  15  16  19 ...
```
```  2   1   3   2   1   3   2   1   3 ...
```