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FAQ - Scale Construction

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(How do I convert a ratio to cents?)
(How do I convert a ratio to cents?)
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==How do I convert a ratio to cents?==
==How do I convert a ratio to cents?==
<font color="#FF8000"> ''' File | scale notation | Calculator ''' </font>
You can do this using the Tune Smithy calculator - go to <font color="#FF8000"> ''' File | scale notation | Calculator ''' </font>
Set the <font color="#FF8000"> ''' Expression ''' </font> to <font color="#0080FF"> ''' Ratios or Decimal ''' </font>
Set the <font color="#FF8000"> ''' Expression ''' </font> to <font color="#0080FF"> ''' Ratios or Decimal ''' </font>
Line 76: Line 76:
Set the <font color="#FF8000"> ''' Value ''' </font> to <font color="#0080FF"> ''' cents ''' </font>
Set the <font color="#FF8000"> ''' Value ''' </font> to <font color="#0080FF"> ''' cents ''' </font>
Type in a ratio here and you will see its value in cents.
Type a ratio into the calculator, and you will see its value in cents.
Alterntively, use the javascript applet in the [[cents and ratios]] page. (You are welcome to add this applet and the others in this help to your own web pages too BTW). </font>
You can also use this javascript applet:
Use <font color="#FF8000"> ''' File | scale notation | Cents ''' </font> to show all the scales in cents.
For more about it see the [[cents and ratios]] page. (You are welcome to add this applet and the others in this help to your own web pages too BTW). </font>
In Tune Smithy, use <font color="#FF8000"> ''' File | scale notation | Cents ''' </font> to show all the scales in cents.
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Revision as of 02:05, 7 July 2008

Feel free to add more questions and answers. If you have a question which isn't answered here - for immediate help contact - and you can also add it to the Questions Pending page.

Old version - auto converted and partly edited FAQ - Scale Construction - auto


How do I make my own scales?


The main notations used for scales are cents and ratios. If you are new to them, see cents and ratios

You can just enter the scale you want to use in cents or ratios notation into the main window Scale text box (You may need to use the More button in the main window to show it).

Here is an example: 235.419 453.56 704.786 927.453 2/1

Be sure to include the 2/1 at the end of the scale if you want it to repeat at the octave. FTS can make non octave scales too, so if you leave out the 2/1 then that scale will repeat at the last interval in the scale - in this case 927.453 cents

This assumes you have File | scale notation | Use SCALA convention: decimal points for cents selected which is the preset - otherwise you need to add the word cents after each entry, like this:

235.419 cents 453.56 cents 704.786 cents 927.453 cents 2/1 - or prefix every number with an apostrophe, FTS recognises '386 as an alternative notation for 386 cents

To enter your scale in Hertz - see the FAQ How do I enter a scale in hertz later in this page.

more options

For more space to enter your scale, show the Bs | Scale and Arpeggio as Text ( Ctrl + T (or Ctrl + 104) ).

You save your scales from File | Save As | Files of Type | SCALA Scales (*.scl) . The Scale / as Text window also has buttons you can use to save your scale.

Making the scale interactively

You can also make your scale interactively, with a click to select notes from other scales. THIS SHOULD LINK TO A PAGE ABOUT HOW TO USE THE "NEW SCALE" WINDOW See Bs | Scale... . Here you will find a drop list of several scales with useful pitches in them - select from any of the scales in the drop list to add its notes to your scale. When your scale is ready, then use the Main win... button below the keyboard picture to copy it into the main window. See also [Seeds.htm#New_scales Make new Scale], and [Seeds.htm#save_scale How to save your scale].

You can add notes from equal tempered scales from this window too, with any number of notes to an octave or non octave. Just type the number of notes you want per octave into the Equal Steps window. You can also [Seeds.htm#equal_divisions_of_non_octave make equal divisions of a non octave] here too.

If you are into meantone scales, then select Mean Tone... from the drop list in the New Scale window, and you will get a dialog to enter the amount of the comma and the position of the wolf fifth. Then click Apply , then click Select All in the Scale... window. See [Seeds.htm#new_meantone_scale New mean tone scale].

common scale constructions

There are other scale construction options from Bs | Scales Options... . See [More_scales.htm Scales]

These are basically just a few common scale construction options, plus some others that I happen to have got involved in. For a thorough and complete scale construction program, use Manuel Op de Coul's SCALA.

See [More_scales.htm#SCALA_scales SCALA scales] .

You can show scales from FTS in SCALA using Bs | Scale Options | SCALA Scales | Show currrent scale in SCALA , e.g. to analyse them, and you can show SCALA scales in FTS using File | @Smithy in SCALA.


Large scales tend to have so many notes that it's normal to work with a small collection of notes from the scale. In FTS you do that using Arpeggios. You may want to make your own arpeggios list for it too possibly. See [Seeds.htm#New_arpeggios Make new Arpeggio], and [Seeds.htm#edit_arp_drop_lists Editing the Arpeggio drop lists].

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How do I enter a scale in Hertz?

First, to show the scales in hertz notation go to File | scale notation | Hertz

Then to enter the scale in hertz use the same notation - all the notations you can use to show scales in FTS can also be used to input the scales. So, just follow the same pattern - add the word Hz after each entry like this:

300 Hz 350 Hz 450 Hz 600 Hz

There's a shortcut too - you can prefix the entire scale with a z:

z 300 350 450 600

See [More_scales.htm#Special_notations Special notations].

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How do I convert a ratio to cents?

You can do this using the Tune Smithy calculator - go to File | scale notation | Calculator

Set the Expression to Ratios or Decimal

Set the Value to cents

Type a ratio into the calculator, and you will see its value in cents.

You can also use this javascript applet:

Ratio / = cents


For more about it see the cents and ratios page. (You are welcome to add this applet and the others in this help to your own web pages too BTW). </font>

In Tune Smithy, use File | scale notation | Cents to show all the scales in cents.

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How do I convert cents to a ratio

Here, the thing is that the cents value will only approximate the ratio - maybe very accurately or maybe less so. So there will be a whole range of ratio values you could convert it to, depending on how close to the cents value you want them to be.

The web page in this help: Find all the best ratios for a scale in cents shows all the approximating ratios found. (this uses javascript, which means the calculations are slower, okay for small quotient ratios.).

In FTS, you can use the calculator again:

Set the Expression to Cents

Set the Value to Same as scale

Then from Ratios Options

enter the desired Tolerance in cents to use for approximate ratios

- say 20 cents by way of example.

You also want to select File | scale notation | Cents or small ratios

Now enter a value in cents, say, 300 . You will see the Value field show 6/5~ . The ~ indicates that it is an approxmate value.

Reduce the tolerance to say 10 cents, and this will show 19/16~

That uses the nineteenth harmonic, which is rather rare. Maybe one is interested in a ratio involving smaller prime factors. To do this, enter, say, 2 3 5 7 in the Ratios Options | Factors to use for approximate ratios - and now it will show 32/27~ - the pythagorean minor third, which is closer to equal tempered one than the just minor third. Set the tolerance to 2 cents, and you find 25/21~ . Set it to 1 cent, and now there are no ratios to find within range so it just shows 300 cents.

Actually there are closer approximations - there always are, but they will involve truly enormous quotients. You can go a little higher using the Max quotient for Approximate ratios box - but this slows down the calculation - and the next one is too high for this anyway for this one. If interested in ratios with really large quotients, I have a tiny MSDOS program to find those - let me know if you want it (I may port it to a windows app. at some point).

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How do I show the scale in various notations, such as hertz,. as cents values or the nearest ratios?

Use File | scale notation and simply change the selection there to the notation you want to use.

Also, note that you can use the [ratios_with_factors.htm Find all the best ratios for a scale in cents] applet to convert an entire scale to / from hertz, ratios, cents and n-et notation.

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How do I look for a particular scale in the SCALA archive from FTS? (tutorial)

First you need to download the SCALA archive, and make the drop list of SCALA scales for FTS to use. See [FTS_and_SCALA.htm Setting FTS up for SCALA and vice versa].

Now go to File | More Scales drop lists , or Scales | More Scales drop lists (look in the drop list of scales) and you should see an entry SCALA Scales . Select that one. The scales drop list will then change to show all the scales in the SCALA archive. You can select any of them from the list.

If you don't see this entry yet, check that you have completed the instructions in [FTS_and_SCALA.htm Setting FTS up for SCALA and vice versa].

Now go to Bs | Scales Options | Search Scales or arpeggios list .

Enter your search term in the words to find field. E.g. if you want to find temperaments, enter temperament here.

Now, maybe you decide that list is too long for your current purposes, and you want to refine it to ones with the word well temperament . Just repeat the search, this time with the word well. Each time you search the main window scales drop list in its current form after the previous searches - i.e. it is a search within the previous results.

That gives a more managable list. Now maybe you want to save it - to do so just use File | Save As | Files of type | Lists of scales (*.lsc) and enter the name you want to use, e.g. " Well temperaments from the SCALA archive ".

To get back to your new drop list of scales at any time, go to File | More Scales drop lists and you will see your new drop list there - or at least you will if you have this check box selected (which is the standard setting) - select it if necessary before you make the drop list of scales:

Bs | Scales Options | SCALA Scales | Make / remake scales / modes drop lists ... (Alt + Shift + 6) | Add scales drop lists to the More Scales drop list whenever they are saved or opened

To get back to the original SCALA scales drop list again ready for a new search, you need to re-open it using e.g. File | More Scales drop lists , and select SCALA Scales again.

You can search for several words in one go - if you had searched for well temperament in the first place it would have had the same effect.

For more about this see [more_scales.htm#Search_scales_or_modes_list Search scales or Arpeggios].


If you plan to do a number of searches, it can be useful to use Bs | Arpeggio... as that makes a copy of the main window scales list. Then in that window, unselect the Sync scale with m. w.

Now you can do searches in the main window and whenever you have refined the list to a small number of scales, you can then get back to the big list again by using the Main win <-... button in your Arpeggio.. . window - which is faster than going to File | More Scales drop lists and then selecting SCALA Scales again from the drop list.

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Why is a distinction made between arpeggios and scales - why not just have scales?

It is to reduce the size of the lists. For instance, if you just used scales, you'd need to supply a different major scale for each key in most of the twelve tone temperaments. Each nineteen tone tuning would require nineteen major scales (apart from the equal tempered tuning) and so on.

It also makes it much easier to temporarily select notes from a larger scale. A composer or musician often has a particular master scale for a composition or group of compositions. Then he or she can make a drop list of arpeggios to select notes from this scale to use for particular movements or sections in a composition, or sections of an improvisation and so on.

The arpeggios get used in an even more general fashion in the fractal tunes. You can make arpeggios that start by going up in pitch, then drop down, then continue up again, or go below the 1/1 at times, or maybe go up to the octave or second octave then down to the fifth before rising to the octave again for the next repeat. This complex interaction of arpeggios and seeds sometimes helps to create some fascinating fractal sound worlds.

Or you can make an arpeggio for a chord, or have notes for a whole sequence of chords one after the other, all in the one arpeggio.

Why call them Arpeggios? Mode sounds much better.

Yes indeed, and these are usually referred to as Modes in tuning circles so this is entirely understandable.

A mode in this sense is any selection of notes from a larger master scale. They can ascend and descend by different routes too, as in Indian or Arabic modes..

However, in FTS I use Arpeggio even more generally than this, since they can change direction mid course, or have repeating notes, or even have negative values too to go below the 1/1 of the master scale - all of which can be useful in fractal tunes. This is perhaps a little too general to be able to call them modes in the tuning sense.

Another reason for calling them arpeggios rather than modes is that there is some risk of confusion with modal music. The use here is similar enough to the tuning concept to be potentially confusing to newbies, e.g. for those who have a background in Irish or Scottish folk music, which still uses the ancient system of modes. Indeed, it could confuse players of Jazz too. In this context a mode is understood as a particular sequence of whole tone and semitone steps, rather than as a selection of notes from a larger scale. See this site: A Beginner's guide to Modal Harmony. Other sites: In search of the Wild Dulcimer (table of contents). For modes as used in Jazz, see. Jazz Modes.

I did originally call them modes in the interface, but changed to Arpeggio to help reduce confusion for newbie users.

Tuning specialists and those who are already familiar with this field can read Arpeggio in this help to mean Mode in nearly all the contexts in which it is used - except for a few cases where the fractal tunes are discussed. In those examples it may be used too generally to count as a mode, and is best understood as an Arpeggio .

In short, the Mode as used in tuning circles can be regarded as a special case of Arpeggio - normally one that starts at the 1/1, ascends to the octave or scale repeat, and optionally descends again to the 1/1 by a different route.

Because the Arpeggio in FTS is so flexible, a more general word is needed here.

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I would like to use arpeggios which ascend one way and descend another way as in Indian or Arabic modes. Can FTS help with this?

Yes it can!

It is an easy matter to add such modes to FTS - you can use the melodic minor as a model for how it works. You can find this in the twelve tone scale arpeggio lists

For details, see How to Make New Arpeggios - that mainly focusses on how it is done interactively in FTS.

To summarise briefly, it's done like this:

0 2 3 5 7 9 11 12 " 0 2 3 5 7 8 10 12

where the numbers are scale degrees, the first half before the double quote is the ascending mode and the second half is the descending one. So this means that it will ascend 7 9 11 12 and descend 12 10 8 7.

Something to be aware of - your arpeggios will need to ascend and descend by the same number of white notes per "octave" of the midi keyboard layout - so if it has fewer notes ascending than it has descending or vice versa, just repeat some of the notes in oneof the directions. You would do it so that the others still align up like this:

   0 2 3 5 7 9 11 12 

"  0 2 3 3 7 8 10 12

(melodic minor with the fourth left out in the descending arpeggio - you just repeat the previous note at that point).

The melodic minor arpeggio is mainly included here for use in fractal tunes. It may not be so very useful for keyboard work in Western music, except for beginner keyboardists - as it is easily fingered. After all in Western music one has only twelve keys and a keyboard designed for the scale one is using, and only one such mode to learn. However it is possible that in Indian and Arabic music, these modes may be rather useful indeed. They would also be exceedingly useful for making fractal tunes too as you could make tunes that automatically follow the traditional modes.

If anyone is interested in compiling such a database in a form useful for performers of Indian or Arabic modes, please contact me. The only database I have is the one for Scala which ignores such differences between ascending and descending modes.

If you have a large database already prepared in some particular notation system, or just prefer to use your own notation system to enter the scales - do contact me first before attempting it. If it is a text based notation system, FTS may be able to recognise it.

I could probably set FTS up to read your database all in one go and make an arpeggios list from it. Just let me know how the notation system works - with a list of all the symbols used and what they mean - and I will let you know if I can program FTS to read it :-).

Don't worry if it seems a little complex either and takes a while for people to learn it - I don't need to learn it myself, only need to know what the rules are so I can program them, and then from then on FTS will be able to read it. It may well be routine to program. Even if it takes a little while, the results would be well worth it.

More information: for arabic modes, The Arabic Maqam World, Near Eastern Modes. For Indian modes, see The sound of India.

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What is a scale?

Some of the things said here may seem confusing if you have been brought up with the conventional idea of a scale in twelve tone music. You talk about practicing arpeggios and scales. Then you say a piece is in say the key of D minor meaning that the home scale for the piece is the scale of D minor.

However, this doesn't tell us whether the tuning of the scale is twelve equal, or quarter comma meantone, or in Werckmeister III - the terminology ignores tuning distinctions. Also what do you do about tunings that don't fit into the normal system of note names and keys at all? What scale are they? E.g. what scale is the tuning with seven equally spaced tones each slightly narrower than a whole tone (seven equal)?

In this field of the study of tuning systems, it is usual to use the word Scale to refer to the precise tuning used. It is hard to think of a better term to use too, even if it is potentially slightly confusing perhaps for a newbie.

Discussions, literature and other programs use the word in this sense, and that is how we will use it. There doesn't seem to be any reasonable alternative. We can reserve the word key for the other usage. Just keep in mind that scale here has aquired a new meaning - the particular precise tuning used, usually presented in cents or ratios notation.

In this way of using the word, the home note of the scale gets called the 1/1, or 0 cents, which can be set to whatever pitch is convenient for you. All other notes are measured from it, e.g. the fifth as 700 cents, or 702 cents or 3/2 etc depending on the tuning, and the octave at 2/1 or 1200 cents. In this use of the word, all the major scales in twelve equal would be regarded as the same scale based on different pitches, 0 200 400 500 700 900 1100 1200 (in cents).

It just takes a bit of getting used to it, and you need to keep the two meanings of the word distinct. Tuning is too broad in its meaning nor it seems is there much risk of confusion in practice once you get used to it.

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