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Superparticular Scales

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Superparticular Scales


Make scales with all the steps superparticular such as 9/8, 5/4 etc...

A scale is superparticular if all the steps are the likes of 3/2, 5/4, 6/5, 7/6, 9/8... or generally, (n+1)/n for some n.

Low numbered superparticular ratios tend to sound particularly harmonious.

Examples of superparticular scales:

just intonation diatonic: 1 9/8 5/4 4/3 3/2 5/3 15/8 2 steps 9/8 10/9 16/15 9/8 10/9 9/8 16/15

pentatonic: 1 9/8 5/4 3/2 5/3 2 steps 9/8 10/9 6/5 10/9 6/5

How to show this window

Ctrl + 13

How to use this window

Click Find all superparticular scales to find a complete list of superparticular scales using David Canright 's algorithm, as described in Superparticular Pentatonics.

Now look in the main window Scales list or Scales Drop List (Ctrl + 110) to find them.

The standard setting here is to find all the 5 note scales with superparticular steps between 81/80 and 5/4 and with the steps adding together as intervals (multiplying as ratios) to give 2/1. Vary the parameters to make other types of superparticular scale.

The steps will be shown in decreasing size, for instance:

5/4 7/6 8/7 8/7 21/20

steps for the scale:

1/1 5/4 35/24 5/3 40/21 2/1

How to use this window to find an interesting superparticular scale

After you have made a list of superparticular scales, choose one that interests you and click Fill main window scales drop list with all re-arrangement of current scale's steps in Scale and Explorations (Ctrl + 17).

In this case, 60 re-arrangements are found. Let's now look for one with a 3/2 in it. Look in the definitions in the Intervals drop list. There are two possibilities, and maybe this one catches your eye:

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


8/7 21/20 5/4 7/6 8/7

Then we are done.

This list is small enough so that one can scan down it by eye. However, for larger lists one could explore the use of Search Scales or Arpeggios (Ctrl + 14) to find particular intervals of interest in the list, such as 3/2 or 5/4 or whatever.

How to add extra steps to make a scale superparticular

Another way to use this window is to make a scale into a superparticular one by adding extra steps. Let's see an example of this.

The Pygmie scale: 1 8/7 21/16 3/2 7/4 2 steps 8/7 147/128 8/7 7/6 8/7 has four superparticular steps, and one wide step which is not superparticular: 147/128

Perhaps we'd like to make the Pygmie scale into a superparticular one by adding an extra note.

Enter 147/128 into the top interval box.

Set the number of notes to 2 , and leave the smallest interval (or range) at its standard setting of 81/80 - 5/4 .

Click the Find all superparticular scales button.

There is only one solution 9/8 49/48

Pygmie scale with an extra note to make the scale superparticular: 1 8/7 9/7 21/16 3/2 7/4 2 steps 8/7 9/8 49/48 8/7 7/6 8/7

Or with the two steps the other way round: 1 8/7 7/6 21/16 3/2 7/4 2/1 steps 8/7 49/48 9/8 8/7 7/6 8/7

Background - superparticular intervals and difference tones

An interval is superparticular if all the steps are the likes of 3/2, 5/4, 6/5, 7/6, 9/8... or generally, (n+1)/n for some n.

Low numbered superparticular ratios tend to sound particularly harmonious.

They are successive notes of a harmonic series. For instance 5/4 is the interval between 5/1 and 4/1, the fifth and fourth notes of a harmonic series. The interval has a difference tone which you may hear as an extra note when you play it on some instruments, at the 1/1 of the harmonic series. For instance notes at an interval of 5/4 played together have a difference tone four octaves below the lowest note of the pair.

This can be heard, and is visibly present in the combined waveform, though since it is a result of combining other higher frequency notes, it doesn't show up when you analyse the sound into its constituent sine waves.

To find out what pitch the difference tone will be for two notes see Beats - Expected beat patterns for pure harmonic timbres (Ctrl + 2).

Some examples of difference tones: For a fifth, the difference tone is an octave below the lowest note. For major thirds, two octaves below. For minor thirds, it is two octaves plus a major third below the lowest note (completes them to make a major chord).

Show as steps in description

Used with the search for all re-arrangements of the current scale...

Each scale found needs a description to show in the Scales list which you use to select the scale. There are two options here - to show the steps of the scale, and to show the intervals from the 1/1.

Select this check box to show the steps in the description, unselect to show the intervals from the 1/1.

This check box only affects the description. You change the way the scale itself is shown by using the Steps button in Scales Drop List (Ctrl + 110) or the Steps check box available in the main window for some of the tasks. The Steps check box or button of course doesn't change the scale descriptions.

Help = F1

Click for help for this window. Or F1. Other opts: Shift , Alt, Ctrl + click...

F1 or click shows the help for the current window in your web browser.

Some windows may have no help yet in which case the help icon is shown crossed out with a red line.

Shift + F1 or Shift + Click brings up the tool tips extra help window (this window) to show any extra help for a tool tip.

You can tell if a tool tip has extra help if it ends ... like this one.

Ctrl + F1 or Ctrl + click takes you to the list of keyboard shortcuts for Tune Smithy.

Alt + F1 or Alt + click (alternatively Caps lock physically held down + F1 or Click) takes you to the on-line page at the web site about the current main window task - which gives a short introduction to it for newbies to the program. If there is no on-line page specific to a task, takes you to the main tune smithy page on the web site.

Since the help for Tune Smithy is currently a bit out of date and needs to be redone completely for the new 3.0 release, then you may find the on-line page for some of the newer tasks particularly useful.

Organise Windows = F2

Or F2 - Reset / save / open for individual windows, right click for cat. list...

Shows the Organise windows window - which you can use to reset all the parameters for the current window - or save them all, or open previously saved parameters for just this window. Also has a drop list of all the windows and their shortcuts.

Scales Dialog Star

Tip of the day - Scales category - right click for neighbouring windows...

Left click for a tip of the day in this category.

Right click to see a menu of neighbouring windows.

The neighbours are the ones you most often move to after this one or within a minute of this one, arranged by popularity.

So as you continue to use FTS, it will learn your habits, and the neighbouring windows listed here, should be the ones you most often visit after this one.

Neighbours, and Previous - Up - Next

N.B. This list of neighbours may change when these pages are updated.

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