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Microtonal Chords

Chord Progression Player Microtonal Chords

Microtonal Chord Symbols - Accepted Note Names - Make new chord symbols using the Extra chords window - Chords for other tuning systems


Microtonal Chord Symbols

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Contact Robert Walker: with any questions or suggestions.

Intro - Tuning Symbols - Adding extra microtonal notes to a chord - Example chords - How to hear these chords in just intonation harmony


You can use microtonal chords with the chord progression player too. These use various modifications of the standard chord symbols to indicate the tuning desired for all the pitches

How the notation works, C chords, Chords based on other notes

How the notation works.

The idea is that just as the normal chord symbols select from twelve equal, microtonal chord symbols can select from various just intonation twelve tone scales. So for instance, Cjm7 is like Cm7 but it choses the most simple just intonation five limit ratios for all the notes, to give 1/1 5/4 3/2 16/9 instead of 0.0 300.0 700.0 1000.0 cents. Similarly, Csm7 gives 1/1 7/6 3/2 7/4 and so on.

The symbols are:

p = Pythagorean, j = Just intonation, k = just intonation with 10/9 and 9/5, in the place of 9/8 and 16/9 (useful for dominant sevenths), , s = Septimal, ud = Undecimal, t = Tridecimal, h = Harmonic series twelve tone scale, eq = reset to twelve equal. Then also you can use o, u, ou, or uo to make various overtone or undertone series based twelve tone scales. For details see Tuning Symbols

If no prefix is given then the chord is understood as tuned to twelve equal pitches, so C then stands for 0.0 400.0 700.0 12000.0 - the twelve equal major chord.

The scales used for these notations are listed in the Symbols section of this help. You can also look them up using a Bs2 | Scale window. In the Name field for the new scale, enter for instance,

#12 j

and you will make the twelve tone just intonation scale used for the chords with a j in the chord symbol. You can use the same method for them all to see what they are, e.g.#12 s to show the septimal scale used, etc.

To see the target tuning for some particular chord, enter

#chord Csm (say)

in the Name field.

You can try this out as you go through the examples in this help, to see how they get tuned. As a shortcut way to show the exact tuning for the currently selected chord in the progression, you can use Shift + Click on the To Arp button - which will use this same method to show the chord in a New Scale window instead of the usual New Arpeggio window.

This way of showin the tuning of the chord in a Bs2 | Scale window works best with C chords. In any case the C chords are rooted on the 1/1 so it is easiest to read their ratios to see what they are. So we will go through the C chords first, then the next section will introduce chords on other roots. C here means the current 1/1 - will be the concert pitch C if you have it set to middle C in the pitch window, and otherwise, whatever pitch you have set for the 1/1.

C chords

As you see, the chord symbol notation specifies exact pitches. . However, the chords in the chord progression normally play the closest available scale pitches in the current scale. So you need to make sure that the scale you are using has sufficiently fine distinctions of pitch to distinguish them - for instance, in most twelve tone scales Cm7 and Csm7 will be identical. To hear a distinction, try them with the larger equal temperaments such as nineteen equal, or thirty one equal or seventy two equal. You can also try some of the larger just intonation scales.

Maybe though you want to hear the exact pitches for the chord symbol - well you can do that too - prefix the chord symbol with 'x' to get the exact pitches. So for instance xCh7 will play the harmonic series seventh chord 1/1 5/4 3/2 7/4 with those exact tunings whatever the current scale. While Ch7 will probably just play the nearest pitches to it in the current scale tuning, probably the normal dominant seventh chord for twelve tone scales..

You can mix notations in a single chord, because the tuning symbol gets used for notes that follow them. Cjmh7 for instance would be the just minor chord 1/1 6/5 3/2 with a harmonic seventh 7/4. The h there only modifies the seventh.

You have to put the symbol before the m in minor chords, so Csm is the septimal minor chord but Cms is the same as Cm (the m here counts as a replacement note you see, replacing the major third in the chord by a minor third, so to retune the result you need to prefix it with the symbol to say how to tune the third).

The s affects the major third if placed immediately after the letter so Cs is 1/1 9/7 3/2. More generally, a symbol placed immediately after the note name works backwards as well as forwards, so Cs affects the notes already made at that point as well as the ones still to come, i.e. C E and G. Any subsequent symbols only work forwards. So in Cjs7 the basic chord is 1/1 5/4 3/2. The j works backwards and forwards, but gets superseded by the s in the forwards direction, so you have added a septimal seventh at 7/4 so the complete chord is 1/1 5/4 3/2 7/4.

So, if you want to use the same scale for the entire chord, the easy solution is to put the symbol right at the start after the note name. Since it works backwards as well as forwards, it will retune the entire chord.

You can also interpret the entire chord in 19, or 31 equal. Prefix the chord symbol by '?' (question mark) for 31 and '*' (asterisk) for 19. So for instance, ?C7 stands for 0 10 18 26 31 in thirty one equal, or whatever are the nearest pitches to those 31-et degrees in the current scale. This notation can only be used at the start of the chord..

You still use the twelve tone numbers with these notations, so /7 means add in the nearest pitch to the seventh, so degree 26 of thirty one equal (depending on what flavour of seventh you are using). If you really want degree 7 of thirty one equal, - a septimal minor third - see Adding extra microtonal notes to a chord,

You don't need any prefix for 72 equal as the sharps and flats are the same as for twelve equal, so it would have no effect on how it gets tuned.

If you use the other notations in the middle of one of these chords, the last used tuning symbol takes precedence - so ?Ch7 would be 0 10 18 in thirty one equal for the major chord, then the 7/4 added in to that at the end as the harmonic seventh.

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Chords based on other notes

When the chord is based on some other note other than C then the entire chord gets transposed to the nearest pitch to the named note in the current scale. By the current scale - this refers to the main window scale in FTS.

So, if you are making chords in a Bs2 | Scale window and want to test these ideas to see how they work, then be sure to set the main window scale to an appropriate scale - the scale to select notes from in the New Scale window will be ignored..

So for instance, Esm is the chord Csm 1/1 7/6 3/2 transposed to the nearest pitch to E in the current scale. Ch is a just intonation harmonic seventh chord 1/1 5/4 3/2 7/4, and Eh7 will transpose that chord to the E of the current main window scale.

Again, you can make these chords using the Bs2 | Scale method by entering

#chord Gh7 (say)

in the Name field.

If you try this with a scale which has the G tuned to 3/2 for instance, then you will find that it makes

3/2 15/8 9/4 21/8 2/1

You will see that it has added a 5/4, 3/2 and 7/4 above the E, as expected, then it adds an extra 2/1 at the end, which isn't in the chord. That is so that when playing these chord notes from the keyboard to try it out in the New Scale window, you can try it out not just in the original position but also at octave repeats above or below it - you need to remember to skip the extra pitches for the 1/1 and octaves when you do that..

It will add in the 2/1 unless it already occurs in the chord. For instance, if you make

#chord Fj

and the scale has a 4/3 on it to use for the F, it will make it as:

4/3 5/3 2/1

since the 2/1 occurs as part of the chord.

But if you make

#chord Fh7

then even though the chord still has a 2/1 in it, it will make it as:

4/3 5/3 2/1 7/3 2/1

That's so that you can play the extra note beyond the octave, and again you are expected to skip the second octave after the 7/3.when trying it out from the keyobard.

Note that if you are using these symbols in a tuning that allows fine distinctions of pitch, you will find that the root of a chord such as say Eb is tuned a bit sharp of the normal twelve equal Eb, and D# a little flat. The notes in increasing pitch are D, D#, Eb E, equally spaced.. That is because the note names follow the nineteen tone system in order to make a distinction between sharps and flats - useful in microtonal contexts with notes of more than twelve notes, and it isn't going to make any difference if the current scale has only twelve notes. If you want particularly to use the twelve equal note system here, then use a tilda: E~bh7, and for finer distinctions than bs and #s you can also use + and - for the 72 equal names and various other possibilities - see the Accepted note names list.

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A few microtonal instrument Links

The nineteen, thirty one etc equal temperaments are favoured by microtonal guitarists because equal temperaments work out naturally as arrangements of guitar frets. See for instance John Starret's microtonal guitar page - Microtonal Guitar Conversion FAQ and Pictures. They work well on other instruments too for the same reason that twelve equal got favoured - if you change the key, you find they are all tuned identically.

Some guitarists use just intonation fret arrangements, which tend to be a bit more complex, things like frets that only go part of the way across the guitar. See Dante Rosatti's Adventures in Just Intonation Guitar. Also David Canright's A Justly Tuned Guitar. Another approach is to just play your guitar fretless.

For other microtonal instruments see John Starrett's list of Microtonal Instruments as a starting point. There are links to other instruments in this help too, mainly electronic ones - see Special microtonal instruments

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Adding extra microtonal notes to a chord

You can use the @ or / notation to add microtonal notes.

You can add any interval at all, so it lets you build up any chord you like as you can add notes as ratios or using cents values. To do that you use r to indicate that it is a ratio or cents value, to be undertsood in the way that FTS interprets scale pitches, e.g. r13/10 etc,

The @ notation is needed for ratios for obvious reasons. So to add a 13/10 for instance you can use @13/10. To add 850.0 cents use @850.0 or /850.0 To add the seventh degree of thirty one equal use @rn(7/31) or /rn(7/31)

In that last example, the 'n' is the n-et notation understood by FTS so n(7/31) stands for the seventh degree of 31-et. In this case, you need to prefix it by r which says that this is to be understood as a scale pitch (ratio or cents or other notation understood by FTS).

You could use the r with ratios like 13/10 etc too but there it can be left out as @r13/10 is the same as @13/10 - the presence of the / shows that it is a ratio anyway and there is nothing it could be confused with. You can also leave it out with the cents values because the decimal point shows that it is a cent value anyway.

You can also add extra notes by name in 19 or 31 equal. See.

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Tuning Symbols

Various twelve tones - Using the symbols with the chord note names - Pythagorean - Harmonic - Overtone -Undertone - Mixed Overtone and Undertone,

Various twelve tones

Note.These aren't all necessarily intended as playing scales particularly. They are to be thought of rather as fertile sources of notes for chords and to use in chord symbols.

x = play exact pitches rather than the closest pitches in current scale- this has to occur before the note name that gives the root of the chord. Root of the chord is tuned to the nearest pitch to it in the curent scale.

y = Same as x, but root of the chord is tuned to the exact pitch specified as well.

eq = Twelve Tone Equal Temperament
h = Harmonic with out of order twenty first and twenty third harmonic:- 17/16 9/8 19/16 5/4 11/8 21/16 3/2 13/8 23/16 7/4 15/8 2/1
- notes are remapped if necessary, to let you use e.g. Ch23 which will play as expected - see Harmonic
j = Just intonation 16/15 9/8 6/5 5/4 4/3 36/25 3/2 8/5 5/3 16/9 15/8 2/1
(uses a 36/25 for the tritone to get nice diminished seventh chords)
k = just intonation with 10/9 as the second: 16/15 10/9 6/5 5/4 4/3 36/25 3/2 8/5 5/3 9/5 15/8 2/1
(this is the same as j except that it has 10/9 and 9/5 instead of 9/8 and16/9)
n = same as h, but played just as is - for techy readers who want to experiment with the h notation to see how it works - see Harmonic.
o = Harmonic Overtone scale 17/16 9/8 19/16 5/4 21/16 11/8 3/2 13/8 27/16 7/4 15/8 2/1 - see Overtone
ou = Mixed Harmonic Overtone and Undertone scale 16/15 8/7 19/16 5/4 4/3 11/8 3/2 8/5 32/19 7/4 15/8 2/1. - see Mixed Overtone and Undertone
p = Pythagorean 2187/2048 9/8 32/27 81/64 4/3 729/512 3/2 128/81 27/16 16/9 243/128 2/1. See Pythagorean
s = Septimal 28/27 8/7 7/6 9/7 4/3 7/5 3/2 14/9 12/7 7/4 27/14 2/1
u = Harmonic Undertone scale 16/15 8/7 32/27 16/13 4/3 16/11 32/21 8/5 32/19 16/9 32/17 2/1 - see Undertone
uo = Mixed Harmonic Overtone and Undertone, with 16/11 for tritone - 16/15 8/7 19/16 5/4 4/3 16/11 3/2 8/5 32/19 7/4 15/8 2/1- see Mixed Overtone and Undertone
ud = Undecimal 33/32 12/11 11/9 14/11 4/3 16/11 3/2 22/14 18/11 11/6 64/33 2/1
t = Tridecimal 13/12 15/13 13/11 16/13 4/3 13/9 3/2 13/8 22/13 26/15 24/13 2/1

The just, septimal, undecimal, and tridecimal scales provide the most simple ratio of the desired type available in the vicinity of the corresponding twelve tone note. The k notation is useful for e.g. dominant sevenths using 9/5 instead of 16/9 for the seventh.

The h, o, ou and u notations give various ways of notating various harmonic series based chors. The o and u notations can have numbers added, e.g. o3 to use the overtone series based on 3/3 instead of 1/1..

The 4/3 and 3/2 get retained in all the systems apart from the harmonic based series notations. If you want your fourths and fifths to be 7, 11 or 13 limit you need to add the notes you need individually, for instance you can use @r13/10 to add in a 13/10 to your chord.

Some of these notations need further comment, - see below, starting with Pythagorean

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Using the symbols with the chord note names

Normally the symbols are only understood as applying to intervals within the chord. So Ej for instance is a just intonation major chord rooted on the closest to the twelve equal E in the current scale. But you can also apply them to the note names too, if you put the symbol before the note name.

So for instance, Ebs will give you a septimal major (super major) chord 1/1 9/7 3/2 on whatever note is closest to Eb in the current scale, and if you look at the exact pitch, then yEbs will give you a septimal major chord on 300.0 cents.

However, if you put the s before the note name, ysEbs will give you a septimnal major (super major) chord on an Eb at 7/6, i.e. 7/6 3/2 7/4, and without the y for the exact pitch, sEbs will find the closest approximation to this chord rooted on on whatever pitch is closest to 7/6 in the current scale.

If instead you wanted a 1/1 5/4 3/2 chord rooted on 7/6 you can do it as ysEbj, which will give 7/6 35/24 7/4 as your chord, and so on.

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p = Pythagorean 2187/2048 9/8 32/27 81/64 4/3 729/512 3/2 128/81 27/16 16/9 243/128 2/1

That first step 2187/2048 is the chromatic Pythagorean semitone - the step that converts a note to the corresponding flat or sharp in pythagorean notation.

So for instance, Eb at 32/27 is a chromatic semitone flatter than 81/64. The D# which you reach if you go up by a chromatic semitone from 9/8 is at 19683/16384, which is actually a little bit sharper than the Eb

In. normal meantone based tuning systems (e.g. quarter comma meantone, or nineteen or thirty one equal), the flat is flatter than the sharp and so the notes go in order D D# Eb E. But in Pythagorean, as we see here, the order of the notes is D Eb D# E. That will happen whenever you have a fifth sharper than 700 cents.

You find this same ordering of the accidentals in schismic temperaments - those are temperaments which include pure or near pure major thirds like meantones, but do it by approximating 8/5 as eight fifths, rather than 5/4 as four fifths as normal meantones do. They also need sharp fifths - and the same happens with other equal temperaments with near pure or sharp fifths such as 29 equal for instance..

Since this is the proper way to interpret the Pythagorean notation then when you use the p notation, you get a distinction between sharps and flats. Then for instance Cp7 (which uses the flat) gives the chord 81/64 3/2 16/9 2/1. There the Bb at 16/9 is 4/3 below the F at 4/3,which is the normal tuning for Bb in Pythagorean scales. This will be distinct from Cp#6 which will give you 81/64 3/2 59049/32768 2/1.

You can also use double sharps and flats etc and they will work the same way. For instance, the second note in the Arabic seventeen tone pythagorean system, is 65536/59049, just a bit flat of 9/8, and close to 10/9. It could be notated using pythagorean accidentals as an Ebb. If you wanted a Csus2 and wanted to use this note in the Arabic seventeen tone system, you could do it as Cpsusbb3 which will make 1/1 65536/59049 3/2 2/1.

These pythagorean double flats can only currently be used withn the symbol and not in the chord root note name. So the extra flat in e.g. pEbb will be ignored.

Later on I will be adding in support for the newly developed Sagittal notation, which extends these pythagorean ideas to let one use any size of accidental one wants, for instance you can notate 5/4, 7/4 etc using suitable Sagittal accidentals as offsets from the pythagorean pitches.

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h = Harmonic with out of order twenty first and twenty third harmonic:- 17/16 9/8 19/16 5/4 11/8 21/16 3/2 13/8 23/16 7/4 15/8 2/1

The idea here is that you can use Ch7 for the harmonic seventh, and e.g. Ch9 for the harmonic ninth 1/1 5/4 3/2 7/4 9/4, then Ch11 for the harmonic eleventh 1/1 5/4 3/2 7/4 9/4 11/4 and so on.

So, it has 7/4 instead of 16/9. Retains the 9/8 as it is useful for the harmonic ninth, and for similar reasons, has an 11/8 instead of 4/3 for h11, also a 13/8 for h13, and more notes for a complete harmonic chord up to Ch23.

Ch23 will be interpreted as

Degree       0    4   7   10   14   17   20    23   25    27     30    33
Name (in C)  C    E   G   Bb   D    F    Ab    B    C#    Eb     F#     A      
Interval    1/1 5/4  3/2 7/4  9/4  11/4 13/4 15/4  17/4  19/4   21/4  23/4

I suppose probably normally you would think of the notes of a fifteenth chord for instance, if you ever had one, as going like this:

Degree       0    4   7   10   14   17   20   24
Name (in C)  C    E   G   Bb   D    F    A    C

notated perhaps as C7/9/11/13/15

But if you can play pure harmonic series pitches, it makes more sense to play the thirteenth and fifteenth as the Ab at 13/4 and the B at 15/4. So these harmonic series chords are interpreted that way if you use them for numbers beyond the eleventh, such as Ch15 etc.

That's as far as this harmonic series notation can go, one might think, up to 19/4. The problem you see is that 21/16 is between 5/4 and 11/8 (for 4/3) so between the E and the F so there is no scale degree unoccupied to assign it to in a twelve tone scale. But if we don't worry about making the scale ascending then we can add in a couple more terms up to 23/16. The idea is to assign them to the F# tritone and the A neither of which is needed for harmonic series chords (and both of which in fact - the sixth and the tritone - are rather rarely used in chord symbols anyway).

So we may as well assign them to something useful for harmonic series chords. So we just assign the remaining ratios we need to the remaining notes in the twelve tone scale, so we assign F# to 21/16, and A to 23/16 (N.B. it would make excellent sense to assign the F# instead to 23/16 which it approximates, so that just the A is out of sequence, but this confuses the part of the player that arranges the arpeggio degrees in ascending order).

Using this scale, we can play harmonic series chords up to Ch23. If you wanted to add in extra harmonics to a chord, you can use the usual method, e.g. Ch9add13 would add in the 13th which in this scale is the 13th harmonic (of 1/4). Ch9add13add23 would add the 13th and 23rd harmonics, and so on.. The player does the necessary translation of 23/16 to the scale degree needed (to get to what one would normally expect to be an A) so you don't need to work that out yourself..

You can also do the even harmonics. It translates those too. So 10/9 is 5/4 transposed up an octave, so scale degree 16. But you just need to do that as Ch9add10 which would add the tenth harmonic.

Indeed you could also add in the 28th harmonic too as that is just the 7th harmonic transposed up a couple of octaves. Do that as Chadd28 which will play 1/1 5/4 3/2 7/1.

Basically you can just think of it as a way of playing harmonic series notes using chord symbols. It lets you play all harmonics up to the twenty third, and you can play any of those in any octave.

If you want to get into the way the notation works, then here are the techy details:

Here is a table for the scale itself:

Degree       0    1    2    3    4   5    6    7    8    9    10    11  12
Interval    1/1 17/16 9/8 19/16 5/4 11/8 21/16 3/2 13/8 23/16 7/4 15/8 2/1

Here is the table for the consecutive harmonics from 5/4 onwards up to 23, and the associated degree numbers that they actually play -

Degree      0     4    7   10   12   14  16    17    19  20   22   23     24  25   26    27   28   30   29   33
           1/1   5/4  6/4  7/4  8/4 9/4 10/4  11/4 12/4 13/4 14/4 15/4  16/4 17/4 18/4  19/4 20/4 21/4 22/4 23/4
Interval   1/1   5/4  3/2  7/4  2/1 9/4  5/2  11/4  3/1 13/4  7/2 15/4   4/1 17/4  9/2  19/4  5/1 21/4 11/2 23/4

We can also play the next one, 24/4 = 6/1 but not the 25/4 because our scale has no ratio involving a 25.

So here is a table for some of the higher harmonics we can play:

Degree     31     32    34    35   36   37   38   39    40   42     41   45   43   44   46   47   48   49   50   51   52   54  53   57   60
Interval  24/4   26/4  28/4  30/4 32/4 34/4 36/4 38/4  40/4  42/4  44/4 46/4 50/4 52/4 56/4 60/4 64/4 68/4 72/4 76/4 80/4 84/4 88/4 92/4 96/4  
           6/1   13/2   7/1  15/2  8/1 17/2  9/1 19/2  10/1  21/2  11/1 23/2 12/1 13/1 14/1 15/1 16/1 17/1 18/1 19/1 20/1 21/1 22/1 23/1 24/1            

n = Harmonic just intonation 17/16 9/8 19/16 5/4 11/8 21/16 3/2 13/8 23/16 7/4 15/8 2/1, but treated just as a twelve tone scale like the others without the special translations for the 23/16 etc. This is a bit strange of course because of the way some of the notes are out of order, for instance, Cn13 is 5/4 3/2 7/4 9/4 11/4 23/8 instead of 5/4 3/2 7/4 9/4 11/4 13/4 - it is included to explore if you feel like getting techy about how the h notation works and to compare the translated chords with the untranslated ones.

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o = select notes from: Harmonic Overtone scale 17/16 9/8 19/16 5/4 21/16 11/8 3/2 13/8 27/16 7/4 15/8 2/1

where all the notes are selected from the harmonic series - with all the notest ransposed down into the same octave. The difference from the h and n scales is that it has 27/16 instead of 23/16, as that is the nearest harmonic series note to the A (with 1/1 at C) - while the h notation simply left out the A and used it as a convenient degree to attach the 23/16 to..

Here though you can specify other overtone scales too. For instance

o3 for13/12 9/8 7/6 5/4 4/3 17/12 3/2 19/12 5/3 11/6 23/12 2/1

That's the overtone series you get from 3/3 4/3 5/3 6/3 ... reduced to the octave. It has some nice harmonic series chords such as 1/1 7/6 4/3 3/2 5/3 11/6 2/1 and so forth.

Similarly, it's o10 for 21/20 11/10 6/5 5/4 13/10 7/5 3/2 8/5 17/10 9/5 19/10 2/1 which is the overtone scale that you get from 10/10 11/10 12/10 13/10... reduced to the octave

This notation constructs a scale using the first note in the specified overtone series within a quarter tone of each twelve equal pitch.

You can also specify the complete ratio to start with instead of just using 10/10 as that last example did. For instance, you can use o12:1 for 17/16 9/8 19/16 5/4 21/16 11/8 3/2 13/8 27/16 7/4 15/8 2/1 which is the overtone scale you get from 12/1 13/1 14/1 ....You can use o12/1 for o12:1 if you prefer. Then o12:2 is actually the same as o12 in terms of the scale you get, but when you get up to o13:3 then that introduces something new over o13:2.and o13.The higher you go up then the more variations you can get by setting the second number here. Anyway you can explore these scales using the #12 o13:3 type notation in the New Scale window.

The motivation for this is that notes of the overtone series are likely to sound harmonious together. The result is more than the effect of individual diads of the chord, for instance if you choose a note like say 19/16 in a harmonic series chord, the combined effect of choosing all the notes fom a harmonic series may well make the complete chord work well and have the result that the note sounds far more harmonious than it does in some other type of chord. Even more so if you transpose it up a couple of octaves - try for instance 1/1 5/4 3/2 19/8 23/8 or some such chord

Incidentally, though this isn't recognised in the chord symbols at present as they are based on twelve tone ideas, when you make these notations in the scale window you can also use it to make scales with other numbers of notes to an octave.

So for instance #19 o makes a nineteen tone overtone scale in the New Scale window:

33/32 17/16 9/8 37/32 19/16 5/4 21/16 43/32 11/8 23/16 3/2 25/16 13/8 27/16 7/4 29/16 15/8 31/16 2/1

But you can't use that one in the chord notations, only to make new scales.

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Undertone notation

u = select notes from: Undertone scale 16/15 8/7 32/27 16/13 4/3 16/11 32/21 8/5 32/19 16/9 32/17 2/1

This notation works just like the overtone one, but using the subharmonic (undertones) series instead of the harmonic (overtones) series. So for instance, u3:13 for 24/23 8/7 6/5 24/19 4/3 24/17 3/2 8/5 12/7 16/9 24/13 2/1 which is the undertone scale you get from 3/13, 3/14 3/15 3/16 ...

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Mixed overtone and undertone notation

ou = select notes from: mixed overtone and undertone scale 16/15 8/7 19/16 5/4 4/3 11/8 3/2 8/5 32/19 7/4 15/8 2/1.

The way these scales work is that the ratios get generated as 1/1, 2/1, 1/2, 3/1, 1/3, 4/1, 1/4, 5/1, 1/5, 6/1, 1/6, 7/1, 1/7 alternating between the overtone and the undertone series. Each is then reduced to the octave and assigned to the nearest twelve tone note if it hasn't yet been given a value. This continues until there are no more notes left to assign. The even numbered terms there have no effect and can be ignored as for instance 6/1 will have already been asigned earlier when 3/1 was done.

So for instance, the 11/8 gets assigned to the F# - then since that note is already asssigned, the 16/11, which would also be an F#, gets skipped. The 13/8 gets skipped because the Ab has already been assigned to 8/5, and the 16/13 gets skipped because the E has already been assigned to 5/4.

Again you can use the likes of uo10 for 20/19 11/10 6/5 5/4 13/10 7/5 20/13 8/5 5/3 20/11 19/10 2/1, or uo10:3 for 13/12 10/9 7/6 5/4 4/3 10/7 3/2 19/12 5/3 11/6 23/12 2/1 etc.This time the scale will be generated using e.g.10/10 11/10 10/11 12/10 10/12 13/10 10/13 14/10 10/14 alternating between the two - this time as it starts at a different place from 1/1 then the even numbered terms such as 12/10 have significance..

uo = same idea but u has preference in each pair: 16/15 8/7 19/16 5/4 4/3 16/11 3/2 8/5 32/19 7/4 15/8 2/1. (only differs in the tritone in this case - 16/11 instead of 11/8)

This time it gets generated using 1/1, 1/2, 2/1, 1/3, 3/1, so it is the same idea as before except that the undertone ratio occurs first in each pair instead of the overtone one, so has precedence.

You can also control the direction in which the search goes. So uo-10 goes backwards as far as it can go (from 10/10) to fill in the scale pitches, rather than forwards:

All of these can be used in chord notations so for instance


selects from

uo-10 = 20/19 10/9 6/5 5/4 13/10 10/7 20/13 8/5 5/3 9/5 19/10 2/1

to give

6/5 10/7 5/3 2/1

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Example chords

C, Cm, C7, Cm7, Cm6, Cdim7

You can try these all out in the New Scale window by entering e.g. #chord Csdim7 in the Name field for the New Scale (see the Intro) as well as use them in the chord progression

Flavours of C

Cj 1/1 5/4 3/2

Cs: 1/1 9/7 3/2

Cud: 1/1 14/11 3/2

Ct: 1/1 16/13 3/2

(and just for fun: Cu5: 1/1 5/4 20/13 - is this a major chord? :-) )

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Flavours of Cm

Cjm 1/1 6/5 3/2

Csm: 1/1 7/6 3/2

Cudm: 1/1 11/9 3/2

Ctm: 1/1 13/11 3/2 2/1

(Cu11m: 1/1 11/9 22/15 - is this a minor chord? :-) )

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Flavours of C7

Cj7 1/1 5/4 3/2 16/9

Ch7: 1/1 5/4 3/2 7/4

Ck7: 1/1 5/4 3/2 9/5

Cs7 1/1 9/7 3/2 7/4

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Flavours of Cm7

Ckm7: 1/1 6/5 3/2 9/5

Cjmh7: 1/1 6/5 3/2 7/4

Cudm7 1/1 11/9 3/2 11/6

Ctm7 1/1 13/11 3/2 26/15 2/1

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Flavours of Cm6 (first inversion of the half diminished seventh)

Cjms6: 1/1 6/5 3/2 12/7
(same as Cu6m6)

Csmj6 1/1 7/6 3/2 5/3
(same as Co6m6)

Co5m6 1/1 6/5 3/2 17/10

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Flavours of Cdim7

Cjdim7: 1/1 6/5 36/25 5/3

Codim7: 1/1 19/16 11/8 27/16

Cuddim7: 1/1 11/9 16/11 18/11

Csdim7: 1/1 7/6 7/5 12/7

Ctdim7: 1/1 13/11 13/9 22/13

Coudim7: 19/16 11/8 32/19 2/1

Co5dim7: 6/5 7/5 17/10 2/1


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How to hear these chords in just intonation harmony

This notation selects the nearest pitch in the current scale for the chord notes; and doesn't change the scale. So for instance, Cj7 is identical to C in most twelve tone scales, but there would be a distinction between them in scales with sufficiently fine distinctions of pitch, such as say 72 equal. The septimal and normal major and minor chords are identical in most twelve tone scales, but in nineteen equal would pick out distinct chords, and so on.

You can see the exact pitches that the chords pick out using the New Scale window as explained in the Intro to this section. But what about the chord progression, how do you hear them there?

One solution is to prefix your chord symbol with an x in which case the current scale gets used only for the chord root and the exact pitches get added on top of that. E.g. xCh11 for the harmonic eleventh and xEh11 for the harmonic eleventh rooted on the E of the current scale etc.

Another solution is to make sure your scale has all the required ratios for the chords in their root position, and use the option Retune scale to chord roots to keep your chords in tune.

Many of the ratios you need are in this scale:

A just 11-limit 31-tone scale

45/44 25/24 16/15 12/11 9/8 8/7 7/6 6/5 11/9 5/4 32/25 21/16 4/3 11/8 7/5 10/7 16/11 3/2 32/21 25/16 8/5 18/11 5/3 12/7 7/4 16/9 11/6 15/8 48/25 88/45 2/1

You enter it using the Scale as Text button or from the New Scale window enter it as the new scale and then press Apply. When using the New Scale window the main window scaledoesn't change immediately because that permits one to have any number of those new scales on the go at once and quickly switch from one to the other in the main window using its Apply button..

As another alternative you can also use a scale with tiny steps such as say 0.0001 cents. That will show up as cents values rather than ratios in any displays in the program such as the Out | Notes Played window set to show intervals, but it is very finely tuned indeed, and will be closer than you can get to specifying the desired pitches in Midi with pitch bends, or even with the MTS sysexes (which can achieve an accuracy of 0.0061 cents)..

Another solution is to simply use a conglomerate scale with all the just intonation intervals you need from the roots of your chords. So for instance if you are using the scales in the notations mentioned so far (apart from the overtone and undertone series scales which are open-ended), you can simply use this conglomerate scale of them all:

Conglomerate Just Intonation

33/32 28/27 17/16 16/15 13/12 12/11 10/9 9/8 8/7 15/13 7/6 13/11 19/16 6/5 11/9 16/13 5/4 14/11 9/7 21/16 4/3 11/8 7/5 23/16 36/25 13/9 16/11 3/2 14/9 8/5 13/8 18/11 18/11 5/3 22/13 12/7 26/15 7/4 16/9 9/5 11/6 24/13 15/8 27/14 64/33 2/1

(in ascending order without repetitions)

You will find both these scales in the New Scale window drop list. To get to them, first click the Scale... button in the player, then select the scale you want from the drop list at the top of the New Scale window. Then use Select All. Finally, edit the result if you need to add any more intervals, and use Apply to put the scale back into the main window after editing.

This works because the chord symbols pick out whatever is the nearest pitch in the scale and in these cases will be able to find the exact pitches desired - so long as you make sure to use Retune scale to chord roots. If you need extra ratios for some chords, say 13/10 or whatever, be sure to include those as well in the conglomerate j.i. scale in order to hear them tuned appropriately.

Be sure to use Retune scale to chord roots with this method for just intonation harmony. If you have it switched off then this conglomerate scale will still retune your C chords to j.i. harmony because it has all the desired pitches for it, but others may be strangely tuned.

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Accepted Note Names

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Intro - Ambiguous naturals, # and b for conglomerate - Specifying the octave,


There are various notation systems you can use for the note names currently (more may be added later, particularly I plan to add Sagittal in the near fu. You can select any of these in the Notations... window, or if you choose Conglomerate in that list then they are all combined into a single notation system.

Conglomerate is the preset. The list includes various common notations for scales up to 72 notes to an octave. The Conglomerate option lets you use any of the other notations in the chords list or even mix notations in the same chord. To resolve ambiguities when you mix notations, see Ambiguous naturals, # and b for conglomerate.

These note names can be used

1. for the root of the chord as in Ab7

2. F or the key signature, e.g. Dminor

3. added to the chord as a pedal note e.g. C/G', using the using the note name system where middle c is c and the octaves are: C''', C'', C', C, c, c', c'', c'''

However don't try to use the accidentals for the note name system in use in combination with the numerals in the chord symbol. These are interpreted as twelve tone degrees, and taken to refer to the nearest pitch in the current scale to the 12-et pitch. The #s and bs in the chord symbol just increment or decrement the relevant twelve tone degrees.

Ex. in nineteen equal, C /D# adds D# to the chord and will differ from C/Eb - but C/2# just adds the nearest note to 300 cents and has the same effect as C/3b because the numerals in the chord are (currently) always interpreted twelve tone fashion.

y notation for exact pitches

Normally these notations will play the closest available pitches in the current main window scale. So if the main window scale is twelve equal they just play twelve equal pitches. If you want them to play the closest available nineteen tone pitches for instance, just change the main window scale to nineteen equal.

If you want the chord to ignore the current scale, and play the exact pitches whatever, then prefix the chord symbol with y. This will give twelve equal pitches for the standard symbols like C, Ab etc.

The y notaiton is probably most needed with the ratio or cents notation. It gets you the exact pitches for all the notes in the chord, including the note name used for the root of the chord, so the chord need not even be rooted in the current scale.

To see how these work, try them out in a Bs2 | Scale window and for the New Scale Name enter #chord followed by the symbol prefixed with y for exact pitches.

So, for instance to try out the distinction of sharps and flats as exact pitches in the current notation you could go to a Scale window and enter this as the new scale name:

#chord yC/eb

And compare:

#chord yC/d#

The notation systems currently available

Twelve tone as letters A to G with the symbol b for flat or # for sharp

Solfeggio: do, di = ra, re, ri = me, mi. fa, fi = se, so, si = le, la, li = te, ti

Solfeggio incidentally, as it has seventeeen note names, might be a nice notation to use for seventeen tone notation - with ra now slightly sharper than di and so on. This implementation has taken this possibility into consideration, and treats these as seventeen tone note degrees - since it won't affect the use of it for twelve tone scales which will be the most common use of the notation - and it may be of interest for notation of seventeen note scales.

Ratio or cents: To specify an exact ratio or cents just use it as a numerical value. So 7/6 specifies the septimal minor. You can prefix it with r if you wish. Use decimal point for cents. You can use this in chord symbols with any of the notation systems. As well as ratios and cents, you can use any other notation recognised by FTS as a way of entering scale pitches, for instance the n notation n(7/31) for the 7th degree of 31 equal etc.

'/' can't be used with the ratios for obvious reasons, so use '@' instead, e.g. add a septimal minor third using @7/6. The first @ encountered in the chord symbol switches the symbol for adding new notes from an '/' to an '@' from that point on in the symbol. So C@13/8/9 is incorrect -the /9 there should be done as @9 instead: C@13/8@

Anything between the ratio and the @ will get ignored (or possibly read as a formula as part of the ratio if it uses symbols FTS recognises as ones for formulae). If there is no '@' then it is played as a major chord and anything not recognised as part of the formula gest ignored..

As with any of these notation systems, the ratio can also be used to set the starting pitch of the chord.

So r5/4 is a major chord on the nearest pitch to a 5/4 above the 1/1 in the current scale.

Since there is such a wide range of possible formulae you could use for the ratio, some means is needed to show where the ratio part of the symbol ends and the chord proper begins. You can use an underline - or just use an '@' again, placed immediately after the end of the ratio, whichever you prefer, to end the formula.

The m of a minor chord will also end the chord root name.

So,. r5/4m is a minor chord starting at 5/4. r5/4m7 is a minor seventh and so on

r5/4_7 or r5/4@7

is a seventh chord rooted on the 5/4.

Then the symbol just continues normally after that.

Be sure to use the y notation if you want the chord to be played based on the root note specified and it isn't in the current main window scale.

Nineteen tone notation -the order of the notes is e.g. C C# Db D so sharps are lower in pitch than the flats of the note above, with E# = Fb and B# = Cb. Again this only has any effect if the scale has intermediate notes so that it has two or more possibilities for the accidentals. The flats and sharps are interpreted as nineteen tone pitches in the conglomerate notation.

Thirty one tone notation - you have to use - or + for half accidentals. An alternative thirty one tone notation uses b for the half accidental so that you need to double them to get the full accidental as e.g. Ebb, but this won't get used here with the conglomerate notation, even if you have the option selected to use that in other situations such as the Tune window - because a note such as Eb in this notation would get confused with the Eb in the nineteen tone notation.

So it runs
C C+ C# Db D- D ... E E+ F- F ... B B+ C- C.
Note that this notation has no E# or Fb or B# or Cb

Then for larger scales still you can use the 72 tone notation, and set your preferred characters for that in the Notation for accidentals window.

So for instance, in the Maneri / Sims notation it runs
C C^ C> C] C#[ C#< C#v C#

In this notation, C# and Bb are interchangeable and are the same as the twelve equal accidentals.

You can also use any of the twelve tone scales in the Tuning Symbols list as a basis for the notation as explained in Using the symbols with the chord note names, e.g. pEb for 32/27.

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Ambiguous naturals and # and b accidentals for the conglomerate notation

In the conglomerate notation, the naturals, and the twelve equal # and b are the same in all the notation systems. The nineteen tone ones get preferred. So in the cases where they may pick out different notes from your scale, you need to indicate which you mean.

This is only for the accidentals using # or b and the naturals. In the conglomerate notation, the + and - accidentals are automatically recognised as 31-et and the special 72-et accidentals are recognised as 72-et.

TWELVE EQUAL AND 72 EQUAL - Enter it as C~# or D~b - prefix the accidental with a tilda (in other contexts you can also use C:# but you can't use that in the chord progression as it will be interpreted as punctuation separating the chords). Enter naturals as C~ etc

You only need to do that for the plain #s. C#^ for instance would always be recognised as in 72 equal because of the ^ at the end.

THIRTY ONE EQUAL - Prefix with ? as in e.g. C?# Enter naturals as C? etc. The other notes using ? are: C?# D?b D?# E?b F?# G?b G?# A?b A?# B?b

D?b/e?b = a sus2 type chord on Db in 31 equal. Note that the e there is case sensitive - if you use a capital E you get the same chord but with the E transposed down an octave - see the next section: .

To take a really complex and maybe artificial example: D?b@e?b@7/4 adds a 7/4 ratio above it.

You can also use '%' instead of '@' as an extra optional note addition symbol for the chord if preferred.

Okay I know this system is a bit clumsy when it comes to 72 and 31 equal; at any rate, it works. There's a small drop list of notation systems to select from in the Notation window, and this is how the chords work if you have "Conglomerate" selected.

If you want to work within e.g. thirty one tone exclusively, select it from the drop list and all the note names in your chord symbols will be interpreted in a thirty one tone way anyway, and you can leave out the ? symbols from them.

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Specifyng the octave

When you want to specify the octave to play for the additional notes, then use lowercase or upper case and the dashes notation. So for instance, C@R13/8 would be a c major chord with a 13/8 below it (i.e. at 13/16) while C@r13/10 would put it at 13/8 because the r is lower case. C@r13/10'' then places it at 13/4 and C@R13/10' places it at 13/32. What matters here is the case of the first letter in the note name so for instance, La is in the octave below the 1/1 and la is in the octave from 1/1 to 2/1. This is only for the additional notes in the chord. If you want to move the entire chord up / down by octaves, then use the octaves notation in the Inversions section.

You can also use the octave shifting symbols / and \ within the chord symbol,as in e.g. C@/r13/8.

You can also use them with the / as a note addition symbol as well, e.g. C//r13/8 where the frst / adds the note, and the second / there is understood as an octave shifting symbol

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Make new chord symbols using the Extra chord Symbols window

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The idea of this window is that you enter the desired chord as scale degrees, and then enter your preferred name for it in the column to the right. This lets you specify any kind of chord you like using scale degrees and give it any name you please. However, since the chord is defined as degrees in this window rather than as pitches in any notation system, then your chord will only work as intended in the scale you design it for. To give an example, if you were to design some chords in a nineteen tone scale, and then changed the scale to a thirty one tone scale,. the 19 of your chords would now no longer be an octave, but some other interval, probably a 3/2 or there abouts, and all the chords would get squashed up accordingly..

So the preset chord notation is better if you can use it, but use this if you are happy making chord symbols unique to a particular scale or number of notes to an octave. It's more flexible because you can use anything you like as the name for the chord. The symbols in this window take precedence over everything else, so you can re-interpret any of the chord symbols already in use using this window, e.g. you could re-interpret the roman numerals notation to mean something suitable for your scale, or whatever. You can save your symbols with the chord progression if you make sure that you select the Save extra chord symbols check box in the main window.

You can use the Find Chords button to find a list of just intonation chords, or chords approximating to just intonation chords for the current main window scale. To set it up you use the J.i. button to the right. You can start by using one of the preset buttons to set it up, e.g. five limit, seven limit or whatever depending on interests. This is the window used for just intonation retuning as you play - so - and ignore the options in that window specific to just intonation retuning such as the dovetailing - it just uses the options which specify which are the acceptable ratios to use. It finds chords from the current scale in which every interval between any pair of notes is just according to the criterion you have set up for it - or if it doesn't find enough just intonation chords, it broadens the criterion and looks for approximate just intonatioin chords.

Once you make your chord, you can then look it up using the Look Up button to see if you can find a chord symbol for it. This just searches through a large list of commonly used chords. It can find the more common chords such as C7 etc - in fact it goes through the list of chords given as examples in the Chord Symbols list plus various microtonal variations on each one such as e.g. Cj7, Ck7, Cs7 etc., and compares the result with the chord you have made. If it doesn't find it, you may well still be able to give it a simple symbol as the range of possible symbols is so large - this is just a short cut way to find a symbol for many of the more common chords.

Often one may want to start from a standard chord and then modify it to make your custom chords. To do that prefix your chord symbol with a ^ sign.

So if you enter e.g. ^E7 as the description, and you are using 31 equal, it will show the chord degrees as:

10 20 28 36 41

because that is the E7 chord in 31 equal (with the final 41 added to make an octave repeating arpeggio).

If you do the harmonic seventh ^Eh7, it will show:

10 20 28 35 41

with the slightly flatter seventh at 35 instead of 36, and so on.

Maybe you want your E7s to always be the more mellow harmonic sevenths in a particular chord progression, to save adding in the h symbols all the time. Well just make an Eh7 like this, then remove the ^ and now you have redefined your E7 chords to have this new meaning.

You can add anything you like after the symbol, as a description of the chord. This will be shown in the field above the chord progression when the chord is played. Only the very first bit of the description up to the first blank space is treated as the symbol proper. So you could have:

E7 = Harmonic seventh chord for thirty one equal.

Only the E7 part there will be used for the chord progression symbol. Indeed you can also use this method just as away of adding a description to the chord, so you could just have:

Eh7 = Harmonic seventh chord

if you want the player to show this whenever it plays the Eh7 chord.

Now lets try something else. This time the idea is to just make a normal twelve tone chord, but use the extra chord symbols as a way of writing it more compactly.

Suppose for instance that you want an Em7 chord, but you want some low notes sounding in the chord.

Well first thing to try is to use the inversions and octave transpositions. Try the likes of ~~Em7 or /.Em7. You also choose whether or not to sound the octave (or second octave for ninths etc) for all your chords from Options | Skip last note (e.g.octave)

But those chords all have the same three notes to them, or four if you choose to sound the octave, and they are just moved up or down in the chord pattern.

Maybe you want a widely spread out chord with more notes, maybe also with gaps in the chord. To start with, maybe you want to keep the high notes as before, and simply add a low B' and E say. Well you can add those notes using the normal notation for the chord progression using: Em7/B'/E

You can make any chords that you want using the standard notations plus inversions and the options to add extra notes and to remove them (using C0 or C(5) to remove the 5 for instance). We could make our entire chord that way.

You need to make up a new chord symbol. Lets call it say, newEm7. You can call it anything you like.

Highlight the first row in the window, or whichever row you want to place it in. If you need more rows then increase the number of rows using the number at the lower left of the window

Since our chord is based on the Em7/B'/E, then lets start by using that. To recover the standard defnition for Em7/B'/E, enter Em7/B'/E in the Symbol and Name field, but prefix it with a circumflex ^


With the main window scale set to a normal twelve tone type scale, the Chord as scale degrees field should now show -13 -8 4 7 11 14 23

If not, make sure you have Any inversion okay unselected in the main window and try again.

So our chord so far is

-13  -8  4  7  11  14  23 
  B'  E  e  g   b   d   b'

Note that the highest note b' here is three octaves above the lowest note B'. Generally you will want this to be a whole number of octaves so that it will work with the figurations if the figuration goes beyond the end of the chord. Anyway the player has done this for us automatically.

Now that we have the numbers there for editing, remove the circumflex and rename the chord to newEm7, the name we want to use. Optionally you can add some extra description to remind you what it is.

newEm7 = with extra B', E and some skipped notes.

Now we want to miss out a few notes. Maybe we want to skip the e and the b, say. Well just remove those numbers 4 and 11 from the Chord as scale degrees field to get

-13  -8  7  14 16  23 
  B'  E  g  d  e'  b'

That then is our new chord.

The easy way to edit it like that is to use the To Arp. button to show the chord in a New Arpeggio window. Then you can click to add and remove notes, play your chord from the keyboard to hear what it sounds like, and then copy and paste the result back to use as in your chord progression. Or indeed, actually as a shortcut way to get it back, if you use Ctrl + click on the To Arp. utton then it will get your edited arpeggio back again,

To use your new chord simply use your symbol newEm7 in the chord progression wherever you want to play it.

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Chords for other tuning systems

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There is some limited support for these already in the preset notations - see the Microtonal Chord Symbols for the tuning of the chords. See also the Accepted note names list for the tuning of the root of the chord. So you could do a major or minor chord in nineteen equal on any root, and a septimal minor chord too, or pick out the various flavours of dominant sevenths in a tuning and so forth.

You can also make new chord symbols using the Extra Chord Symbols window. See the previous section Make new chord symbols using the Extra chords window which works through a twelve tone example - you can use the same methods for scales with other numbers of notes.

Note though that your new symbols won't respond to changes in the numbers of notes per octave and the tuning in the same way as the preset chord symbols. If you use the preset ones such as Am7 etc or the microtonal variations on those such as Asm7 etc, the nearest available pitches get used, so it will sound pretty much the same whatever scale you choose, with pitches of the chord notes shifted a little if necessary to match the nearest pitches in the current scale.

When you make your own chord symbols however, these are recorded as scale degrees only - the scale does not get recorded as part of the symbol. This means that when you change the main window scale, the tuning of all the notes in the chord changes. Sometimes that may be what you want, for instance if the chord is a chord for twelve tone or nineteen tone equal or almost equal scales etc.

However, if you make a chord symbol for a chord in nineteen equal and then change the scale to thirty one equal, you will find that the sound of the chord changes. The intervals span the same number of scale degree steps as before, and as each step is smaller, the intervals will get compressed and become smaller and the chord will sound completely different.

Later I have plans to add in the very extensive Scala chords list, and some way of finding and specifying them for the current scale. See Scala | View | List of Chords. The plan is that those will have the preferred scale or tuning of the chord recorded in some way, and so work like the standard symbols and pick out appropriate pitches in any scale. But I don't know yet how exactly this will work, or what types of symbols will be used or how they will get specified.

So - anyway though there is really only rather limited support for the more exotic chords, using the Extra Chord Symbols system at least one can define your own chords for particular tuning systems you happen to be working with. You can use all the special microtonal notations to notate the chords and if one of those can describe the chord you want to play then that will apply to any scales that have that chord. You will find a Find Chord button in the window which can be used to find a suitable microtonal notation chord.