View source for User:VectorGraphics/Vector's introduction to 15edo/Chords

Note: this will use a slightly different notation scheme. {▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒} is a blank set of 15 notes with no notes being played. In case a chord spans multiple octaves, a separator will be used: {▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒|▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒}. 

The highest note of a chord will always be followed by a blank space, so a dyad spanning a perfect fifth will look like: {▓▒▒▒▒▒▒▒▒▓▒}.

And only the root note of a chord will be marked in the darkest shade, so if the first note is the root, the chord is: {█▒▒▒▒▒▒▒▒▓▒}.

We will start with the diatonic scale.

== Section 1: 1-3-5(-7) chords ==
These are the chords found in the diatonic scale, formed from a root, third, and fifth, and their positions in the Ionian scale.

=== Right-handed chords ===
█▒▒▒▒▓▒▒▒▓▒ - major triad (on 1, 4, 5)

█▒▒▒▓▒▒▒▒▓▒ - minor triad (on 3, 6)

█▒▒▓▒▒▒▒▓▒ - wolfminor triad (on 2)

█▒▒▒▓▒▒▓▒ - major diminished triad (on 7)

=== Left-handed chords ===
█▒▒▒▒▓▒▒▒▓▒ - major triad (on 1, 4)

█▒▒▒▓▒▒▒▒▓▒ - minor triad (on 2, 3, 6)

█▒▒▒▒▓▒▒▓▒ - wolfmajor triad (on 5)

█▒▒▓▒▒▒▓▒ - minor diminished triad (on 7)

The wolfminor and wolfmajor triads can be seen as tenser than their corresponding chords with fifths. As such, the LH Ionian's Vwmajh7 chord works as a strong resolution to the stable tonic chord.
Note that each scale only contains five perfect fifths instead of six.

There are also many types of seventh chord available!

=== Right-handed chords ===
█▒▒▒▒▓▒▒▒▓▒▒▒▒▓|▒ - major 7th chord (on 1, 4)

█▒▒▒▓▒▒▒▒▓▒▒▒▓▒|▒ - minor 7th chord (on 3, 6)

█▒▒▓▒▒▒▒▓▒▒▒▓▒▒|▒ - wolfminor harmonic 7th chord (on 2)

█▒▒▒▒▓▒▒▒▓▒▒▓▒▒|▒ - harmonic 7th chord (on 5)

█▒▒▒▓▒▒▓▒▒▒▒▓▒▒|▒ - major half diminished 7th chord (on 7)

=== Left-handed chords ===
█▒▒▒▒▓▒▒▒▓▒▒▒▒▓|▒ - major 7th chord (on 1, 4)

█▒▒▒▓▒▒▒▒▓▒▒▒▓▒|▒ - minor 7th chord (on 2, 6)

█▒▒▒▓▒▒▒▒▓▒▒▓▒▒|▒ - minor harmonic 7th chord (on 3)

█▒▒▒▒▓▒▒▓▒▒▒▓▒▒|▒ - wolfmajor harmonic 7th chord (on 5)

█▒▒▓▒▒▒▓▒▒▒▒▓▒▒|▒ - minor half diminished 7th chord (on 7)

Some of these chords produce other chords in inversions:

█▒▒▒▒▓▒▒▓▒▒▒▓▒▒|▒ - wolfmajor harmonic 7th chord 

▓▒▒▓▒▒▒▓▒▒█▒▒▒▒|▒ - minor diminished augmented chord 

▓▒▒▒▓▒▒█▒▒▒▒▓▒▒|▒ - major half diminished 7th chord

▓▒▒█▒▒▒▒▓▒▒▓▒▒▒|▒ - wolfminor diminished 7th chord

-

█▒▒▒▓▒▒▒▒▓▒▒▓▒▒|▒ - minor harmonic 7th chord

▓▒▒▒▒▓▒▒▓▒▒█▒▒▒|▒ - wolfmajor diminished 7th chord

▓▒▒▓▒▒█▒▒▒▓▒▒▒▒|▒ - slendric augmented tetrad

▓▒▒█▒▒▒▓▒▒▒▒▓▒▒|▒ - minor half diminished 7th chord

-

█▒▒▓▒▒▒▒▓▒▒▒▓▒▒|▒ - wolfminor harmonic 7th chord

▓▒▒▒▒▓▒▒▒▓▒▒█▒▒|▒ - harmonic 7th chord 

▓▒▒▒▓▒▒█▒▒▓▒▒▒▒|▒ - major diminished augmented chord

▓▒▒█▒▒▓▒▒▒▒▓▒▒▒|▒ - slendric diminished 7th chord


These include the harmonic seventh chord, which reflects the harmonic series the same way the normal major chord does.

Note that there aren't any normal dominant chords or diminished chords in any of these. However, the normal harmonic seventh chord (and the wolf major harmonic 7th chord) work as a resolution to the tonic, since they both contain a tritone interval resolving inwards, even if it's not constructed the same as a dominant chord. To construct more equable dominant and diminished chords, we'll need to leave the scale, but that's for later. Now, let's check out suspended chords.

== Section 2: Suspended triads and the slendric chord ==
Let's first meet the sus4 triad, so I can explain:

█▒▒▒▒▒▓▒▒▓▒ - tendo triad (on 1, 3, 5 RH, 1, 3, 6 LH)

In 15edo, we call the sus4 triad a "tendo triad", which indicates its special characteristics. Notably, the fourth is flat enough that it kind of functions as a supermajor third in a way, and can be seen as an augmented third if we take the major second between the fourth and the fifth as a wolf third. 

Its counterpart, the sus2 triad, is called an "arto triad".

█▒▒▓▒▒▒▒▒▓▒ - arto triad (on 1, 4, 6 RH, 2, 4, 6 LH)

Here the middle note can be seen as a large whole tone, or as a wolf third.

The reason they get these special names is because if you play both of the middle notes of the triads in the same chord, you get...

█▒▒▓▒▒▓▒▒▓▒ - slendric tetrad (on 1 RH, 6 LH)

The distance between the second and the fourth is exactly the same as between either and its closest bounding note - that is, the notes split the fifth perfectly into three. That's why we call this a "slendric tetrad", because each of these 240c steps represents a large whole tone, and three of them make a perfect fifth.

More on this phenomenon at [[Extraclassical tonality]].

We can extend this to the slendric pentad:

█▒▒▓▒▒▓▒▒▓▒▒▓▒ - slendric pentad (not in diatonic scale).

You may notice that this is exactly the same as the 5edo scale! This is actually where the term "slendric" comes from, as the Indonesian "slendro" tuning uses a similar 5-note scale.

== Section 3: Non-diatonic chords ==
Of course, as seen in my scales section, there are numerous other scales available in 15edo, and these are some chords you might find in some of them:

█▒▒▒▓▒▒▒▓▒ - diminished triad

█▒▒▒▓▒▒▒▓▒▒▒▓▒ - diminished harmonic seventh tetrad

These diminished chords result from stacking the same minor third interval. The diminished seventh tetrad is the root chord of the darkest mode of onyx (other dark modes contain a version with a sharper seventh, {█▒▒▒▓▒▒▒▓▒▒▒▒▓▒}) Note that the diminished seventh tetrad does not actually close at the octave; a diminished ninth triad would look like this:

█▒▒▒▓▒▒▒▓▒▒▒▓▒▒|▒▓▒ - diminished ninth pentad


Stacking a wolf third results in the slendric tetrad and pentad. But you can also get this nice chord by closing at the fourth:

█▒▒▓▒▒▓▒ - slendric triad


The chord that does close the octave is the augmented chord:

█▒▒▒▒▓▒▒▒▒▓▒▒▒▒|▓▒ - augmented triad/tetrad

This augmented chord is actually tuned identically to in 12edo. 


█▒▒▒▒▓▒▒▒▓▒▒▒▓▒|▒ - dominant 7th chord

We can use our diminished triad to build a dominant 7th chord. This uses the sharp minor seventh (1040c) instead of the flat one (960c). It's actually a weaker resolution than the harmonic seventh, because the highest note drops 2 steps to resolve instead of 1, but it mirrors the structure of the conventional dominant seventh chord, so it might see some use as a "friendlier" alternative to the normal major seventh chord, or when a strong resolution isn't desired. It also appears as the root chord in the brighter modes of onyx.