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== Constructing chords and splitting steps ==
== Constructing chords and splitting steps ==
A more interesting way to construct scales is to stack chords rather than single intervals. For example, if we stack 3 major chords, we get:
█▒▒▒▒█▒▒▒█▒▒▒▒▒█ (major chord on root)
+ ▒▒▒█▒▒▒▒▒█▒▒▒▒█▒ (major chord on fifth)
+ █▒▒▒▒▒█▒▒▒▒█▒▒▒█ (major chord descending from root)
= █▒▒█▒██▒▒█▒█▒▒██ - major
Surprise! Here's the zarlino scale again.
Let's try minor chords. This corresponds to flatting the seventh, third, and sixth degrees by one step.
█▒▒██▒█▒▒██▒▒█▒█ - minor
In fact, this is actually how the normal major and minor scales are constructed in diatonic, so it's neat to see that it checks out here.
So instead, let's try a different chord (let's say, for some reason, you cared a lot about wolf chords).
█▒▒▒▒█▒▒█▒▒▒▒▒▒█ (wolf major chord on root)
+ ▒█▒▒▒▒▒▒█▒▒▒▒█▒▒ (wolf major chord on wolf fifth)
+ █▒▒▒▒▒▒█▒▒▒▒█▒▒█ (wolf major chord descending from root)
= ██▒▒▒█▒██▒▒▒██▒█ - wolf major
You can call this the "wolf major scale" because of how it's constructed.
Let's try a kind of diminished chord, called a wolf diminished chord, made of two different sizes of minor third. Since the chord is a lot narrower, we'll need 4 chords instead of 3 to build a reasonable scale.
█▒▒█▒▒▒█▒▒▒▒▒▒▒█ (wolf diminished chord on root)
+ ▒▒▒▒▒▒▒█▒▒█▒▒▒█▒ (wolf diminished chord on dim fifth)
+ █▒▒▒▒▒▒▒█▒▒█▒▒▒█ (wolf diminished chord descending from root)
+ ██▒▒█▒▒▒█▒▒▒▒▒▒█ (wolf diminished chord descending from aug fourth)
= ██▒██▒▒██▒██▒▒██ - diminished
This scale also contains a "major wolf diminished" chord, which corresponds to the harmonic series sequence 5:6:7, much like a normal major chord corresponds to 4:5:6.
As can be seen, there's a lot of possibilities here.
Another way to build scales is by choosing a few key intervals, and splitting the jumps between them into steps. For example, we might decide that our key notes are {█▒▒▒▒██▒▒█▒▒▒▒▒█}.
Here, we have two large jumps that can be split into steps, and there's a couple ways to do this, including ones that just result in zarlino and ones that result in scales we haven't seen before.
█▒▒█▒██▒▒█▒▒█▒▒█ - zaretan
█▒█▒▒██▒▒█▒▒█▒▒█ - legatus
█▒▒█▒██▒▒█▒▒██▒█ - decurion
█▒█▒▒██▒▒█▒▒██▒█ - kaiser
If we choose a different set of key intervals, we get a different set of possible scales:
█▒▒█▒█▒▒▒▒▒▒█▒▒█ =
█▒▒█▒█▒▒█▒█▒█▒▒█ - anhedonia
█▒▒█▒█▒█▒█▒██▒▒█ - mok
█▒▒▒▒█▒█▒▒▒▒█▒▒█ =
█▒▒█▒█▒█▒█▒▒█▒▒█ - amsel
█▒▒▒██▒█▒▒▒██▒██ - drossel
== Periodicity ==
Let's bring up the harmonic table:
{| class="wikitable"
|'''0c'''
|400c
|800c
|'''0c'''
|400c
|800c
|'''0c'''
|400c
|800c
|'''0c'''
|400c
|800c
|'''0c'''
|-
|480c
|880c
|80c
|480c
|880c
|80c
|480c
|880c
|80c
|480c
|880c
|80c
|480c
|-
|960c
|160c
|560c
|960c
|160c
|560c
|960c
|160c
|560c
|960c
|160c
|560c
|960c
|-
|240c
|640c
|1040c
|240c
|640c
|1040c
|240c
|640c
|1040c
|240c
|640c
|1040c
|240c
|-
|720c
|1120c
|320c
|720c
|1120c
|320c
|720c
|1120c
|320c
|720c
|1120c
|320c
|720c
|-
|'''0c'''
|400c
|800c
|'''0c'''
|400c
|800c
|'''0c'''
|400c
|800c
|'''0c'''
|400c
|800c
|'''0c'''
|-
|480c
|880c
|80c
|480c
|880c
|80c
|480c
|880c
|80c
|480c
|880c
|80c
|480c
|-
|960c
|160c
|560c
|960c
|160c
|560c
|960c
|160c
|560c
|960c
|160c
|560c
|960c
|-
|240c
|640c
|1040c
|240c
|640c
|1040c
|240c
|640c
|1040c
|240c
|640c
|1040c
|240c
|-
|720c
|1120c
|320c
|720c
|1120c
|320c
|720c
|1120c
|320c
|720c
|1120c
|320c
|720c
|-
|'''0c'''
|400c
|800c
|'''0c'''
|400c
|800c
|'''0c'''
|400c
|800c
|'''0c'''
|400c
|800c
|'''0c'''
|-
|480c
|880c
|80c
|480c
|880c
|80c
|480c
|880c
|80c
|480c
|880c
|80c
|480c
|-
|960c
|160c
|560c
|960c
|160c
|560c
|960c
|160c
|560c
|960c
|160c
|560c
|960c
|-
|240c
|640c
|1040c
|240c
|640c
|1040c
|240c
|640c
|1040c
|240c
|640c
|1040c
|240c
|-
|720c
|1120c
|320c
|720c
|1120c
|320c
|720c
|1120c
|320c
|720c
|1120c
|320c
|720c
|-
|'''0c'''
|400c
|800c
|'''0c'''
|400c
|800c
|'''0c'''
|400c
|800c
|'''0c'''
|400c
|800c
|'''0c'''
|-
|480c
|880c
|80c
|480c
|880c
|80c
|480c
|880c
|80c
|480c
|880c
|80c
|480c
|-
|960c
|160c
|560c
|960c
|160c
|560c
|960c
|160c
|560c
|960c
|160c
|560c
|960c
|-
|240c
|640c
|1040c
|240c
|640c
|1040c
|240c
|640c
|1040c
|240c
|640c
|1040c
|240c
|-
|720c
|1120c
|320c
|720c
|1120c
|320c
|720c
|1120c
|320c
|720c
|1120c
|320c
|720c
|-
|'''0c'''
|400c
|800c
|'''0c'''
|400c
|800c
|'''0c'''
|400c
|800c
|'''0c'''
|400c
|800c
|'''0c'''
|}
In 15edo, as I mentioned in the Intervals section, there are four types of semitones: the diatonic, chromatic, ptolemaic, and compound semitones.
{| class="wikitable"
|80c - ptolemaic semitone
|480c
|880c
|80c - compound semitone
|-
|560c
|960c
|160c
|560c
|-
|1040c
|240c
|640c
|1040c
|-
|320c
|720c
|1120c
|320c
|-
|800c
|'''0c'''
|400c
|800c
|-
|80c - diatonic semitone
|480c
|880c
|80c - chromatic semitone
|}
The standard zarlino scale can be defined by having the chromatic and ptolemaic semitones as its "chromas": if we create a tile that repeats at each multiple of these two chromas, the intervals within that tile are precisely those in the zarlino scale. The zarlino scale may be called the chromatic-ptolemaic scale because of that.
However, we can choose a different pair of semitones as our chromas, and get a different scale. Here's the chromatic-compound scale.
█▒▒███▒██▒█▒████ - elena
█▒▒███▒█▒██▒████ - kee'ra
There are two variants as, since the compound semitone can be created by stacking two augmented fourths, the augmented fourth and diminished fifth fall precisely on the edge.
And the ptolemaic-diatonic scale:
█▒▒▒█▒▒█▒▒█▒▒█▒█ - myn
== Other scales made in interval space ==
[[File:Chair 2424242.png|thumb|The chair lattice forms the shape of a chair]]
Common to the construction of scales (and the parent concept for periodicity) is the "lattice", a type of scale formed by "connecting the dots" in some interval space.
This is the scale generated by "connecting the dots" to form the shape of a chair in the 7-limit in 15edo. This is called the "Chair of Mr. Bob".
███▒█▒██▒████▒▒█ - kee'ra (Chair of Mr. Bob)
In 15edo, this tuns out to exactly be the kee'ra scale (though on a different mode), due to the archy temperament.




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WIP
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