User:VectorGraphics/Vector's introduction to 15edo/Scales

From Xenharmonic Wiki
Jump to navigation Jump to search

One of the cool things about 15edo is how many different kinds of scales it gives you access to. Here, a "scale" refers to a pattern of notes, regardless of mode, key, or root note. So the A minor scale, the F major scale, and so forth are all the same scale, called diatonic.

Now, 15edo has its own kind of diatonic, but it looks a little different from the diatonic scales you might be used to. Usually, a diatonic scale is a kind of moment-of-symmetry scale, which is a kind of scale that can be constructed by stacking one interval. In this case, it's a fifth. But something a little funny happens when you try to stack fifths in 15edo: it loops around on itself after only 5 notes. This means that the scale you end up with doesn't really look like diatonic at all: {█▒▒█▒▒█▒▒█▒▒█▒▒█}. Here, fully shaded boxes represent notes that are in the scale, and partially shaded boxes represent notes that aren't in the scale.

So instead, we need to construct a different kind of diatonic scale, which is different in that it has two different sizes of whole tones. This is called the zarlino scale, and its pattern is this: {█▒█▒▒██▒▒█▒█▒▒██}. In this scale, each interval has 3 sizes: as I mentioned, seconds and sevenths have small and large major sizes, while thirds, fourths, fifths, and sixths have new "wolf interval" varieties.

Non-diatonic scales

We first start with 15edo's chromatic scale. In 15edo, the chromatic scale is also called the "valentine scale": 3 chromatic steps equals a major tone, 5 is a major third, and 9 is a perfect fifth. Being the chromatic scale, valentine contains all 15 notes: {████████████████}

Next, there are the porcupine scales: onyx (█▒█▒█▒█▒▒█▒█▒█▒█) and pine (██▒█▒█▒█▒█▒█▒█▒█). The steps of the porcupine scales are minor tones. In onyx (pine is really just a version of onyx with a "blue note"), four notes have perfect fifths above them, and four notes have perfect fourths, overlapping at one note which has both. Onyx has a special relationship to the diatonic scale, however, and this can be found by sharping the third and seventh degrees of onyx to produce the zarlino diatonic scale.

The third set of scales we'll look at is the kleismic scales, called smitonic (██▒▒██▒▒██▒▒█▒▒█) and kleistonic (███▒███▒███▒██▒█), with the latter sort of functioning as a chromatic scale of the former. Smitonic contains a singular perfect fifth, with a major and minor third available on the same note; in general, smitonic contains many intervals of the 5-limit and 7-limit.

The fourth set of scales is the tritone-generated scales: pentic (███▒▒▒▒▒██▒▒▒▒▒█), peltonic (████▒▒▒▒███▒▒▒▒█), balzano (█████▒▒▒████▒▒▒█), and semiquinary (██████▒▒█████▒▒█).

Surprisingly, the structure of peltonic is exactly analogous to that of 12edo's diatonic! It contains four perfect fifths, which correspond to 12edo's major sixths; 12edo's perfect fifths correspond to large tritones. Its extension, balzano, which I find the most reasonable of the four tritone-generated scales to use, has six fifths, all of which can be used to build some kind of chord.

After this, we have two special families of scales: augmented and blackwood. We will start off with blackwood.

Blackwood

Blackwood is based on our "fifth-generated diatonic" {█▒▒█▒▒█▒▒█▒▒█▒▒█}. If we split each step of this scale into a larger and smaller portion, we get the blackwood scale proper: {█▒██▒██▒██▒██▒██}. This scale has only two modes: major and minor, but it is extremely significant if you're used to diatonic harmony, since it has the unique property of always having either a major triad or a minor triad on any given note. In fact, by adding "missing" notes to zarlino to ensure that (which happen to themselves be in the shape of a major chord), we get blackwood.

Augmented

Augmented is, conversely, based on major thirds. The basic augmented scale is {█▒▒▒▒█▒▒▒▒█▒▒▒▒█}. We can follow the same formula as blackwood and split this scale into large and small steps to get the following possible combinations:

- █▒▒▒██▒▒▒██▒▒▒██ - the augmented scale proper, which contains fifths on exactly three of the six notes. 12edo contains a tuning of this scale, but we usually don't talk about it in 12edo.

- █▒▒█▒█▒▒█▒█▒▒█▒█ - the wholetone scale, which alternates the two sizes of wholetones and which contains many 7-limit intervals, but no fifths. This is similar to 12edo's wholetone scale, but since there are two different kinds of wholetones in 15edo, it uses both.

- █▒█▒██▒█▒██▒█▒██ - the hyrulic scale, which is a way of combining the two prior scales to form a 9-note scale. This can be thought of as having perfect thirds, and major and minor fifths.

- █▒▒███▒▒███▒▒███ - the tcherepnin scale, which is another combination of hard and soft triwood. This can also be thought of as having similar harmony to hyrulic, but with the minor fifth one step flatter.

Diatonic scales

The zarlino scale can be generated similarly to the 12edo diatonic scale. But because our fifth is so sharp, the "fifth" between two of our notes needs to be a large tritone instead. This tritone is called a "wolf interval". When this is tuned justly, the wolf interval sounds out-of-tune and dissonant, but in 15edo it's the same thing as a large tritone, so it's more usable. The standard tuning of zarlino places the wolf interval either three or four generators up. But we can get a different kind of diatonic if we put the wolf interval 5 steps up: {█▒▒█▒██▒▒█▒▒█▒██}. I call this the "Didymian diatonic" due to its similarity to Didymus' diatonic scale from ancient Greece. In fact, various Greek scales can be tuned in 15edo:

- █▒█▒▒██▒▒█▒█▒▒██ - zarlino, such as Ptolemy's intense diatonic or Archytas' diatonic

- █▒▒█▒██▒▒█▒▒█▒██ - didymian, such as Didymus' diatonic or Ptolemy's soft diatonic

- █▒█▒█▒█▒▒█▒█▒█▒█ - onyx, which is similar to Ptolemy's equable diatonic

- █▒▒▒███▒▒█▒▒▒███ - similar to Greek enharmonic or chromatic scales

MODMOS structures

Let's take the minor scale:

█▒▒██▒█▒▒██▒▒█▒█ - minor

A common way to alter the minor scale is to sharpen the sixth and/or seventh degrees, resulting in the harmonic and melodic minor and variations of Dorian:

█▒▒██▒█▒▒██▒▒▒██ - harmonic minor

█▒▒██▒█▒▒█▒█▒▒██ - dark melodic minor

█▒▒██▒█▒▒█▒▒█▒██ - bright melodic minor

█▒▒██▒█▒▒█▒█▒█▒█ - "diatonyx" dorian

█▒▒██▒█▒▒█▒▒██▒█ - didymian dorian

A scale with a single generator (octave-equivalent, of course) is called a "MOS", and all these scales, including normal zarlino, are "MODMOSes" of onyx, formed by sharping or flatting a few notes of it respectively. While technically, because this is an equal tuning, any 7-note scale can be a MODMOS of onyx, I try to reserve it for scales that have harmonic commonalities with diatonic or with onyx.

█▒█▒█▒█▒▒█▒█▒█▒█ - onyx

Here's the onyx scale, which we'll be using as an example to explore what kinds of scales you can build like this.

First, let's reorient it so all the small steps are next to each other. Here's the darkest mode of onyx:

█▒█▒█▒█▒█▒█▒█▒▒█ - pandian

Now, we can create zarlino by sharping the third and sixth degrees of this scale.

█▒█▒▒██▒█▒▒██▒▒█ - mixolydian

But we can sharp or flat other degrees as well. For example, if we sharp only the sixth degree, we get a kind of "diatonyx" which is a hybrid between the two scales.

█▒█▒█▒█▒█▒▒██▒▒█ - diatonyx (type 1)

This happens when sharpening the third degree only as well:

█▒█▒▒██▒█▒█▒█▒▒█ - diatonyx (type 2)

By choosing a different interval or pair of intervals to sharp, we can get scales that are like diatonic, but different.

█▒█▒█▒█▒▒█▒▒█▒██ - didymonyx (type 1)

█▒▒█▒██▒▒█▒█▒█▒█ - didymonyx (type 2)

█▒█▒█▒▒█▒█▒▒█▒██ - ?????

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:

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.

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

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.


WIP

Retrieved from "https://en.xen.wiki/index.php?title=User:VectorGraphics/Vector%27s_introduction_to_15edo/Scales&oldid=198035"