15edo

The 2-dimensional harmonic table that serves as 15edo's defining feature.

15edo, or 15 equal divisions of the octave, is the equal tuning featuring steps of (1200/15) = 80 cents, 15 of which stack to the perfect octave 2/1. It is notable for its acceptable but rather distant approximation of the 11-limit featuring a near-isoharmonic 4:5:6, and for its contorted mappings.

Theory

Edostep interpretations

15edo's edostep has the following interpretations in the 2...11 subgroup:

• 16/15 (the difference between 5/4 and 4/3)
• 25/24 (the difference between 5/4 and 6/5)
• 81/80 (the difference between 9/8 and 10/9)
• 36/35 (the difference between 5/4 and 9/7)
• 22/21 (the difference between 7/6 and 11/9)

JI approximation

15edo has roughly 10-20% error on prime harmonics 3 through 11, which is a deviation from just intonation significant enough to severely affect its structure, without fully compromising the function of the prime harmonics. It is best seen as a crude approximation of the 11-limit. Because it is not a meantone system, the best diatonic to use for 5-limit harmony is the Zarlino diatonic scale (LMsLMLs), tuned in 15edo as 3-2-1-3-2-3-1. Note that 15edo lacks a standard MOS diatonic scale due to its fifth being 720 cents. Significantly, 15edo is 5 x 3, and inherits its tunings of 3 and 7 from 5edo, and 5 from 3edo. This requires either a chain of 11/8s or 23/16 or a 2-dimensional lattice be used to visualize 15edo's structure in a similar manner to the circle of fifths in 12edo.

Approximation of prime harmonics in 15edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	0.0	+18.0	+13.7	-8.8	+8.7	+39.5	-25.0	+22.5	+11.7	+10.4	-25.0
	Relative (%)	0.0	+22.6	+17.1	-11.0	+10.9	+49.3	-31.2	+28.1	+14.7	+13.0	-31.3
Steps (reduced)		15 (0)	24 (9)	35 (5)	42 (12)	52 (7)	56 (11)	61 (1)	64 (4)	68 (8)	73 (13)	74 (14)

Thirds in 15edo

Quality	Minor	Major
Cents	320	400
Just interpretation	6/5	5/4

Chords

15edo contains 5edo's suspended triads, now functioning as a kind of "tendo and arto" triads. However, it adds to 5edo standard major and minor triads. Its major triad is especially notable for being close to an isoharmonic 50:63:76 triad, a property not shared by either other 5n-edos like 25 or 12edo. Additionally, the wolf chords coming with the Zarlino diatonic have a wolf fifth of 640 cents, which is also the tuning for 16/11 and thus significantly more functional than the wolf fifth in diatonic is in general. Additionally, 15edo approximates the harmonic tetrad 4:5:6:7 as [0 5 9 12]. 9:10:11:12 is equidistant (spanning a perfect fourth), and so is 6:7:8:9 (spanning a perfect fifth).

Intervals

Interval categories

Here is a table of 15edo's intervals:

Name (Vector's theory)	Name (ADIN)	Degree	Cents	Approximate Ratios
Unison	Unison	0	0	1/1
Semitone	Pentaminor second	1	80	25/24, 16/15
Minor tone	Pentamajor second	2	160	10/9
Major tone, wolf third	Supermajor second, subminor third	3	240	8/7, 9/8
Minor third	Pentaminor third	4	320	6/5
Major third	Pentamajor third	5	400	5/4
Perfect fourth	Perfect fourth, supermajor third	6	480	4/3, 21/16
Small tritone, diminished fifth, wolf fourth	Pentafourth	7	560	11/8, 7/5
Large tritone, augmented fourth, wolf fifth	Pentafifth	8	640	16/11, 10/7
Perfect fifth	Perfect fifth, subminor sixth	9	720	3/2, 32/21
Minor sixth	Pentaminor sixth	10	800	8/5
Major sixth	Pentamajor sixth	11	880	5/3
Wolf sixth, narrow minor seventh	Subminor seventh, supermajor sixth	12	960	7/4, 16/9
Wide minor seventh	Pentaminor seventh	13	1040	9/5
Major seventh	Pentamajor seventh	14	1120	48/25, 15/8
Octave	Octave	15	1200	2/1

Let's take a look at the zarlino diatonic scale. Zarlino is an MV3 scale, meaning that there are at most 3 sizes of any given interval.

So, let's lay out all the modes of zarlino, and see where our scale degrees fall:

1..2.34..5.6..71 // Ionian
1.2..34..5.6..71 // Ionian
1.23..4.5..67..1 // Dorian
1..23..4.5..67.1 // Dorian
12..3.4..56..7.1 // Phrygian
12..3.4..56.7..1 // Phrygian
1..2.3..45..6.71 // Lydian
1..2.3..45.6..71 // Lydian
1.2..34..5.67..1 // Mixolydian
1.2..34.5..67..1 // Mixolydian
1..23..4.56..7.1 // Aeolian
1..23.4..56..7.1 // Aeolian
12..3.45..6.7..1 // Locrian
12.3..45..6.7..1 // Locrian
. 

As you can see, except for seconds, the main diatonic major/minor dichotomies remain intact, albeit with the occasional exception. Specifically, for thirds, fourths, fifths, and sixths, you find "wolf intervals" on two of the fourteen modes for each, which for fifths are between perfect and diminished, and for thirds, they are below minor. Each of these types of wolf intervals is special because they correspond to the prime harmonics we have access to in 15edo, but not in 12edo: the wolf sixth, for example, represents the harmonic seventh ratio of 7/4 (and the wolf third represents its complement 8/7) (but note that these are also our largest seconds and smallest sevenths), and the wolf fourth represents the undecimal tritone, 11/8 (and of course, the wolf fifth represents its complement 16/11.)

The harmonic table

In 12edo, we often arrange intervals on a "circle of fifths", which outlines a lot of the harmonic structure we use in that tuning. In 15edo, a circle of fifths doesn't end up covering all the intervals (as it repeats after 5 steps), so we might think to use a circle of major thirds, the next simplest interval in terms of JI, but as it turns out, that ALSO doesn't cover all the intervals, repeating after only 3 steps. And with harmonic sevenths, we're back to looping after 5 again. As such, 15edo can be described as "contorted" in 2.3.7 and 2.5. So, the only solution (other than stacking the harmonic 11) is to use a two-dimensional "circle", which forms a torus-shaped "harmonic table":

0c	400c	800c	0c	400c	800c	0c
480c	880c	80c	480c	880c	80c	480c
960c	160c	560c	960c	160c	560c	960c
240c	640c	1040c	240c	640c	1040c	240c
720c	1120c	320c	720c	1120c	320c	720c
0c	400c	800c	0c	400c	800c	0c
480c	880c	80c	480c	880c	80c	480c
960c	160c	560c	960c	160c	560c	960c
240c	640c	1040c	240c	640c	1040c	240c
720c	1120c	320c	720c	1120c	320c	720c
0c	400c	800c	0c	400c	800c	0c

We call this a torus because it wraps around from one side to the other, like if you were to print this on the surface of a donut.

Here,

• fifths are found by stepping 1 step up the Y-axis.
• harmonic sevenths are found by stepping 2 steps down the Y-axis.
• major thirds are found by stepping 1 step right along the X-axis.

Instead of two types of semitones, 15edo has four:

• the limma, found by stepping down and to the left by 1 on the table. This separates intervals belonging to different classes.
• the classical chroma, found by stepping down 1 and right 2 on the table. This separates intervals with the same class, but different qualities.
• the syntonic comma, found by stepping up 4 and left 1 on the table. This separates standard intervals from "wolf" intervals.
• the diaschisma, found by stepping up 4 and right 2 on the table. This separates fifth-generated "diatonic" intervals and their augmented or diminished zarlino counterparts. Note that this semitone is actually the complement of the just diaschisma, since the diaschisma proper is a negative interval in 15edo.

Scales

One notable thing about 15edo is how many different kinds of scales it gives you access to.

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: 15. 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: 15. In this scale, each interval has 3 sizes: 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: 15

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" 15. If we split each step of this scale into a larger and smaller portion, we get the blackwood scale proper: 15. 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. Thus, the blackwood scale can serve as an extremely powerful alternative to the absent MOS diatonic of this edo - it is, in a sense, a "purer" version of said system. (In other edos, this scale's relation to zarlino generalizes to that of blackdye, which loses some of its structural elegance in favor of a more direct diatonic connection and more in-tune intervals.)

Augmented

Augmented is, conversely, based on major thirds. The basic augmented scale is 15, which is equivalent to 12edo's augmented triad. 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.

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.

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 15.

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 blocks

In 15edo, there are four primary ways of reaching the semitone on the harmonic table, corresponding to different copies of the semitone repeating periodically near the unison. It can be found as the difference between a stack of two fifths and a minor tone (the syntonic comma), the difference between a major third and a perfect fourth (the limma), the difference between the two tritones (the diaschisma), and the difference between a major and minor third (the classical chroma).

The standard zarlino scale can be defined by having the classical chroma and syntonic comma 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.

However, we can choose a different pair of semitones as our chromas, and get a different scale. Here's the classical chroma x diaschisma scale.

├──┴┴┴─┴┴─┴─┴┴┴┤ - elena

├──┴┴┴─┴─┴┴─┴┴┴┤ - kee'ra*

*Note: in just intonation, the kee'ra scale is a 7-limit "chair" scale, not a 5-limit periodicity block.

There are two variants as, since the diaschisma can be created by stacking two augmented fourths, the augmented fourth and diminished fifth fall precisely on the edge.

And the classical chroma x limma scale:

├───┴──┴──┴──┴─┤- myn

Regular temperaments

15edo shares Porcupine with 22edo, Augmented with 12edo, Semaphore with 19edo, and Blackwood with 10edo.

Notation

Due to MOS-diatonic-based notations being nonfunctional with edos that have multiple chains of fifths (except for ups and downs notation, and even that requires E and F be treated as enharmonic), they are somewhat inconvenient for working with 15edo. Notation is often KISS notation based on onyx or pentawood, or notation based on the Zarlino diatonic scale.