# Rank-2 temperaments by mapping of 3

Following is a list of octave-repeating rank-2 temperaments, organized by the mapping of the prime 3.

The motivation for this is that 4/3 and 3/2 are the most consonant intervals within an octave, so it makes sense to look for temperaments where they occur often. Moreover, if two temperaments fall into the same category on this page, they not only have similar MOS structure, but also the consonant intervals 4/3 and 3/2 will appear in the same places in each MOS they have in common, so an important part of the harmonic structure is similar as well as the melodic structure. See also pergens.

## Complexity 0

Temperaments in this category temper out a 3-limit comma, so 3 is mapped to an interval of some equal temperament, and unequal intervals are used only for higher primes such as 5 or 7.

The canonical example is blackwood, in which all 3-limit intervals are approximated by 5edo, and the unequal subdivisions of those steps are only used to represent the prime 5.

Technically there is an infinite number of possible mappings in this category, because there is an infinite number of EDOs you could choose to map 3 to. However, the only practically useful ones are based on EDOs that are both small, and contain relatively accurate mappings of 3.

[<5 8...], <0 0...]>

[<7 11...], <0 0...]>

[<12 19...], <0 0...]>

[<29 46...], <0 0...]>

## Complexity 1

Temperaments in this category have octaves as periods and good old fourths and fifths as generators. Therefore they can be faithfully notated with standard Western notation, unlike temperaments in all the other categories on this page.

Mandatory MOSes include 3, 5, and *almost* 7 (except father is all screwed up and has an 8-note MOS instead).

[<1 2...], <0 -1...]>

## Complexity 2

### Period 1

Temperaments in this category split either 4/3 or 3/2 into two equal parts.

The first splits 4/3 into two ~250 cent intervals. Mandatory MOSes include 4, 5, and 9.

[<1 2...], <0 -2...]>

And the second splits 3/2 into two ~350 cent "neutral thirds". Mandatory MOSes include 3, 4, and 7.

[<1 1...], <0 2...]>

### Period 2

Temperaments in this category split the octave into two ~600 cent intervals, and have both 4/3 and a ~100 cent interval as generators. Mandatory MOSes include 4, 6, 8, 10, and 12.

[<2 3...], <0 1...]>

## Complexity 3

### Period 1

First we have "the porcupine category". 4/3 is divided into 3 equal parts of ~166 cents. Mandatory MOSes include 7, 8, and 15.

[<1 2...], <0 -3...]>

Next we have temperaments in which 3/2 is divided into 3 equal parts of ~234 cents. Because this is so close to 8/7, most if not all of these are in the Gamelismic clan. Mandatory MOSes include 5, 6, 11, and 16.

[<1 1...], <0, 3...]>

Finally we have temperaments in which 8/3 is divided into 3 equal parts of ~566 cents (or equivalently 3/1 into 3 parts of ~634 cents). (Interesting fact: This also implies that 9/8 is divided into 3 equal parts.) Mandatory MOSes include 5, 7, 9, 11, 13, and 15.

[<1 3...], <0, -3...]>

### Period 3

These pretty much have to temper out 128/125 to be any good. See Augmented family. Mandatory MOSes include 6, 9, 12.

[<3 5...], <0 -1...]>

## Complexity 4

### Period 1

Generator ~125 cents. Mandatory MOSes: 9, 10, 19.

[<1 2...], <0 -4...]>

Generator ~175 cents. Mandatory MOSes: 6, 7, 13, 20.

[<1 1...], <0 4...]>

Generator ~425 cents. Mandatory MOSes: 3, 5, 8, 11, 14, 17.

[<1 3...], <0 -4...]>

Generator ~475 cents. Mandatory MOSes: 3, 5, 8, 13, 18, 23, 28, and probably some more.

[<1 0...], <0 4...]>

### Period 2

Generator ~50 cents. Mandatory MOSes: all even numbers up to 22.

[<2 3...], <0 2...]>

Generator ~250 cents. Mandatory MOSes: 4, 6, 10.

[<2 4...], <0 -2...]>

### Period 4

Generator ~100 cents. Mandatory MOSes: 4, 8, 12.

[<4 6...], <0 1...]>

## Complexity 5

Generator ~100 cents. Mandatory MOSes: everything up through 12.

[<1 2...], <0 -5...]>

Generator ~140 cents. Mandatory MOSes: 8, 9, 17.

[<1 1...], <0 5...]>

Generator ~340 cents. Mandatory MOSes: 3, 4, 7, 11, 18, 25, 32, 39.

[<1 3...], <0 -5...]>

Generator ~380 cents. Mandatory MOSes: 3, 4, 7, 10, 13, 16.

[<1 0...], <0 5...]>

Generator ~580 cents. Mandatory MOSes: all odd numbers up through about 25.

[<1 4...], <0 -5...]>

## Complexity 6

### Period 1

Generator ~83 cents.

[<1 2...], <0 -6...]>

Generator ~117 cents.

[<1 1...], <0 6...]>

Generator ~283 cents.

[<1 3...], <0, -6...]>

• I can't think of anything.

Generator ~317 cents.

[<1 0...], <0 6...]>

Generator ~483 cents.

[<1 4...], <0, -6...]>

• I can't think of anything.

Generator ~517 cents.

[<1 -1...], <0 6...]>

### Period 2

Generator ~34 cents.

[<2 3...], <0 3...]>

• I can't think of anything.

Generator ~166 cents.

[<2 3...], <0 -3...]>

Generator ~234 cents.

[<2 2...], <0 -3...]>

### Period 3

Generator ~50 cents

[<3 5...], <0 -2...]>

Generator ~150 cents

[<3 4...], <0 2...]>

### Period 6

[<6 10...], <0 -1...]>

## Higher complexity

[<1 0...], <0 7...]>

[<1 -1...], <0 7...]>

[<1 1...], <0 8...]>

[<1 -1...], <0 8...]>

[<1 1...], <0 9...]>

[<1 4...], <0 -9...]>

[<1 -1...], <0 10...]>

[<2 4...], <0 6...]>

[<2 5...], <0 -6...]>

[<2 1...], <0 6...]>

[<1 5...], <0 -13...]>

[<1 4...], <0 -15...]>

[<1 -1...], <0 16...]>

[<1 -5...], <0 17...]>

[<3 6...], <0 -6...]>

[<9 15...], <0 -2...]>