# User:IlL/Using pairs of edos to define temperaments

I like to think of (rank 2) temperaments not so much as a set of commas, but as a property that you can extract from two edos that represent similar JI subgroups; it's the thing responsible for two edos being similar. (Technically, this operation that you do on two edos is called the wedge product, but that's beside the point.) This helps identify the most natural subgroup temperaments and MOS scales.

So for sake of reference, this page lists the results of the following procedure:

1. Find two reasonable-sized edos that represent many of the same JI intervals (i.e. accurately represent a shared JI subgroup, preferably consistently).
2. Use the Temperament Finder to get the optimal generator size and the commas. (1 period per octave unless stated otherwise.)
3. Stack up the generator to get an interval chain; analyze what MOS scales are good for what chords.

## 10&16

Lemba: a 2.3.7.13 temperament

POTE gen. 233.6044¢; 2 periods/oct

## 11&15

Orgone: a 2.7.11 temperament

POTE gen. 323.2801¢

## 11&17

Machine: a 2.7.9.11 temperament

POTE gen. 214.3843¢ (approx. 9/8~8/7)

## 11&20

Joan: a 2.7.11.15 temperament

POTE gen. 541.5530¢ (approx. 11/8~15/11)

## 13&17

Lovecraft: a 2.9.11.13 temperament

POTE gen. 281.1364¢ (approx. 13/11)

## 13&18

A-Team: a 2.9.5.21 temperament

POTE gen. 464.1396¢ (approx. 21/16)

## 16&19

(a Magic extension): a 2.5.7.13 temperament

POTE gen. 378.4481¢ (approx. 5/4~16/13)

## 16&27

a 2.5.7.13 temperament

POTE gen. 223.3392¢ (approx. 8/7)

## 17&20

a 2.7.11.13 temperament

POTE gen. 421.3091¢ (approx. 14/11)

## 17&31

a 2.3.7.11 temperament

POTE gen. 425.2436¢ (approx. 14/11~9/7)

## 19&27

Sensi: a 2.3.5.7.13 temperament

POTE gen. 443.322¢ (approx. 9/7~13/10)