# Meantone temperaments

## 1/4-comma meantone

The 1/4-comma meantone temperament was the most commonly used meantone temperament (prior to well-temperaments). The perfect fifth is reduced by 1/4 of a syntonic comma so that the major third is pure (5/4) as well as its complement the minor 6th (8/5). Here is an example of 1/4-comma meantone by defining the chromatic notes with cents:

Here is the same music showing the meantone calculations, which are stacks of perfect fifths that are reduced by 1/4 of a syntonic comma to make pure major thirds (5/4):

Below is an example where the calculation to reduce the fifth by
1/4 of a syntonic comma are not reduced. The comma is `81/80`

, and
the expression `(3/2)*(81/80)^(-1/4)`

takes the perfect fifth `(3/2)`

and reduces it by 1/4 of a syntonic comma.

## 1/3-comma meantone

The perfect fifth is reduced by 1/3 of a syntonic comma so that the minor third is pure (6/5) as well as its complement the major 6th (5/3).

Comparing 1/4-comma and 1/3-comma:

## 1/2-comma meantone

1/2-comma meantone creates 9/5 minor sevenths, and inversely 10/9 major seconds). The major thirds are 100/81, or 385 cents (compared to 386 cents for 5/4 major thirds, or 400 cents for equal-tempered major thirds).

## 1/5-comma meantone

## 1/6-comma meantone

## 2/7-comma meantone

## References

- Meantone article on Wikipedia
- Meantone article on Xenharmonic Wiki
- 2/7 comma meantone article at Tonalsoft encyclopedia of microtonal music theory.