# whole-tone / whole step / major-2nd / M2

[Joe Monzo]

A musical interval whose diatonic pitch-distance always traverses 2 adjacent letter-names inclusively, which may or may not involve a change of accidental, and which is approximately 200 cents, give or take about 50.

Some examples of major-2nds:

```Abb : Bbb
Ab : Bb
A : B
A# : B#
Bbb : Cb
Bb : C
B : C#
B# : Cx

etc.
```

The interval designations "whole-tone", "whole step", and "major-2nd" (or "M2") are synonymous. Modern theorists also sometimes use a small triangle to indicate "major".

Also synonymous with the definition of tone which designates roughly a 9/8 ratio or an interval with a size of approximately 200 cents or 1/6 of an octave.

The standard 12-edo major-2nd is 2(2/12) = exactly 200 cents.

The ancient pythagorean tuning gives a major-2nd with the ratio 9:8 = ~203.9100017 cents.

The 5-limit just intonation adds 10:9 (~182.4037121 cents) to the Pythagorean 9:8, so that it has two typical sizes for the major-2nd.

Having two different common sizes for the major-2nd is often seen as a problem in JI. A very successful solution to this during the "common-practice" era (c. 1500 - 1900) was meantone tuning, which, as its name indicates, gives diatonic scales which have only one size of major-2nd, its size falling midway between the two just intonation pitches (~193.156857 cents in 1/4-comma meantone).

7-limit just intonation also provides the large major-2nd of ratio 8:7 = ~231.1740935 cents.

11-limit just intonation also adds the very narrow major-2nd of ratio 11:10 = ~165.0042285 cents.

Successively closer small-integer rational approximations of the 12-edo "whole-tone" are:

``` ratio    ~cents

9:8   203.9100017
28:25  196.1984787
37:33  198.0710955
46:41  199.2119417
55:49  199.9798433 (about 1/50 cent narrower than 2(2/12))
```
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