Baseball

Baseball is a temperament using a large tritone, around 638.5 cents, as a generator, and an octave period. This results in a temperament with very wide fifths (around 714 or 715 cents); the fifth being three generator steps away.

The temperament is named in honor of Babe Ruth and Hank Aaron; Ruth having had 714 career home runs, and Aaron breaking Ruth's record with his 715th; both numbers being possible widths of the baseball fifth in cents. The baseball fifth is about as sharp as the 12edo major third, and provides the characteristic sound of baseball. Although sharp, it is not at all unpleasant, and it's relatively easy to get accustomed to. A 12edo fifth played after a series of baseball fifths may actually sound flat because the listener will have gotten accustomed to the latter.

Apart from the wide fifth, another characteristic of baseball is that the 5th harmonic is approximated in two different ways (one of which is also very sharp, the other is slightly flat). Both are usable for harmony which gives an unusual situation in which two noticeably different intervals actually have the same just approximation.

Baseball forms MOSes of 15, 17, 32, 47, etc. In fact, baseball temperament was discovered during an attempt to modify 17edo to allow 5-limit harmonies. Baseball[17] has the property that all three types of thirds (corresponding to 4, 5, and 6 steps of 17edo), have very high complexity using the baseball generator, and as a result, each of them "splits" into two different types which are roughly the same distance apart. This enables the 5-limit major and minor thirds, each of which is almost halfway between two scale degrees of 17edo, to be approximated in two different ways; "from above" and "from below".

While baseball does not provide a good approximation of the Pythagorean whole tone (9:8) or the 7:9 supermajor third, it does provide good approximations of both 8:7 and 10:9.

Baseball also provides workable approximations of higher limits, too. In particular, since its generator is so close to 13:9, it does a lot of tridecimal intervals very well, and can approximate the barbados triad (10:13:15) with relative ease. Although not as close, the generator can also be taken as a 16:11 or as a 10:7.

An example of a baseball[17] scale in which the fifths are exactly 715.5 cents wide is given below:

0
77
154
231
308
385
462
539
616
638.5
715.5
792.5
869.5
946.5
1023.5
1100.5
1177.5

As can be seen, all steps are 77 cents wide except for two much narrower (22.5 cent) steps. Baseball[17] is thus a 15L+2s scale. In this scale there are two different approximations of 6:5 (at 331.5 and 308 cents), and two of 5:4 (at 407.5 and 385 cents). The sharper of the two major thirds can also double as a 14:11, and the sharper minor third can also function as 11:9, depending on the context. The interval of 253.5 cents can function either as a flat 7:6 (for example, in the triad 6:7:8) or else as a 15:13.