Temperament again

One difficulty with the book is that Isacoff is writing a book on a subject he doesn't seem to completely understand himself. It is a short book, and skimpy on substance, but he does try to explain temperament on pages 60-67. His conclusion is that we can have pure octaves and some pure fifths by means of Pythagorean tuning, but that any attempt to have both pure thirds and pure octaves must fail, because 128/125 > 1, and any attempt to have pure minor thirds and pure octaves must fail, because 648/625 > 1. He apparantly does not see the inconsistency of this with either his discussion of the Pythagorean tuning or his subsequent discussion of meantone. He also seems to think that the main difficulty with Pythagorean tuning are the wolf fifths; it "won't work" because of this, though in fact it was the desire for better thirds which caused people to temper to meantone, and it was then that the fifths became the main problem.

As I noted before, he is also uncertain about his mathematics; he tries to explain what an irrational number is, but what he says (that it has "no definable limit" and can't be expressed in terms of integers) is nonsense; and to say it "simply goes on forever" sounds as if he thinks it is the same as a number with a non-terminating decimal expansion. He also seems to think that the Pythagorean theorem is some sort of calculation, and that the Pythagoreans did not know about the fundamental theorem of arithmetic, despite the fact that this is the basis for showing there are algebraic irrationalities, which they famously discovered.

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:

/tuning/topicId_35069.html#35069

> As I noted before, he is also uncertain about his mathematics; he
tries to explain what an irrational number is, but what he says (that
it has "no definable limit" and can't be expressed in terms of
integers) is nonsense; and to say it "simply goes on forever" sounds
as if he thinks it is the same as a number with a non-terminating
decimal expansion. He also seems to think that the Pythagorean
theorem is some sort of calculation, and that the Pythagoreans did
not know about the fundamental theorem of arithmetic, despite the
fact that this is the basis for showing there are algebraic
irrationalities, which they famously discovered.

***Thanks, Gene. This was helpful as I read this puppy...

jp