Question about enharmonics

Thank you, Daniel Wolf, for helping clarifying the tuning evolution with
the following paragraph:

>The familiar notation system used for 12tet predates 12tet. This notation
>is basically pythagorean, extending indefinitely by fifths in either
>direction through the addition of sharps and flats. When mapped onto
>meantone, which has one size of fifths, this indefinite extension is
>preserved but the relationship of sharps to flats flips over (i.e. in
>meantone db is higher in pitch than c#). When mapped onto 12tet, so-called
>enharmonic equivalence is introduced and the chain of fifths becomes a
>closed cycle of 12 members.

However, as a tuning "newbie" (I 'fess up...") I need a little bit more
elaboration and clarification of this whole "enharmonic" problem. I know
this has to be "easy old hat" stuff to you guys... Enharmonics are
obviously equivalent in 12-tET, but please describe in more detail what
they were before... If this is too dumb for the list, please post me off-
list. However, I doubt that it is, since I happen to know there are
probably 100-200 "lurkers" out there on the list who aren't posting
probably simply because they don't really know the answers to such
questions either....

Thanks!

Joseph Pehrson

I think that Joe Monzo has written about this extensively, but I give a
quick response:

I prefer to save "enharmonic" for the classical Greek tetrachordal genus
with a major third and a semitone ("pyknon") divided into two pitches,
broadly approximable by quartertones (see Chalmers, _Divisions of the
Tetrachord_).

Because the interval between, say c# and db, in either pythagorean, just or
meantone, was of microtonal magnitude, it was considered an "enharmonic"
interval. With the application of a tempered mapping where c# was equal in
frequency to db, the two pitches became "enharmonically equivalent."

For beginners: please note that none of this has anything to do with
"inharmonic".

Daniel Wolf

Joseph Pehrson wrote,

>Enharmonics are
>obviously equivalent in 12-tET, but please describe in more detail what
>they were before...

Roughly speaking, in the West until ~1420, due to Pythagorean tuning, sharps
were higher than flats: e.g., G# was higher than Ab, D# higher than Eb, etc.
From ~1480 to ~1790, due to meantone tuning, the situation was reversed, and
Ab was higher than G#, Eb was higher than D#, etc. Many keyboards were built
which allowed these distinctions, particularly in the 16th-17th centuries.
As we know, though, the 18th-19th century shift to closed 12-tone systems
was kind of messy, like all revolutions in tuning.

> [Joseph Pehrson, TD 531.9]
> I need a little bit more elaboration and clarification of this
> whole "enharmonic" problem. I know this has to be "easy old hat"
> stuff to you guys... Enharmonics are obviously equivalent in
> 12-tET, but please describe in more detail what they were
> before...

Joe, I discuss this a little in my book. Focus on the chapters
describing 3 and 5: the big switch happened when theory shifted
from a Pythagorean paradigm to a 5-limit JI one, c 1480.

You'll also find interesting information about the various
kinds of semitones - diatonic, chromatic, and enharmonic -
on my Marchetto webpage:
http://www.ixpres.com/interval/monzo/marchet/marchet.htm

-monz

Joseph L. Monzo Philadelphia monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
--------------------------------------------------

________________________________________________________________
YOU'RE PAYING TOO MUCH FOR THE INTERNET!
Juno now offers FREE Internet Access!
Try it today - there's no risk! For your FREE software, visit:
http://dl.www.juno.com/get/tagj.