Indeed strange.  It seems that the author had his/her personal definitions.

"Pythagorean, Just Intonation, Tempered Pythagorean, Meantone,
Well-tempered, and Equal Temperament. "


   - *Just intonation*, i. e the grid of pure fifths and thirds
includes *Pythagorean
   tuning* (a chain of fifths) as as a subset or special case.
   The same is true for the case that *Pythagorean tuning* in practice was
   tuned as the virtually identical chain of eight fifths of the standard
   scale based on the three hexachords Bb-F-C-G-D-A-E-B,** to which the thirds
   of some of the main modes are added: F#, C# and G#.  The twelfth not can
   equally serve as Eb or D# *** (though B doesn't have a usable fifth! The F#
   is flat by a syntonic comma, ca 1/5 of a semitone). ****

   - "tempered Pythagorean" is a misnomer of something that is not
   identified

   -  *Wohltemperiert* (*Well-tempered*)  includes *Equal Temperament* as a
   subset or special case, as Werckmeister (1645-1706) called *Equal
   temperament* a "wohl temperirte Harmonia" in his "Paradoxal discourse",
   1707 (published posthumously)
   Source:
   http://reader.digitale-sammlungen.de/en/fs1/object/display/bsb10527832_118.html
   ).


** Or only the "natural" hexachord CDEFGA with its lower and upper
semitones (B - CDEFGA -Bb)

*** D# (*dis*) has remained the note name in organ tablature and remains up
to today the standard in organ building.  Both long after Pythagorean
tuning ...

**** I find the explanation of *Pythagorean tuning* via the
hexachord-system and the main church modes much more compatible with the
development of music theory of the time.
It would be much easier in practice to develop a tuning from the
hexachords. That the resulting tuning is virtually the same as what we
define as strict Pythagorean tuning (a chain of eleven fifths),
Gb-Db-Ab-Eb-Bb-F-C-G-D-A-E-B. The notes  Gb, Db, and Ab were however not


Kind regards

Ibo

* * * * * *


2018-09-05 23:09 GMT+02:00 Kelzenberg, David C <[log in to unmask]>:

> That's very strange.  Does anybody know what the difference is between
> "Pythagorean" and "Straight Pythagorean" might be?  Is regular Pythagorean
> gay?  (I never knew there were different Pythagorean tunings--perhaps
> putting the wolf between different 5ths?)  And, really...  would anyone
> really be using Pythagorean tuning (of any sort) at that time?
>
> dk
>
>

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