Yes, the wide fifths
creates bad thirds
And what creates
purer thirds are narrow fifths.
If you only have
narrow fifths it is not space for so many of them.
5 x - 1/5 PC (-144
TU) provides two almost pure thirds on 84 TU, not more
And then you have used up the -720 TU , the pythagorean comma.
And then you have used up the -720 TU , the pythagorean comma.
By having wide
fifths we can have more tempered fifths creating more good
thirds.
Here's an example.
(The fifths must fall with -720 TU from left to right . Then the last tone in the diagram is the same as the one to the left . Ab= G# or Gb=F# in the diagram above. Horizontal line is a pure fifth)
(The fifths must fall with -720 TU from left to right . Then the last tone in the diagram is the same as the one to the left . Ab= G# or Gb=F# in the diagram above. Horizontal line is a pure fifth)
If we have three wide
fifths of + 90 TU,
we can get 7 fifths
of -144 TU (-1/5 PC) like this. (Diagram now from Ab)
OR (not so steep)
8 of -120 TU (1/6 PC) ,with one good third in addition but each of them is less pure.
The purpose and the advantage is to be able to play in keys with few key signature with very nice thirds in the centre and with the possibility to use all
thirds (no wolfs). But the bad chords should never be used as the
primary key, and the temperament becomes very abrupt.
Regarding Haugsand 6
Hz so it is true that we get a third wider than a pythagorean . But it is not so bad. because the fifths are not so wide in 5.5/6 Hz.
But I therefore proposed
to tune Gb-Db pure afterwards, especially if you want to play a piece in D flat major. But how wide these fifths are depends on which Hz you choose.
If you choose a bit over 7 Hz you get no wide fifths, but 4 pure ones .
If you choose a bit over 7 Hz you get no wide fifths, but 4 pure ones .
In the link below you can see how the wide fifths influence on the temperaments as a whole. The most extreme temperaments are on the top down to almost circulating.
On the other hand, 6 pure fifths as in
Vallotti, Kellner or Werckmeister III is also problematic, with 3 adjacent Pythagorean thirds (660 TU )
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