More than 1600 diagrams , 175 tunings !

(Pinned post ) IMPORTANT :

For those of you who are familiar with cent values in several temperaments, please use the diagram called «Minor cent» in the beginning.

Here you will find cent values for fifths, major thirds and minor thirds within the same graph.

The red vectors is deviation from pure major third and blue vectors for minor third.

In the other diagrams I am using TU, but simply divide values by 30 to get a precise enough cent value.

Pythagorean comma = 23.46 cents = 720 TU

Syntonic comma = 21.51 cents = 660 TU.

(These numbers are easy to divide in 2,3,4,5,6...)

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The easiest way to get an overview of historical and newer tunings ! Read the Introduction (1) carefully and the diagrams will tell you much more than all the cent tables on the web.
Many of those who have found the key to the diagrams use this webside a lot.
If you have questions , let me hear, click HERE

To Holger , about wider fifths and what you gain from them

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.



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)

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 .



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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