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...)

.................
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

V STANDARD DEVIATION



Here I will briefly show how to measure a tunings extremity

The function is called Standard Deviation and you can find it on calculators with statistics and in spreadsheets.

Put simply it calculates the deviations from the average.

If you have 10 apples and divide it equally among five people all of them get 2 apple (average).
Then Standard Deviation is 0
  .
But if a greedy person takes 8 apples and two others get 1 each and two none, then it becomes a high Standard Deviation, (3.39)
SD on how the wealth in the world is divided between the world's population would unfortunately become a very big number. :(

In our case, we take an SD number of the 12 fifths and the second number of the 12 major thirds.

If we again look at Vallotti we have the fifths as follows:

0,0,0,0,0, -120, -120, -120, -120, -120, -120.0

If we enter these 12 numbers and use Standard Deviation then we get (approximately) 63

Thirds are
660,660,540, 420,300,180,180,180,300,420,540,660

Taking SD of these thirds we get (about) 192
.
If we take SD on the most extreme tuning in this collection that is 1/3 Meantone , we get
respectively 554 on the fifths and 945 on the major thirds !!

Average amount of fifths is always -60 and the major third 420 TU.
SD thus shows deviation from this average.
SD on the fifths and thirds often follow each other to a certain degree.
 If the discrepancy is large in fifths then it is generally the same also in the case of thirds.

But let's look at an example:



Here we see that every third is 420 TU(like ET) , thus a SD on thirds like 0!
But the fifths splays. 9 pure fifths and three impure fifths of -240 TU get an SD like 109.

For simplicity, I add SD on fifth and the third together and when we get a kind of average for extremity degree.
In the blog I have ordered tunings consecutively from high to low with this SD sum as the number I
relate to.

SumSD = (SD of the fifths + SD of the maj thirds) as indicated on each diagram.

A tip when browsing these is to start at the bottom and go upwards towards the extreme diagram which is a little harder to understand.
Duplicates are detected at once with this system. Young II and Vallotti get the same values ​​(same structure).


I have a letter in front indicating type temperament.
M = Meantone, definition: 11 equal fifths


P = Pythagorean, definition 11 or 10 pure fifths

MM =Modified meantones means here that one or more thirds are greater than 660 TU (pyth ters)

C = circulating. Here is no thirds greater than 660 TU



J = Just intonation


CU = Curves. These I have made a bit for fun designed to create as smooth transition as possible between neighbor thirds . These are not proven in practice, but certainly works good.But I think the tuner needs an app to tuning the instrument with these temperaments.

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