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.
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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