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

IV FOR THE MORE ADVANCED, (whole tone, semitone)


Behind the perception of intervals we will find  mathematical relationships.
Each interval has its fraction or deviation from such fraction.
When two tones in the main diagram is horizontal in relation to each other, they form a precise fraction and its inversion (eg major third/ minor sixth)
We will now find all known relationships (unrelated to the 7th and 11th overtones)
Vector tip would have been on the top tone if the diagram had focused on the inverted interval.
I have prioritized 3/2, 5/4 and 6/5. If Ihad chosen the fourth, 4/3 instead of 3/2 (forth instead of fifth) then  the tip of the vector would always point the opposite way.

Pictures:










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I want to focus at the relationship between 256/243 (also called Pythagorean Limma) and 16/15 .
The Pythagorean Limma we get after five pure fifths in succession.
But also of course in a chain like this :






A pure 16/15 semitone is 660 TU wider than 256/243 (Limma :horizontal dotted line)
Sometimes I also get a little confused and mix together deviation of an interval and the interval itself. Besides, it can be difficult to get hold of which direction of the vector would have been in these more distant intervals.
 If we take a look at F# 7 in the previous diagram we see that B / A # is the interval pyth.limma, the dotted horizontal line through five fifths. It gives a deviation of 660 TU from the interval 16/15 (the oblique dotted line) It can look like the size of half tone (16/15) increases when the tone A# rises, but it shrink to be smaller than the 16/15 ratio, ie A# ladders, lead tone becomes higher, and we get an interval smaller than 16/15
If I  have a vector to 16/15 interval, then should the arrow here been turned down! Which in turn would have meant that this interval is less than 16/15.
This can be good to have consciously done before we look at some simple dominant-tonic progressions. And even more importantly it will be in our next diagram called «do-re-mi-fa-so».
A Pythagorean major third (660 TU) is no interval to rest in , but it gives a high leading-tone and works extra good in such a function.
We will now see (hear) some simple chord progressions


Here we are hearing a pythagorean third, B/A # has the pythagorean limma interval.
The minor triad Bm is nice.
B-D = - 300 TU
D-F # = 300 TU
B-F # = 0 TU
The major triad is  worse.
B-D # = 660 TU
D # -F # = - 660 TU
B-F # = 0 TU
So we take an example where the interval-quality of minor and major has swapped.




Here we see that the leading-tone has a longer way (low lead tone) up to the high lead tone (Limma) (dotted line)

A minor triad with large deviations from pure minor third
Bb Db = - 660 TU
Db-F = 660 TU
Bb F = 0 TU
However, the major third are here quite good
Bb-D = 300 TU
D- F = -300 TU
Bb F = 0 TU

Major thirds are even better here:


Major third is
F A = ​​180 TU
A-C = -300
The fifth is not pure here
F-C = -120 TU
Finally


This allows you to analyze yourself. :)
I have left out the numbers for seventh chords, but I think that the reader now manages this.

Here I will mention that seventh position is quite different in the examples.



We see here that the seventh is much higher in A7 than F7.

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TETRA chords (do-re-mi-fa-so)
These things I have made to get an overview of the size of the whole tones and semitone steps.
I focused on pyth.limma largely due to this diagram, where I do a twist. I place the upper graph (constantly based on fifth-chains) so that having   five pure fifths in succession (here Db-Ab-Eb-Bb-F-C),  the first five notes of the scale from Ab will get in line.
The C-Db will be a pyth.limma (no violet vector)


If the first two whole tones are small ((and if the forth between do and fa is quite pure) then  the half tone between the 3rd and 4th (the semitone)will be correspondingly larger as it is reflected in the violet vector. It shows the deviation from pyth. Limma (256/243). A long arrow upward reflects that the interval is a good amount larger than limma.


256/243 which is the ratio between the upper and lower graph shown in purple vector is not a consonant interval but I have chosen it to more easily assess half and whole-tone step relatively. By studying the pattern on the first five tones we get a sense of (or using exact numbers) the relationship between the notes of the scale.


(The major third in the highlighted notes relate to 81/64 and not 5/4 (the minor third 32/27))
Here comes a Kirnberger III variation.


And here comes the scales in different keys:


Additionally, we can see visualized that 10/9 whole tone is not included in particularly useful tunings.
And again the syntonic comma . A pure 9/8 (whole tone) is 660 TU greater than 10/9.
To obtain a whole tone at 10/9 the 2 fifths must take -330 TU each(average), and that is a very impure fifths.
It's Kirnberger II


And here we see a whole tone (D-E) as a 10/9


Here we see the impossibility of 10/9 whole tone without highly impure fifths.

10/9 is found more frequently in "just intonation".



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