QUICK START

Click :For those who are familiar with commas etc

Click :   Video : Quick start  

There can be a myriad of calculations even in a simple logic temperament as Vallotti.

But how clear and simple can a diagram be so that a quick look can show us the whole idea .. I have pondered on  this for a long time.

The technical side of this I started to develope one year ago.

A main diagram must focus on the fifth and the major third. The minor third  hangs with in tow. If the fifth and major third are pure,then  is also the minor third in the triad pure .

Circle of fifths are natural, but since my wish is to see clearly when the fifth is pure or how tempered it is, I leave a horizontal line correspond to a pure fifth. This means that the graph will slope downwards as a whole and Gb and F # will come in different levels.

But we become quickly accustomed to the fact that the two enharmonic notes (identical notes with different names, e.g. here G♭ and F#) are not in line.

(Gb and F # as a series of 12 fifths has a very distant mathematical and musical relationship,  531,441 / 524,288  )

The red graph above is 4 fifths ahead and a syntonic comma above the lower graph and the red graph/vectors shows the deviation from a pure major third.

The focus on the fifth prim/octave I have also with regard to the measuring unit. 

Temperament Units (John Brombaugh) is based on the Pythagorean comma (720 TU) instead of octave (1200 cents). The amazing thing is that  the syntonic comma then will be 660 TU, and both of these values ​​can  willingly be factorized.

A fluke !!

 720 and 660 can be split in 
2,3,4,5,6 (Both PC and SC)
8 and 9 (PC only)
10 (Both)
11 (only SC)
12 (Both)

(( Here is a temperament that is based on -1/5 SC tempered fifths  Hawkes 1807))

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

But now Vallotti.It will then be like this :

Picture (Link)

6 pure fifths (horizontal lines) and 6  tempered fifths of -120 TU / - 1/6 PC.

Regarding to the major third, one can notice the number 420 TU which is the major  third size  in Equal T. and 660 TU which is the Pythagorean third.

On the diagram we can see quickly three good thirds at 180 TU, 2 at 420  TU  (EqualT) and 3 Pythagorean thirds (660). And also important:  a clear pattern !

The latter of the three thirds can be  parts of triads with  pure fifths, horizontal lines.

Gb-Bb = 660 TU, Gb-Db = 0. (Bb Db = - 660 TU) (exuse my use of = )

If we will have a larger section or focus on chords around Gb or F# we can choose this diagram :

2 octaves

This is the second of 10 diagrams that accompany each tunings.

The minor third   and its deviation  from pure is obtained by summing the TU from red vector tip vertically to green vector tip.

C major. C E = 180 TU, C-G = 120 TU. In all normal tunings are the minor third  smaller than pure..

E-G are here thus - 300 TU.

Like this:

A visualization of E-G

This value is also down below on this diagram (blue):

Minor values (vs equal , dotted lines)

Those who  will focus on minor third as a whole we have this diagram:

Minor third

Here we see that all the blue vectors  are  pointing downwards.

The diagram is also mathematically correct compared to other intervals. The vertical distance (Vallotti) between eg A (red) and Bb (dark green) (in relation to the interval 16/15) is - 180 TU, like this:

16/15

Only when we look at several tunings comparing them you get an overview and see what regularities that are unavoidable. Basically variations are endless, but logically and not least in practice they are considerably more limited. There are many duplicates circulating on internet  and there are many that are close to each other. 

I have arranged in groups and I allowed myself to call temperaments with 11 even fifths for Meantones.

Modified Meantones are temperaments which one or more thirds are wider than 660 TU, with less than 11 even  fifths.

Circulating:  no major thirds are wider than 660 TU

These are again arranged , allowing the mildest temperament according to  Standard Deviation to be placed on the bottom of the page. .

The number of SD you will find  on each diagram, the first number in brackets is the SD of the fifths, the second of the thirds. These two follow each other more (or a few times less) , so I allowed myself to add them (unmathematical) together for convenience and sort them by this sum.

When we becomes accustomed to the diagrams it may be clarifying to see for example how the tempering in groups as  Circulating and Meantones ends up in Equal temperament. (with Windows I can click on a diagram and use the arrow keys to scroll through the temperaments) 

There are links everywhere and also above  each diagram  is there a link to 10 diagrams  (Not absolutely all of them has these 10)

Another diagram  includes the third important comma, Diesis (1260 TU), where we can get a quick overview of how this comma is divided into 3 thirds. The three commas are appearing  in the diagram lik this, (still Vallotti) :

Diesis and three commas

There is also a diagram for beats which are often necessary when to tune .

Beats

And then one diagram  that is designed to focus on the whole tones and semitones:

do-re-mi-fa-so

There is also a table where the cent-values are located.

Table

.

In the introduction you can follow  educational videos showing graphs combined with tone-examples.

I also have a link to LibreOffice ("Play around..") that I no longer use but that shows the graphs, a file for each tuning. After downloading it, you can move the tones up and down  and see how it affects the  graph and values.

You can also put your own cent-values into the spreadsheet.

But read the introduction to this first  (and the others).

I hope you will enjoy  to get acquainted with both the way the diagrams are made and then get an survey  of the tunings  as well.