The graphical guide to tunings -a new blog

http://thegraphicalguidetotunings.blogspot.no/

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 TUThese 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.

On Tue, Jan 12, 2016 at 11:07 PM, Gunnar Tungland
gunnnar.tungland@... [TUNING] <TUNING@yahoogroups.com> wrote:

> http://thegraphicalguidetotunings.blogspot.no/

In summary, if you draw a line whose slope represents the temperament
of the circle of fifths, then the temperament of various other
intervals is given by the vertical separation between that line and
various shifted versions of that line -- because other intervals are
expressible as various numbers of fifths.

Neat!

In the blog, temperaments are expressed in TU, where a TU is 1/720 of
a Pythagorean comma. Allow me to re-express some of them in mils,
where a mil (a.k.a. a prima) is 1/1023 of a 12EDO semitone, or 3.00005
TU (see http://beta.briefideas.org/ideas/50bd78ca85c5e2523c11f83250e851c3
& http://beta.briefideas.org/ideas/db017aafc9ef680d0549e46e25aa8964 ).
The mil is so named because it is approximately 1/1000 of a semitone.

> Temperament Units (John Brombaugh) is based on the Pythagorean comma (720 TU)

239.996 mils ~= 240 mils

> instead of octave (1200 cents).

12276 mils.

> The amazing thing is that the syntonic comma then will be 660 TU

more precisely, 660.039 TU = 220.009 mils ~= 220 mils,

> and both of these values can willingly be factorized.

> 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)

With the mil, we lose a factor of 3, but:

240 (PC) and 220 (SC) can be split in
2,4,5 (Both PC and SC)
3, 6, 8 (PC only)
10 (Both)
11 (only SC)
12 (only PC)

> But now Vallotti.It will then be like this :
> ...
> 6 pure fifths (horizontal lines) and 6 tempered fifths of -120 TU / -1/6 PC

40 mils.

> Regarding to the major third, one can notice the number 420 TU

140 mils

> which is the major third size in Equal T.

That's more precisely 420.039 TU = 140.0107 mils.

> and 660 TU

220 mils

> which is the Pythagorean third. On the diagram we can see quickly three good thirds at 180 TU

60 mils,

> 2 at 420 TU

140 mils

> (EqualT) and 3 Pythagorean thirds (660)

220 mils.

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

220, 0, -220 mils.

> C major. C E = 180 TU, C-G = 120 TU...

60 mils, 40 mils.

> E-G are here thus - 300 TU

100 mils.

> 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

-60 mils.

> Circulating: no major thirds are wider than 660 TU

220 mils.

> Another diagram includes the third important comma, Diesis (1260 TU)...

more precisely, 1260.118 TU = 420.03 mils ~= 420 mils.

1/4 syntonic meantone is 55-mil meantone.
1/5 syntonic meantone is 44-mil meantone.
1/5 ditonic meantone is 48-mil meantone.
1/6 ditonic meantone is 40-mil meantone.
12EDO is 20-mil meantone.

In the email message, all the linked diagrams in TU are convertible to
whole numbers of mils. Some of the diagrams on the blog (e.g. at
http://thegraphicalguidetotunings.blogspot.no/p/vallotti-ny.html ) are
NOT convertible to whole numbers of mils, but the mil is such a small
unit that rounding to the nearest mil would make negligible
difference.

OTOH, "Mil You" is a lousy title compared with "TU You". :D

Congratulations and kind regards,
Gavin R. Putland.

On Wed, Jan 13, 2016 at 10:48 AM, Gavin R. Putland <grputland@...> wrote:

> where a mil (a.k.a. a prima) is 1/1023 of a 12EDO semitone, or 3.00005
> TU (see http://beta.briefideas.org/ideas/50bd78ca85c5e2523c11f83250e851c3
> & http://beta.briefideas.org/ideas/db017aafc9ef680d0549e46e25aa8964 ).

P.S.: Make sure you don't include the '&' in either link. Behavior may
depend on your browser or email client.

--- Gavin R. Putland.

Hi
Thanks for your contribution!
If I had known about your idea I'd probably considered it carefully before I had made a choice.

The benefits of your idea is:
Fewer digits, had given more space in the diagrams.
Corresponds easy to cents, roughly.
1/4 SC = 55
1/5 SC = 44
these important values are easy to remember.
.......
Mil is a bit less flexible due to a factor of 3 is missing.

From ancient times we have 360 degrees of the circle, and many are accustomed to different divisions of it, and therefore many users will already be acquainted with numbers from the factorizing.
120 is thus slightly better than 40. 240 has no such tradition.
This applies not to the SC. So there the units are equal.
1/3 SC meantone is a historical temperament, so this tempered fifth is perhaps the one I would miss the most.
I have otherwise informed in the introduction that the users can easily find cents roughly , by dividing with 30.
The disadvantage of mil is that the whole blog is based on images (and spreadsheets must be a bit redone.), that job I will not do again ... :)
Another disadvantage is that the TU despite that it is not so widely used is known in circles where one goes deeper into this stuff. A new unit will be to some confusion and make this wonderful idea that TU is ,
not so easy to come into more common use.
But here are three temperaments in MIL:

https://drive.google.com/open?id=0B13tOzJL2zRpQXdJWlR0c0ZLWkk https://drive.google.com/open?id=0B13tOzJL2zRpQXdJWlR0c0ZLWkk

Regards, Gunnar

..is it a way to editing the messages....... :(:(

Dear Gunnar,

On Wed, Jan 13, 2016 at 9:47 PM, gunnnar.tungland@... [TUNING]
<TUNING@yahoogroups.com> wrote:

> But here are three temperaments in MIL:
>
> https://drive.google.com/open?id=0B13tOzJL2zRpQXdJWlR0c0ZLWkk

Thank you very much.

The meantone example, at a glance, shows one "wolf" fifth, four "wolf"
major thirds (with one duplicated) and three "wolf" minor thirds (with
one duplicated). Your diagrams not only convey that sort of
information quickly, but also show the pattern in it.

Regards,
--
Gavin R. Putland
http://www.grputland.com
https://t.co/MRg6aSp6lr .

Hello again, Gavin

Thanks for your positive feedback!
If you have any wish to get some tunings visualized, calculated in mil, then you can just tell me, I will do it. (within reasonable limits :)
Regards, Gunnar