| Welcome to | |
| "The Graphical Guide to Tunings" | For those who are familiar with commas ! |
| (There are links everywhere on this blog , in a kind of folder system. ) | |
| (Video w/audio: Quick start ) | |
| Click here for QUICK START |
As an introduction please listen to my little composition that is made to get pure intervals and no comma problems through the whole piece, except in the end where you will hear the syntonic comma really sound. And that comma is one of two important commas that will be shown clearly in all of these diagrams when you have been familiar with them. Innumerable attempts has been done to "solve" the comma problem, but then you have to deal with more than 12 tones pr. octave. I got my idea and that resulted in :
IN TUNINGLAND (Click)
Have you ever longed to play with exclusively pure chords on your harpsichord using two manuals, if nothing else just to hear how the harpsichord sounds with purity in focus, or just meditate a little and get some peace of mind? Pure chords from Ab to E (from the view of circle of fifths)
Download this SETUP (click), and open it in Excel.
Let C7=0 for totally pure chords, but with a more audibly comma shift. C7= -2 for a more smooth version, but fifths like in 12ET. Or C7= -1 for a tuning in the middle of these.
The video below with totally pure intervals is from my second blog which is all about the syntonic comma. See and and hear this video now and welcome back to this website too.
The objective of these graphs is to give some insight and a visual presentation instead of the jungle of numbers which so often are characterizing historical presentations of tunings. This includes 10 diagrams for each tuning, with identical concepts but visualizing this from various angles and various types/amounts of information.
a) I hope this introduction can assist you to understand the three most important commas; Pythagorean, syntonic and diesis. The videos with graphs and audio will give an insight as to what both the cembalist and the mathematician can relate to.
b) Insight: When you are conversant with the commas and how they are visualized in the graphs, it is my intention that you will understand the tunings you chose to venture into. A good idea is to print (or split the screen) two or more tunings to make comparisions easier .
c) Overview : I have attempted to systemize the temperaments through the assistance of the standard deviation tool used in statistics. The selected tuning category and the standard deviation values are included in all the graphs and using these numbers is a good way to sort the tunings.
THE PYTHAGOREAN COMMA
https://en.wikipedia.org/wiki/Pythagorean_comma
https://en.wikipedia.org/wiki/Pythagorean_comma
The idea of this model is based on the circle of fifths. This facilitates that those chords with the greatest affinity are close together – a logical choice.
The other element here is also logical, however I have not seen this done previously: I let a fifth sequence become a straight horizontal line.
Due to the Pythagorean comma this has a special consequence for the diagrams.
12 pure fifths following each other will look and listen like this:
Video : 12 pure fifths (only the first 50 seconds)
Notice that F# ends up over the initial G♭. This difference (approximately ¼ semitone) is what we call the Pythagorean comma.
The Pythagorean comma is due to our choice of 12 tones in the octave, and not due to any mathematical affinity which is the case with the syntonic comma (next chapter).
If we had chosen to have Gb and F # at the same height since they were the same tone,
the graph would look like this: The pythagorean comma displayed at the cost of the fifths
Therefore since G♭ and F# are always equal in pitch, and we in our diagrams prioritize the fifths that vary, we need to lower F# with 23.46 cents in the diagram which is the Pythagorean comma.
the graph would look like this: The pythagorean comma displayed at the cost of the fifths
Therefore since G♭ and F# are always equal in pitch, and we in our diagrams prioritize the fifths that vary, we need to lower F# with 23.46 cents in the diagram which is the Pythagorean comma.
In this example there are 11 pure fifths (horizontal line), but with a very flat B-F# fifth.
So we accord priority to the fifth – the pure ones becomes horizontal – and 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.
TEMPERAMENT UNITS
The most frequently used measurement for music intervals is cent. 1 octave with 12 semitones = 1200 cent. In an equal temperament tuning (the way the piano is tuned) there will be 100 cent for each semitone.
In the video you just saw I introduced the measurement unit TU.
In the video you just saw I introduced the measurement unit TU.
This ingenious idea, Temperament Units (TU), stems from John Brombaugh, an organ builder. I first read about this unit on Bradley Lehmans website.
Instead of basing the calculations on the prime/octave which in theory always is perfect, we let it become 36828,628199 TU!
Why??
Because then the Pythagorean comma acquires a value of 720 which has the same accessible traits as 360 degrees in a circle. The number can be factorized in many ways, which we will soon see.
The ratio cent/TU is nearly 1/30, or more precisely 1/ 30.6905235 . Those who are more comfortable with cent numbers should simply divide the TU value by 30, more exactly use a calculator (/30.6905) or use the cent diagram (Main cent) related to each tuning.
The ratio cent/TU is nearly 1/30, or more precisely 1/ 30.6905235 . Those who are more comfortable with cent numbers should simply divide the TU value by 30, more exactly use a calculator (/30.6905) or use the cent diagram (Main cent) related to each tuning.
In order to avoid using the imperfect B-F# fifth (previous video) we can subdivide 720 TU on several fifths.
Here are some examples of how the comma can be subdivided:
They of course do not need to follow each other sequentially. Kellner includes a perfect fifth in between (E-B), but is still a 1/5 Pythagorean comma.
If we divide the comma in 6, we can for instance get the Vallotti tuning which I will use as a point of departure. There is actually an Anti-Vallotti group on Facebook :). But I will use Vallotti for purely pedagogical purposes.
Should we subdivide the Pythagorean comma equally on all 12 fifths we will get the equal temperament tuning, in which most pianos are tuned.
----------------------------------------------------------------------------------------------
THE SYNTONIC COMMA
The previous number was 12 due to the number of keys on the piano.
(Ratio F#(12 pure fifths) / G♭ = 531 441 / 524 288 )
Simplified said the next number is 4 (Diesis is 3 explained in Introduction part II)
The syntonic comma is a comma which also others than keyboard players need to relate to (in practical terms, not mathematical terms), and actually more then them because they are forced to intonate.
Choirs can fall in pitch when they think they are singing pure. From my blog IN TUNINGLAND (each chord is totally in tune):
Here are the two mathematical frequencies which separately are very close but which cannot be unified.
Choirs can fall in pitch when they think they are singing pure. From my blog IN TUNINGLAND (each chord is totally in tune):
And there is also a danger in rising when singing "pure":
A pure major third E/C = 5 / 4 ( The 5. harmonic/4.harmonic )
If we take four pure fifths successively, C-G-D-A-E the calculation will be
3/2 x 3/2 x 3/2 x 3/2 = 81/16, transposed down 2 octaves = 81/64
This gives a margin of error of (81/64) / (5/4) = 81/80, which is the syntonic comma.
We therefore use a small segment of the previous graph, only 4 fifths, C-G-D-A-E.
The third C-E becomes disturbingly imperfect – too high – shown on the red vector at 660 TU.
To avoid confusion with the previous section I would have preferred it to be called the syntonic third, but that is not the name - it is called the Pythagorean major third.
In Temperament Units it equals 660,03925726. With a clear conscience we define it to be 660 with an error margin of only 1/1000 cents. (Syntonic comma in cents is 21.51).
The number 660 can be factorized into fractions in several ways, and I will return to this. It is also really fortunate that the syntonic comma acquires this value in combination with the Pythagorean 720. Both are divisible by 2,3,4,5,6 etc… (for instance 720(PC)/5 = 144 , 660(SC)/5= 132).
This makes TU especially convenient for simple calculations. And it often results in considerably easier numbers to work with than cents.
Not only is it simple in mathematical terms, but it also speaks the same language that cembalists have used for several hundred years; they have always referred to fractions of comma, the first measure unit for deviation from a pure fifth.
One should be aware that the size of the pythagorean and syntonic commas are relatively equal. Therefore they previously often used to be confused.
..............................
The green E and the red E will always have the same interval on the y axis. If we want a perfect C-E we need to lower both the green and red E with 660 TU.
The video shows an audio example of what transpires when we lower the E.
The video starts with 4 perfect fifths and eventually shows how the comma can be subdivided between the 4 fifths while the third is completely perfect. The end result is an equal distribution – 165x4 and a final mix of 1/3 SC (220) and 1/6 SC (110). (The comma does not have to be equally distributed).
.................................
We often see referrals to 1/5 or 1/6 of syntonic comma. If we compare with how we subdivided the various fractions of the Pythagorean comma, this is not as logically evident when it comes to the syntonic comma. The basic question is how do we subdivide on 5 fifths when the definition of this comma describes 4 fifths.....
Here we see that when the accumulated sum of imperfections on 4 fifths becomes different from (-) 660, the difference is added to the major third: 132 TU imperfect. This is how I choose to visualize this to highlight the 5 fifths.
Here are two thirds with calculations (labeled with red and green):
It may be useful to memorize these fractions of the commas, and it may also be useful to memorize how the various fifth values impact the major third. In the next video the green fractions are related to the Pythagorean comma (-720 TU) and the red fractions are related to the syntonic comma (660 TU) (without audio).
When the same values are used on 11 fifths with nearly perfect thirds this is called meantone. The point of departure (and most well-known) is ¼ SC= -165 TU. The line will look like this. It will fall 720 TU from the start to the end of the graph, but will take a detour through a peak.
Question☺
Based on what we have discussed, how does this impact the thirds?
When you have read Introduction I, you will understand the answer looking at this diagram
........................................................................
When you have read Introduction I, you will understand the answer looking at this diagram
........................................................................
THE MAJOR THIRDS
So far I have only focussed one third.
So far I have only focussed one third.
For simplicity’s sake we now choose the Vallotti tuning. This tuning is based on 1/6 PC, -720 TU subdivided on 6 fifths. And then we now include all the vectors.
And here’s the whole diagram:
We have seen that the vertical interval between red and green notes (eg red E and green E) are constant (660 TU) . We have entered a red graph along the vector points located 4 fifths after the green one and 660 TU higher – but otherwise identical.
"Vallotti" lets the fifths fall from green F , and the red graph do have a similar fall from red F and cause the C-E third to become quite good , 180 TU deviation from pure.
I have tried to maintain a minimal level of information.
Many temperaments are very simple. However, they can be very confusing if one only views the cent values.
Regardless, I still choose to include one superfluous element which has entered into some of the graphs, namely green vectors which indicate the deviations from the pure fifth.
I have also considered including the note labels on the green vector points:
However, this becomes too cluttered and unnecessary for most. (But this way it becomes easier to see how we can read major chords vertically).
MINOR THIRDS
This introduces us to the next issue. This graph can be used mathematically correct for more intervals. The most important one is the minor third.
The sum of the vertical length of the red and green vector equals the deviation from a pure minor third.
Notice that the blue vector is pointing downwards with the third on the vector point. This is due to the fact that the minor third is here smaller than a perfect minor third.
And this is how we can read 2 minor triads from the main graph:
For those who wish to study the minor chords more directly I have included another diagram (Minor TU , Minor cent). In this graph I have adjusted the upper fifth graph one notch to the right. We then find the minor chords to be only vertical. The colour on this graph is changed to blue, which I will use as the colour for minor thirds.
The minor chords are then evident in this way:
Again I leave out some information in order to have a clearer presentation. This time the fifth in the minor chord is left out.
In Vallotti it is easy to find the values as all imperfect fifths are -120 TU, the rest are 0 TU.
In a more obscure tuning we can find the fifth in two ways. The first is to summate the vertical values (be aware of the minus/plus values). The second is to find the fifth on the lower graph.
...........................................
We then move back to the major thirds and find this:
Vallotti contains 3 Pythagorean thirds (660 TU)
Gb-Bb
Db-F
B-D#
Vallotti contains 3 good thirds,
F-A ,
C-E, and
G-B at 180 TU
And then it contains 2 thirds that are completely identical to the equal temperament, 420 TU (notice this value)
Eb-G
A-C#
Let us listen:
I have also a Youtube channel with a little music wandering through the circle of with while you are looking at the diagram.
VALLOTTI, Youtube
The rest of the 20 selected temperaments you can chose on the right side of the video
...................................
Let us include a slightly more advanced example.
VALLOTTI, Youtube
The rest of the 20 selected temperaments you can chose on the right side of the video
...................................
Let us include a slightly more advanced example.
First the circle of fifths in Padua.
Here we see that 2 fifths have a width of 135 TU. This impacts Gb – Bb, it becomes 930 TU.
A characteristic of Padua is that it contains more than 4 fifths with ¼ SC (-165 TU) due to the 2 wide fifths.
As we have seen several times the consequence of 4 such fifths becomes a pure third. With 6 such fifths we get 3 perfect thirds.
What we also should notice is the considerable agravation of the thirds from Bb-D, Eb-G up to Ab-C . The red and black graphs go their separate ways.
And then we again find a couple of minor thirds in the same tuning, G-Bb /C-Eb.
Notice that + and – are done this way as the minor third is negative (the third is underneath). You can simply measure the interval (just remember the opposite plus/minus signs) and then put the minus sign in front of the final result. If the minor third is underneath(almost always), the value is negative.
And then I would recommend that you take a close look at these 6 minor chords, and finally how you can find these values in the "Minor TU (Picture) ".
...................................
THE DIATONIC SCALE
THE DIATONIC SCALE
Finally we will see that it is not possible to do a perfect tuning of the 7 tones in the scale (C major in the example).
We have to choose either to have a perfect G-D or D-A fifth. The difference is the well-known value 660 .
I have as mentioned included as little information as possible in order to have a clear and understandable diagram.
I have used some examples of the minor third which are not immediately evident in the main diagram.
This is especially useful when we shall eventually take a look at another expanded graph (Diesis) in which the green vectors are removed.
For those who need assistance with regards to the calculations for the minor third I have another diagram ("Vs equal , minor values") in which the value for the minor third in the major chord (E-G in C-major) is indicated below in blue.
In addition I here have a dotted line which indicates where 420 TU is on the column, which is the equal major third. This makes it easier to compare the thirds as the whole graph falls and to see the deviation from our well known impure major third in Equal temperament.
I have also included a dotted line for the equal minor third. This one is to be compared with the green vector point. If the dotted line goes beneath the green vector point the minor third is more perfect than the equal third.
In order to understand the graphs it is always smart to take a look at "equal temperament".
Notice that the lower dotted line in this example intersects with the green points the whole way.
I have a separate diagram for those who prefer cents which is similar to the previous one.
As a conclusion to the introduction to the first two commas I present another graph which covers 2 octaves, which is one of 10 graphs for each tuning.
Here we see that the Pythagorean comma is not just between Gb and F# .
I here indicate -720 (PC) and 660 (SC) at several points.
________________________________________________________________________________
If you have followed these calculations it is my hope that the visualizations themselves yield an overview of a temperament, as you now understand the basis for the graph development.
