"diluted" temperaments (long)

[Ron Koval]

Hi everyone

With all the recent talk about temperaments, I thought I'd chime in and 
include some graphs to make understanding the temperaments a little easier.  
I'll do the best I can with the ascii (text) format.

Many thanks to both Ed Foote and Bill Bremmer for coming to Chicago and 
sharing what they do with our chapter.  Without the PTG, I would not have 
the information needed to sort through all this stuff!

So, on to some temperaments!

First, let's look at ET, the quest for most techs.  All major thirds are 
tuned wide of pure, by 13.7 cents.


major 3rds built on bottom note, scale of graph runs from zero on left 
(pure), to a bit over 22 cents wide(about the widest third still usable)
  0.........................ET...........20 cents wide
C :.........................*
C#:.........................*
D :.........................*
D#:.........................*
E :.........................*
F :.........................*
F#:.........................*
G :.........................*
G#:.........................*
A :.........................*
A#:.........................*
B :.........................*

Notice how 13.7 is closer to unusable than pure, yet this is the sound we 
are used to hearing.   In the other temperaments, some thirds are closer to 
pure, which forces other thirds to be farther from pure.  Traditionally, the 
temperaments ran the 'edgy' keys all the way up to over 20 cents from pure 
to get the maximum payoff in the 'smoother' keys.

Take a look at Young's temperament from 1799:

C :..........*
C#:.....................................*
D :.................*
D#:.........................*
E :.................................*
F :..............*
F#:.........................................*
G :..............*
G#:.................................*
A :.........................*
A#:..................*
B :.....................................*

Woah, it's one train wreck after another, if we play thirds by half steps!  
This is how we test for ET, why is it such a mess in THIS temperament, we 
can't make music with this!

Well, it just depends on perspective.  Use the circle of fifths analyze this 
temperament and it looks like this:

still major thirds with the lower note listed
pure........................ET...............

C :..........*
G :..............*
D :.................*
A :.........................*
E :.................................*
B :.....................................*
F#:.........................................*
Db:.....................................*
Ab:.................................*
Eb:.........................*
Bb:..................*
F :..............*
C :..........*

All right, now it begins to make a little more sense, it's easy to see the 
symmetry.

But what if that F# is just too much for my taste?  Ah, the beauty of 
computers and tuning with offsets.

Youngs temperament offset from ET:
C 5.9
C# 0
D  2
D# 3.9
E -2
F  5.9
F# -2
G 3.9
G# 2
A  0
A# 5.9
B -2

You want something with color, like the symmetry of Youngs, but a little 
weaker?  Simple, just pick a number to multiply by....

How about 50%?  Multiply each offset by .5 And it looks like:
pure........................ET....................

C :.................*
G :...................*
D :.....................*
A :.........................*
E :.............................*
B :...............................*
F#:.................................*
Db:...............................*
Ab:.............................*
Eb:.........................*
Bb:.....................*
F :...................*
C :.................*

"Best" third is 9.7 cents wide, "worst" is now 17.7 cents wide.

Or come up with one of your own that has the characteristics you like!
Try this for a mild one
C 1.8
C# 0
D 1.1
D# 1.5
E  -.7
F  2.1
F#  -.3
G  1.7
G# .8
A  0
A# 2
B  -.6

C :....................*
G :.....................*
D :.......................*
A :.........................*
E :............................*
B :..............................*
F#:..............................*
Db:..............................*
Ab:...........................*
Eb:..........................*
Bb:........................*
F :.....................*
C :....................*

thirds from 11.2 - 16 cents wide.

Now come on, do you really expect to tune to the nearest tenth cent?

Well, you have to if you are trying for ET, otherwise the width of your 
thirds will vary.  This is my "Penny" temperament (maximum offset one cent), 
the closest to ET that I try to tune. This way my target is lined up with 
the circle of fifths! (after all, that's the structure of music, and were 
working for the music, right?)

Koval "Penny" temperament

C 1
C# 0
D  0
D# 0
E -1
F  1
F# -.7
G  .5
G# 0
A  0
A# .7
B -1

looks like:

C :.....................*
G :......................*
D :........................*
A :.........................*
E :...........................*
B :...........................*
F#:............................*
Db:...........................*
Ab:...........................*
Eb:..........................*
Bb:........................*
F :.......................*
C :.....................*

thirds from 11.7 - 15.1 cents wide.

Even one cent can make a difference, if placed in the right location.

Use a little math, graph out the temperaments and see what they're designed 
to do.  Young's is a well temperament, which is different from meantone, or 
pythagorean, they all have a different structure to them.

That's all for now!  Back to work (lots of churches this week!)

Ron Koval

Chicagoland
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