Contents

- A Chain of Perfect 3rd Harmonics
- Acoustic Landmarks
- 12-equal, Meantone(s), and Irregular Temperaments
- Ideas for further work
- Source code to generate the diagrams and tables
- Home page and contact information

Temperament is the art of cleverly compromising between competing, and mutually exclusive, acoustical and musical goals. I have struggled for a long time to understand meantone temperaments, why exactly certain keys should be unplayable, and what does it mean when people say things like "¼-comma meantone is the same thing as 31-equal temperament", or if that is even true. The following represents my best attempt to make sense of it all, with what I believe are some novel diagrams and mathematical expressions [but if anything is original here, there is also a good chance that it is a mistake, so I would appreciate any corrections!].

I try to draw as many pictures and give as many tables as possible.
The focus is on physics, not number theory, so there is a lot of talk about harmonics
and iterative calculation by computer replaces a discussion of continued fractions.
By going a different route, you see different scenery!
You could generate the Pythagorean scale fractions from
(3/2)^{n} with octave equivalence rescaling by powers of 2, but I find it
more pedantically sound to work with 3^{n} with rescaling 2, since that
brings out sharply that it is the 3rd harmonic underneath it all.
I don't want to give a cookbook of the various values in cents of the notes
in a bunch of different tunings, which is very well done in many places, most notable
J.M. Barbour's *Tuning and Temperament*.
I want to understand the general principles behind them. But don't worry; there
is nothing here more advanced than high school algebra.

This discussion is based in a large part on
*Mathematics and Music* by Dave Benson
Specifically, portions of chapter 5 and 6 and Appendicies E and J.
Indeed, the presentation I give grew out of attempting to understand
Benson's material, and could be considered a reworking and extension of what he
has.

I would like to thank Manuel Op de Coul and Pierre Lewis for reading this and pointing out certain specific errors to me; any remaining ones I am responsible for.

Largely of historical interest today (although it is still played by
proponents of early music) the Pythagorean scale is, for us, a theoretical
springboard; the simplest case which illustrates all the others.
The following diagram is usually called the *Spiral of 5ths*. "5th" refers to
adjacent notes in the spiral being 5 notes in a diatonic scale apart: **g** is 5 scale
steps up from **c**, which is 5 steps up from **f** and so on. I think it is
much clearer to observe that what is really going on is that the pitch of **g**
is the pitch of the 3rd harmonic of **c**, rescaled by an apporpriate factor of 2
(in this case 1/2) so that all the notes lie in the same octave. This holds for all
the other notes as well. Below I will give explicit formulas for calculating these
values.

The *spiral* is to emphasize that notes that are equal in pitch (enharmonic)
in the standard 12-equal temperament are not equal here ( thus **g**^{##}
is not quite the same pitch as **a**, which in turn is not the same pitch as
**b**^{bb}).

The Pythagorean generating function is, for an integer n:

P_{y}(n) = 3^{n} 2^{κ(n; 3)}

where κ(n; 3) is the unique integer in the interval
* [ n log(3)/log(2) - 1, n log(3)/log(2) ] *

Thus 2^{κ(n; 3)} is the octave rescaling factor. You
can see it depends also on the parameter 3 (the 3rd harmonic);
this will be modified when we discuss meantone temperament below.

Here is a table I have calculated of the values of the Pythagorean function
*P _{y}(n)*, which I have found frequently useful for reference purposes.
You can see how the power of 2 does not vary in any clean, simple way with the power
of 3. (The source code to generate this table as large as you like is available at the
end of this document).

Index | Three | Two | Fractional Value | Note Name |
---|---|---|---|---|

20 | 3^{20} |
2^{-31} |
3486784401 / 2147483648 | F^{###} |

19 | 3^{19} |
2^{-30} |
1162261467 / 1073741824 | B^{##} |

18 | 3^{18} |
2^{-28} |
387420489 / 268435456 | E^{##} |

17 | 3^{17} |
2^{-26} |
129140163 / 67108864 | A^{##} |

16 | 3^{16} |
2^{-25} |
43046721 / 33554432 | D^{##} |

15 | 3^{15} |
2^{-23} |
14348907 / 8388608 | G^{##} |

14 | 3^{14} |
2^{-22} |
4782969 / 4194304 | C^{##} |

13 | 3^{13} |
2^{-20} |
1594323 / 1048576 | F^{##} |

12 | 3^{12} |
2^{-19} |
531441 / 524288 | B^{#} |

11 | 3^{11} |
2^{-17} |
177147 / 131072 | E^{#} |

10 | 3^{10} |
2^{-15} |
59049 / 32768 | A^{#} |

9 | 3^{9} |
2^{-14} |
19683 / 16384 | D^{#} |

8 | 3^{8} |
2^{-12} |
6561 / 4096 | G^{#} |

7 | 3^{7} |
2^{-11} |
2187 / 2048 | C^{#} |

6 | 3^{6} |
2^{-9} |
729 / 512 | F^{#} |

5 | 3^{5} |
2^{-7} |
243 / 128 | B^{} |

4 | 3^{4} |
2^{-6} |
81 / 64 | E^{} |

3 | 3^{3} |
2^{-4} |
27 / 16 | A^{} |

2 | 3^{2} |
2^{-3} |
9 / 8 | D^{} |

1 | 3^{1} |
2^{-1} |
3 / 2 | G^{} |

0 | 3^{0} |
2^{0} |
1 | C |

-1 | 3^{-1} |
2^{2} |
4 / 3 | F^{} |

-2 | 3^{-2} |
2^{4} |
16 / 9 | B^{b} |

-3 | 3^{-3} |
2^{5} |
32 / 27 | E^{b} |

-4 | 3^{-4} |
2^{7} |
128 / 81 | A^{b} |

-5 | 3^{-5} |
2^{8} |
256 / 243 | D^{b} |

-6 | 3^{-6} |
2^{10} |
1024 / 729 | G^{b} |

-7 | 3^{-7} |
2^{12} |
4096 / 2187 | C^{b} |

-8 | 3^{-8} |
2^{13} |
8192 / 6561 | F^{b} |

-9 | 3^{-9} |
2^{15} |
32768 / 19683 | B^{bb} |

-10 | 3^{-10} |
2^{16} |
65536 / 59049 | E^{bb} |

-11 | 3^{-11} |
2^{18} |
262144 / 177147 | A^{bb} |

-12 | 3^{-12} |
2^{20} |
1048576 / 531441 | D^{bb} |

-13 | 3^{-13} |
2^{21} |
2097152 / 1594323 | G^{bb} |

-14 | 3^{-14} |
2^{23} |
8388608 / 4782969 | C^{bb} |

-15 | 3^{-15} |
2^{24} |
16777216 / 14348907 | F^{bb} |

-16 | 3^{-16} |
2^{26} |
67108864 / 43046721 | B^{bbb} |

-17 | 3^{-17} |
2^{27} |
134217728 / 129140163 | E^{bbb} |

-18 | 3^{-18} |
2^{29} |
536870912 / 387420489 | A^{bbb} |

-19 | 3^{-19} |
2^{31} |
2147483648 / 1162261467 | D^{bbb} |

-20 | 3^{-20} |
2^{32} |
4294967296 / 3486784401 | G^{bbb} |

log( 3^{n} 2^{κ(n)} ) |
= | n log(3) + κ(n) log(2) |

= | Q(n) - R(n) |

where R(n) := -κ(n) log(2)

In **Figure 2**, this is illustrated for

Fig. 2

It isn't practical to have an infinite number of notes, so we have to cut
off the progression. We have to stop at some point on the Sharps branch and
at some point on the Flats branch, and we have to join the branches together.
Cutting off the progression of notes arbitrarily produces wolf intervals
(which we shall discuss in detail later) What are good closing points?
Ideally, two notes would be equal and we wouldn't have an infinite progression.
Let us show that this can never happen. Suppose, for n≠m, two notes would be equal:

3^{n} 2^{κ(n)} = 3^{m} 2^{κ(m) }

then

3^{(n - m)} = 2^{(κ(m)- κ(n) )}

But this is impossible, because the left hand side is prime factored, and it
contains only **3**s while the right is prime factored into only **2**s,
so they can never be equal! Thus the pure 3rd harmonic sequence never repeats itself,
it is infinite (although if you generate it far enough, the numbers get
arbitrarily close) Let us examine some good cutoff points:

Observe that 3 = 2^{α} for α = log(3)/log(2)

then 3^{m} = 2^{α m}. Let p be the nearest whole number
to αm. Then we need only use a calculator to get the values for
αm for each positive integer m; the ones that are close to integers
are good oportunities for the cycle of 5ths to close:

m (power of 3) | value α m | p (power of 2) |
---|---|---|

12 | 19.0195 | 19 |

41 | 64.98 | 65 |

53 | 84.003 | 84 |

306 | 484.9985 | 485 |

Now let's compare this very simple method with a brute force calculation using a Perl script genPythag.pl

perl -w genPythag.pl 30 CLOSING > table.htmland look at the output file

Count of notes generated |
m = notes in scale |
New note name |
New note value |
Best match value |
Best match name |
Distance in cents |
---|---|---|---|---|---|---|

6 | 5 | A^{} |
3^{3} 2^{-4} |
3^{-2} 2^{4} |
B^{b} |
90.2249956730629 |

13 | 12 | G^{b} |
3^{-6} 2^{10} |
3^{6} 2^{-9} |
F^{#} |
23.4600103846492 |

42 | 41 | C^{###} |
3^{21} 2^{-33} |
3^{-20} 2^{32} |
G^{bbb} |
19.844964519116 |

54 | 53 | F^{4#} |
3^{27} 2^{-42} |
3^{-26} 2^{42} |
D^{4b} |
3.61504586553314 |

307 | 306 | G^{22b} |
3^{-153} 2^{243} |
3^{153} 2^{-242} |
F^{22#} |
1.76973519144986 |

So what do we see comparing Table 3 with Table 2?

Observe that if we generate 13 notes in the symmetric manner described above,
then **g**^{b} almost coincides with **f**^{#}. This is
the *that exact distance* from its enharmonic equivalent until we get to
the count of 42; (that is **g** with 22 flats ) comes extremely close
to the +153rd one ( **f** with 22 sharps ). And you can keep on going, if
you are so inclined...

Count | New note name | new note value | best match value | best match Name | Distance in cents |
---|---|---|---|---|---|

4 | D^{} |
3^{2} 2^{-3} |
3^{0} 2^{0} |
C | 203.910001730775 |

5 | B^{b} |
3^{-2} 2^{4} |
3^{0} 2^{0} |
C | 203.910001730775 |

6 | A^{} |
3^{3} 2^{-4} |
3^{-2} 2^{4} |
B^{b} |
90.2249956730629 |

7 | E^{b} |
3^{-3} 2^{5} |
3^{2} 2^{-3} |
D^{} |
90.2249956730627 |

8 | E^{} |
3^{4} 2^{-6} |
3^{-1} 2^{2} |
F^{} |
90.2249956730629 |

9 | A^{b} |
3^{-4} 2^{7} |
3^{1} 2^{-1} |
G^{} |
90.2249956730627 |

10 | B^{} |
3^{5} 2^{-7} |
3^{0} 2^{0} |
C | 90.2249956730629 |

11 | D^{b} |
3^{-5} 2^{8} |
3^{0} 2^{0} |
C | 90.2249956730631 |

12 | F^{#} |
3^{6} 2^{-9} |
3^{1} 2^{-1} |
G^{} |
90.2249956730629 |

13 | G^{b} |
3^{-6} 2^{10} |
3^{6} 2^{-9} |
F^{#} |
23.4600103846492 |

14 | C^{#} |
3^{7} 2^{-11} |
3^{-5} 2^{8} |
D^{b} |
23.460010384649 |

15 | C^{b} |
3^{-7} 2^{12} |
3^{5} 2^{-7} |
B^{} |
23.4600103846492 |

16 | G^{#} |
3^{8} 2^{-12} |
3^{-4} 2^{7} |
A^{b} |
23.460010384649 |

17 | F^{b} |
3^{-8} 2^{13} |
3^{4} 2^{-6} |
E^{} |
23.460010384649 |

18 | D^{#} |
3^{9} 2^{-14} |
3^{-3} 2^{5} |
E^{b} |
23.460010384649 |

19 | B^{bb} |
3^{-9} 2^{15} |
3^{3} 2^{-4} |
A^{} |
23.4600103846492 |

20 | A^{#} |
3^{10} 2^{-15} |
3^{-2} 2^{4} |
B^{b} |
23.460010384649 |

21 | E^{bb} |
3^{-10} 2^{16} |
3^{2} 2^{-3} |
D^{} |
23.4600103846492 |

22 | E^{#} |
3^{11} 2^{-17} |
3^{-1} 2^{2} |
F^{} |
23.460010384649 |

23 | A^{bb} |
3^{-11} 2^{18} |
3^{1} 2^{-1} |
G^{} |
23.460010384649 |

24 | B^{#} |
3^{12} 2^{-19} |
3^{0} 2^{0} |
C | 23.460010384649 |

25 | D^{bb} |
3^{-12} 2^{20} |
3^{0} 2^{0} |
C | 23.460010384649 |

26 | F^{##} |
3^{13} 2^{-20} |
3^{1} 2^{-1} |
G^{} |
23.460010384649 |

27 | G^{bb} |
3^{-13} 2^{21} |
3^{-1} 2^{2} |
F^{} |
23.460010384649 |

28 | C^{##} |
3^{14} 2^{-22} |
3^{2} 2^{-3} |
D^{} |
23.460010384649 |

29 | C^{bb} |
3^{-14} 2^{23} |
3^{-2} 2^{4} |
B^{b} |
23.460010384649 |

30 | G^{##} |
3^{15} 2^{-23} |
3^{3} 2^{-4} |
A^{} |
23.460010384649 |

31 | F^{bb} |
3^{-15} 2^{24} |
3^{-3} 2^{5} |
E^{b} |
23.460010384649 |

32 | D^{##} |
3^{16} 2^{-25} |
3^{4} 2^{-6} |
E^{} |
23.460010384649 |

33 | B^{bbb} |
3^{-16} 2^{26} |
3^{-4} 2^{7} |
A^{b} |
23.460010384649 |

34 | A^{##} |
3^{17} 2^{-26} |
3^{5} 2^{-7} |
B^{} |
23.460010384649 |

35 | E^{bbb} |
3^{-17} 2^{27} |
3^{-5} 2^{8} |
D^{b} |
23.4600103846492 |

36 | E^{##} |
3^{18} 2^{-28} |
3^{6} 2^{-9} |
F^{#} |
23.460010384649 |

37 | A^{bbb} |
3^{-18} 2^{29} |
3^{-6} 2^{10} |
G^{b} |
23.460010384649 |

38 | B^{##} |
3^{19} 2^{-30} |
3^{7} 2^{-11} |
C^{#} |
23.460010384649 |

39 | D^{bbb} |
3^{-19} 2^{31} |
3^{-7} 2^{12} |
C^{b} |
23.460010384649 |

40 | F^{###} |
3^{20} 2^{-31} |
3^{8} 2^{-12} |
G^{#} |
23.460010384649 |

41 | G^{bbb} |
3^{-20} 2^{32} |
3^{-8} 2^{13} |
F^{b} |
23.460010384649 |

42 | C^{###} |
3^{21} 2^{-33} |
3^{-20} 2^{32} |
G^{bbb} |
19.844964519116 |

43 | C^{bbb} |
3^{-21} 2^{34} |
3^{20} 2^{-31} |
F^{###} |
19.8449645191159 |

44 | G^{###} |
3^{22} 2^{-34} |
3^{-19} 2^{31} |
D^{bbb} |
19.8449645191158 |

45 | F^{bbb} |
3^{-22} 2^{35} |
3^{19} 2^{-30} |
B^{##} |
19.8449645191159 |

46 | D^{###} |
3^{23} 2^{-36} |
3^{-18} 2^{29} |
A^{bbb} |
19.8449645191158 |

47 | B^{4b} |
3^{-23} 2^{37} |
3^{18} 2^{-28} |
E^{##} |
19.8449645191159 |

48 | A^{###} |
3^{24} 2^{-38} |
3^{-17} 2^{27} |
E^{bbb} |
19.844964519116 |

49 | E^{4b} |
3^{-24} 2^{39} |
3^{17} 2^{-26} |
A^{##} |
19.8449645191159 |

50 | E^{###} |
3^{25} 2^{-39} |
3^{-16} 2^{26} |
B^{bbb} |
19.8449645191158 |

51 | A^{4b} |
3^{-25} 2^{40} |
3^{16} 2^{-25} |
D^{##} |
19.8449645191159 |

52 | B^{###} |
3^{26} 2^{-41} |
3^{-15} 2^{24} |
F^{bb} |
19.8449645191158 |

53 | D^{4b} |
3^{-26} 2^{42} |
3^{15} 2^{-23} |
G^{##} |
19.8449645191159 |

54 | F^{4#} |
3^{27} 2^{-42} |
3^{-26} 2^{42} |
D^{4b} |
3.61504586553314 |

55 | G^{4b} |
3^{-27} 2^{43} |
3^{26} 2^{-41} |
B^{###} |
3.61504586553324 |

56 | C^{4#} |
3^{28} 2^{-44} |
3^{-25} 2^{40} |
A^{4b} |
3.61504586553314 |

57 | C^{4b} |
3^{-28} 2^{45} |
3^{25} 2^{-39} |
E^{###} |
3.61504586553324 |

If we have a straight diagonal line like this,
then obviously we can slide the right angle "elbow" piece up and down the line and the
relationship between *m*, Δ_{p} and δ still holds. Thus:

δ = *distance( f^{#}, g^{b} )* =

The determining factor is the range of allowable values for the length *m*;
for *m* ≤ 40*m = 12**p = 19**Pythagorean comma* ρ) but allow *m*
to reach as high as 53 and then *p = 485*

Perl source code genPythag.pl to generate these tables.

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