On this page, I define equal temperaments using my matrix formalism. If you don't understand matrices, you might still get something out of this page. If you don't know about the overtone series, this is unlikely to be meaningful to you.
This page generally follows from the one on linear temperaments because I wrote that first. As equal temperaments are that bit simpler, you may want to read this page first. Hopefully, everything here will make sense without it.
Equal temperaments use a matrix H' to approximate the harmonic coordinates H. As an example, here is the matrix for 31-equal in the 11-limit:
( 31)
( 49)oct
H' = ( 72)---
( 87) 31
(107)
This tells you that a perfect octave is 31 steps, 3/1 is approximated by 49 steps, 5/1 by 72 steps, 7/1 by 87 steps and 11/1 by 107 steps. All other intervals can be calculated from this, so 11/8 is 107-3*31 =14 steps, 7/5 is 87-72 = 15 steps and so on.
If you want to find out how many steps there should be to approximate each prime number ratio, and you know how many notes you want to an octave, multiply H by this and round off to whole numbers:
(log(2) ) ( 31.000) ( 31)
31 H 31 (log(3) ) ( 49.134) ( 49)
--- = ------(log(5) ) = ( 71.980) ~ ( 72)
oct log(2)(log(7) ) ( 87.028) = ( 87)
(log(11)) (107.242) (107)
This usually works, but not always, as explained below. To see how good an approximation this is, take an error matrix dH=H'-H. As we have perfect octaves, it's simpler to take the octave equivalent dh:
[ ( 49) ( 49.134) ] oct (-0.134)oct
dh = [ ( 72) - ( 71.980) ] --- = ( 0.020)---
[ ( 87) ( 87.028) ] 31 (-0.028) 31
[ (107) (107.242) ] (-0.242)
For comparison with other temperaments, you can divide by through by 31. However, to see how good a temperament 31-equal is, this matrix is better. A large number of steps will always give good approximations. The best temperaments are those that have a good approximation with a small number of steps. The relevant quantity here is the error as a proportion of step size.
Taking the 11-odd-limit, the worst tuned interval is 9/5, which is 0.020 + 2*0.134 = 0.288 steps flat. You know it's flat because the signed error (2*-0.134)-0.02 = -0.288 is negative.
As another example, try 24-equal:
(log(2) ) (24.000) (24)
24 H 24 (log(3) ) (38.039) (38)
--- = ------(log(5) ) = (55.726) ~ (56)
oct log(2)(log(7) ) (67.377) = (67)
(log(11)) (83.026) (83)
[ (38) (38.039) ] oct (-0.039)oct
dh = [ (56) - (55.726) ] --- = ( 0.214)---
[ (67) (67.377) ] 24 (-0.377) 31
[ (83) (83.026) ] (-0.026)
Some intervals here are extremely good. Take that 11/6: only 0.039-0.026 = 0.013 steps sharp. However, some are very bad. 7/5 is 0.214+.377 = 0.591 steps flat. That it is more than half a step out is extremely important. It means the interval could be better approximated in a different way in the same temperament.
Now, I need to explain some terminology. I would say that the interval 7/5 is inconsistent in this temperament. Or, more generally, that 7/1, 5/1 and 7/5 together are inconsistent in any mapping of 24-equal. What most people say (after Paul Erlich) is "24-equal is inconsistent in the 7-limit." Paul Hahn had comprehensive lists of consistency levels.
Inconsistency is always a problem. It means there is no way of playing some chords so that all their intervals are as well tuned as they should be. The obvious solution is to avoid inconsistent intervals. Alternatively, you could choose your H' and stick to it. I would describe this as a consistent mapping. So, all intervals are consistent with each other, but some are very poor. This would run into confusion, though, calling an inconsistent temperament consistent.
When you have a lot of steps to the octave, consistency is less important. For example, I quite often use this:
( 384)
( 607)oct
H' = ( 892)---
(1078)384
Why? Because it's a good 7-limit meantone and it can be tuned exactly with my TX81Z's 768 step tuning tables. The fact that 3/1 would be better represented by 609 steps is not important.
An interesting scale to define with matrices is 88CET. This is where each step is 88 cents. In this tuning, an octave is poorly represented, so defining it by an integer H' will not do. It's discoverer, Gary Morrison, gives the intervals 9/7, 11/9, 7/6, 7/4 and 10/9 as the primary consonances. So, we can say
( 1 -2 1 0 0) ( 2) (-1 -1 0 1 0) ( 3)88oct ( 0 -2 0 0 1)H' = ( 4)----- ( 0 2 0 -1 0) ( 5)1200 (-2 0 0 1 0) (11)
Unfortunately, this can't be solved for H'. It is not a basis. (0 2 0 -1)H = (-2 0 0 1)H - 2*(-1 -1 0 1)H. So, let's substitute 9/7 for the less melodically useful 3/1. Then,
( 1 -2 1 0 0) ( 2)
(-1 -1 0 1 0) ( 3)88oct
( 0 -2 0 0 1)H' = ( 4)-----
(-2 0 0 1 0) (11)1200
( 3 0 0 0 0) (41)
( 1 -2 1 0 0)-1( 2) ( 41)
(-1 -1 0 1 0) ( 3)88oct 1( 65)88oct
H' = ( 0 -2 0 0 1) ( 4)----- = -( 95)-----
(-2 0 0 1 0) (11)1200 3(115)1200
( 3 0 0 0 0) (41) (142)
This is equivalent to every third step from 41-equal. For 88CET, H' is not a basis as not all intervals derived from it are in the scale. However, you can use the 5-dimensional basis above. This is something that wouldn't work with octave equivalent matrices, of course.
So, what if you don't know what step size you want, but have a set of commas that you want to vanish? The procedure is much the same as for linear temperaments, but you swap one of the perfect intervals for another comma. For example, in 19-equal the small diesis (-10 -1 5)H or (-1 5)h vanishes, along with the syntonic comma (-4 4 -1)H. 19-equal can then be defined:
( 1 0 0) (1 0 0) ( -4 4 -1)H' = (0 0 0)H (-10 -1 5) (0 0 0)
(1 0 0) (19)oct
H' = (36/19 0 0)H = (36)---
(49/19 0 0) (49) 19
Which completely describes the temperament in the extended 5-limit. You can add more commas for higher limits; the same procedure applies. For octave based equal temperaments, the number of steps to an octave is the determinant of the octave equivalent defining matrix:
|4 -1| = (4*5) - (-1*-1) = 20-1 = 19 |-1 5|
A fact first documented by A. D. Fokker. Using the octave specific definition matrix and inverting it is still better, as it tells you the step sizes for H', and so all intervals defined on it. This is especially useful for choosing between different mappings of inconsistent temperaments.
The 88CET mapping above can be derived from the following commas:
( 4 -12 2 0 3) ( -3 2 -1 2 -1)H (-10 1 0 3 0) ( -1 5 0 0 -2)
We also need to define the size of an interval to get a matrix that will invert. I'll choose the triple octave:
( 3 0 0 0 0) (3) ( 4 -12 2 0 3) (0) ( -3 2 -1 2 -1)H' = (0)oct' (-10 1 0 3 0) (0) ( -1 5 0 0 -2) (0)
Here, oct' is the size of a stretched octave, which can be chosen as whatever you like. The solution is:
( 3 0 0 0 0)-1(3) ( 41)
( 4 -12 2 0 3) (0) ( 65)oct'
H' = ( -3 2 -1 2 -1) (0)oct' = ( 95)---
(-10 1 0 3 0) (0) (115) 41
( -1 5 0 0 -2) (0) (142)
This is a general mapping for a scale that works the same as 88CET. The determinant of the defining matrix is 123. So, we only use every 123/41=3 steps. The triple octave is 41 steps. In 88CET, that is 88*41=3608 cents. That gives an oct' of 1202.67 cents or oct' = 1.0022 oct.