User:CritDeathX/Sam's 19-note Well Temperament

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Explaining Why I Wanna Temper 19EDO To The Feds

"Sam, 19EDO is already close enough to JI! Why would you need to make a well temperament for it?!?!?!?!!!" Good question, I'll answer that.

First off, I don't even know if this is an actual well temperament. It probably is, but the way I get to it certainly doesn't scream it to me. So if it isn't, then I'm sorry.

Secondly, the short answer is simple: I wanna make it more accurate. Yeah, 19EDO is close to 1/3-comma meantone, but the 5/4 is -7c. For a tuning that everyone says is so pure, I'd rather have a -3c 5/4 than a -7c 5/4. A couple of other JI intervals are also wonky enough to question whether they should be used to approximate the intervals listed, like 4\19 for 7/6 (-14c) and 15\19 for 7/4 (-21c). Like I said for my 17-note well temperament, I don't want to use JI nor use a large EDO. The answer to this dilemma is to make your own temperament.

We know what we need to do. So we choose what intervals we want and then we make our tuning, right? Well, I find that there's a bunch of temperaments that don't give me the intervals I want to use. A bit of the temperaments listed for 19EDO have something making it to where around 250c represents 7/6, 950c represents 7/4, etc etc. This doesn't feel right with me, which is kind of a shame. So what do we do if we can't use a straight generator?

Well, we use two.

The Math Behind It

Now, since we want to use two generators, we must need to figure out what generators to use by this point. The intervals I have in mind as generators are 9/7 and 5/4, just so that we can keep everything in check. Lets use 6\19 for 5/4 and 7\19 for 9/7, per tradition.

Starting with 5/4, we can choose the intervals we want and end up with this equation:

(1log2(5/4) + 2log2(14/9) + 4log2(12/5) + 5log2(3/1)) / (1^2 + 2^2 + 4^2 + 5^2)

Multiplying our answer by 1200, we get 380.18447121c, which admittedly is still a bit flat, but its better than what 19EDO gives.

For 9/7, because I'm lazy, lets assume we just want to approximate 9/8 & 7/4. We get this equation:

(1log2(9/7) + 5log2(32/9) + 13log2(28/1)) / (1^2 + 5^2 + 13^2)

Multiplying our answer by 1200 gives 443.129594209c. This is around the same as 19EDO.

The next question comes to what stacks do we use from each generator? Well, lets see what our generators give:

1/1 5/4 14/9 27/14 6/5 3/2 15/8 7/6 13/9 9/5 9/8
0.0c 380.18c 760.37c 1140.55c 320.74c 700.92c 1081.11c 261.29c 641.48c 1021.66c 201.85c
1/1 9/7 5/3 14/13 25/18 9/5 15/13 3/2 27/14 5/4 13/8
0.0c 443.13c 886.26c 129.39c 572.52c 1015.65c 258.78c 701.91c 1145.04c 388.17c 831.3c

Just by this simple interval chain alone, we can notice a couple things:

  1. Our 9/7 generator has a more accurate 5/4 than our 5/4 generator, strangely enough.
  2. We have not one, not two, but four different types of major 2nds; in our 9/7 chain, we can either go 5 generators down (184.35c) or go 14 generators up (203.82c). In our 5/4 chain, we can go 10 generators up (201.85c) or 9 generators down (178.34c)
  3. Some intervals here differ in very small amounts, with our P5s in both chains only differing by a cent.

By these observations, we realize that we've run into a problem: there's too many intervals. How do we decide what stacks go to which interval? I'll do the work for y'all.

Choosing Our Intervals

To demonstrate what intervals we're gonna use, I'll represent our 5/4 chain using A and our 9/7 chain using B. How ever many generators up and down will be represented by an integer beside it.

Degree Generator Stack
0
1 B11
2 B3
3 B14
4 A7
5 A4
6 B9
7 B1
8 B-7
9 B4
10 A8
11 B7
12 A2
13 B10
14 B2
15 A12
16 B5
17 A6
18 A3

And we have our wonderful, unequal scale; our 19-note well temperament tuning.

The Scale

Degree Cents Difference JI Approximation 19EDO Difference
0 0.0 +0 1/1 0.0 +0
1 74.43 +3.76 25/24 63.16 -7.48
2 129.39 +1.1 14/13 126.32 -1.97
3 203.81 -0.1 9/8 189.47 -14.44
4 261.29 -5.58 7/6 252.63 -14.24
5 320.74 +5.1 6/5 315.79 +0.15
6 388.17 +1.86 5/4 378.95 -7.36
7 443.13 +8.13 9/7 442.11 +7.11
8 498.09 +0.04 4/3 505.26 +7.24
9 572.52 +3.8 25/18 568.42 -0.03
10 641.48 +10.2 36/25 631.58 +0.3
11 701.91 -0.04 3/2 694.74 -7.21
12 760.37 -4.55 14/9 757.89 -7.03
13 831.29 -9.24 13/8 821.05 -19.48
14 886.26 +1.9 5/3 884.21 -0.16
15 962.21 -6.62 7/4 947.37 -21.46
16 1015.65 -1.94 9/5 1010.53 -7.06
17 1081.11 -7.16 15/8 1073.68 -14.59
18 1140.55 +3.51 27/14 1136.84 -0.2
19 1200 +0 2/1 1200 +0

Little Observations

I think I'm confident enough to say that this belongs in the 2.3.5.7.13 subgroup, the same as 19EDO.

The average difference for 19EDO is 7.649444...c, while our temperament's average is 4.1444...c.