User:CritDeathX/Sam's 17-note Well Temperament

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Random Ramblings To Reason Why I Did This

Okay, so lets get one thing straight (not me): I love my Neo-Gothic intervals. The sound of a 14/11 somehow works and calls back to my 12EDO-self as its rapid warbles make my brain excited. 13/11 also works for the same reason.

The problem for me is that I don't wanna use a large EDO to approximate these well, nor do I wanna compose in JI. So, the natural assumption (for me, at least) is to look at 17EDO and see if that works.

Well, here's the catch: we don't have a 14/11. Instead, we get 9/7, which does not sound good in a normal triad. In my honest opinion, it rubs a bit too close to being a interseptimal interval to be considered a consonance. I believe its one of the reasons why George Secor said that 6:7:9 was most consonant chord in his own 17-note well temperament, which we could assume its close to 17EDO. Considering that its tones line up 4 times, its as consonant as 4:5:6, so I can't exactly blame him on that front either.

But we're not trying to get 6:7:9, we're trying to get a tuning with 14/11 and 13/11 without using a large EDO or using JI. So how are we going to achieve that?

Easy; we make our own version.

The Math Behind This

We have our main intervals in mind; 14/11 & 13/11. Lets assume we also want the prime intervals that make up those intervals (7, 11, & 13). Assuming you (the reader) are thinking of the same steps as I am for these intervals, we end up with this little equation:

(4log2(13/11) + 6log2(14/11) + 8log2(11/8) + 12log2(13/8) + 14log2(7/4)) / (4^2 + 6^2 + 8^2 + 12^2 + 14^2)

(in case you're confused by this, a similar equation can be found on the Wikipedia page for Wendy Carlos' alpha scale)

With this, we get 0.05797208485 as our answer so far. That's only on log2 though, so we multiply this answer by 1200, giving us 69.5665018267c, a cent flat from one step of 17EDO. Next thing you do is you load this up on the Rank-2 temperament thing on the Scale Workshop website, load it up for 17 notes, and you got yourself a 1L16s scale, which we'll be calling our little 17-note well temperament (at least if i understand this enough).

The Scale

Degree Cents Difference JI Approximation 17EDO Difference
0 0 +0c 1/1 0 +0c
1 69.57 -1.1c 25/24 70.59 -0.8c
2 139.13 +0.5c 13/12 141.18 +2.7c
3 208.69 +4.8c 9/8 211.76 +7.9c
4 278.27 -10.9c, -3.1c 13/11, 20/17 282.35 -7.5c, +1c
5 347.83 +0.8c 11/9 352.94 +5.5c
6 417.39 -0.1c 14/11 423.53 +6c
7 486.97 -11c 4/3 494.12 -3.9c
8 556.53 +5.2c 11/8 564.71 +13.4c
9 626.09 -2.2c 23/16 635.29 +7c
10 695.67 -6.3c 3/2 705.88 +3.9c
11 765.23 +0.3c 14/9 776.47 +11.5c
12 834.79 -5.7c 13/8 847.06 +6.5c
13 904.36 +1.5c 27/16 917.65 +11.8c
14 973.93 +5.1c 7/4 988.24 +19.4c
15 1043.49 -5.9c 11/6 1058.82 +9.4c
16 1113.06 +12c, +1.9c 17/9, 19/10 1129.41 +28.4c, +18.4c
17 1200 +0c 2/1 1200 +0c

Little Observations

Admittedly, 13/11 in my scale is more flat than 17EDO, but at least we can get cooler intervals in replacement of that.

Something to note here is that similar to pseudo-semaphore, there's two different types of fifths; you got 486.97c (713.03c) and 695.67c.

The comma this tempers out is ((25/24)^6)/(14/11), for which the monzo is |-19 -6 12 -1 1>. It also tempers out 99/98, which I certainly find interesting.

The best limit I could think of is 2.3.6.7.11.13.23. I imagine there's probably a better way to demonstrate this limit, though.

The average difference from JI for 17EDO was 8c (taking the lowest difference for intervals with 2 approximations), but the average for my temperament was only 3.5c. Impressive, honestly.

An alternative generator could be 69.9628521248c, though this doesn't make much of a difference (to my knowledge).

Retuning

I retuned Rachmaninoff's first prelude of the 13 Preludes into this temperament. That's it.

Closed Up

This was a piece I decided to whip up cause Xenwolf mentioned that retuning 12EDO pieces probably isn't the best way to demonstrate this. I've made sure to try and at least use all the notes and show examples of chords that could be made (including a huge harmonic series cluster!). Enjoy this little thing.