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== Random Ramblings To Reason Why I Did This ==
== Random Ramblings To Reason Why I Did This ==
Okay, so lets get one thing straight (not me): I love my [http://dkeenan.com/Music/NobleMediant.txt Neo-Gothic] intervals. The sound of a [[14/11]] somehow works and calls back to my [[12edo|12EDO]]-self as its rapid warbles makes my brain excited. [[13/11]] also just works for some strange reason, as well.
Okay, so lets get one thing straight (not me): I love my [http://dkeenan.com/Music/NobleMediant.txt Neo-Gothic] intervals. The sound of a [[14/11]] somehow works and calls back to my [[12edo|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 [[Just intonation|JI]]. So, the natural assumption (for me, at least) is to look at [[17edo|17EDO]] and see if that works.
The problem for me is that I don't wanna use a large [[EDO]] to approximate these well, nor do I wanna compose in [[Just intonation|JI]]. So, the natural assumption (for me, at least) is to look at [[17edo|17EDO]] and see if that works.
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<small>''(in case you're confused by this, a similar equation can be found [[wikipedia:Alpha_scale|on the Wikipedia page for Wendy Carlos' alpha scale]])''</small>
<small>''(in case you're confused by this, a similar equation can be found [[wikipedia:Alpha_scale|on the Wikipedia page for Wendy Carlos' alpha scale]])''</small>


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 [https://sevish.com/scaleworkshop/ the Scale Workshop website], load it up for 17 notes, and you got yourself a 16L1s scale, which we'll be calling our little [https://sevish.com/scaleworkshop/?name=Rank%202%20scale%20(69.5665018267%2C%202%2F1)&data=69.566502%0A139.133004%0A208.699505%0A278.266007%0A347.832509%0A417.399011%0A486.965513%0A556.532015%0A626.098516%0A695.665018%0A765.231520%0A834.798022%0A904.364524%0A973.931026%0A1043.497527%0A1113.064029%0A1200.000000&freq=440&midi=69&vert=1&horiz=3&colors=&waveform=triangle&ampenv=organ 17-note well temperament] (at least if i understand this enough).
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 [https://sevish.com/scaleworkshop/ the Scale Workshop website], load it up for 17 notes, and you got yourself a 1L16s scale that starts on C, which we'll be calling our little [https://sevish.com/scaleworkshop/?name=Rank%202%20scale%20(69.5665018267%2C%202%2F1)&data=69.566502%0A139.133004%0A208.699505%0A278.266007%0A347.832509%0A417.399011%0A486.965513%0A556.532015%0A626.098516%0A695.665018%0A765.231520%0A834.798022%0A904.364524%0A973.931026%0A1043.497527%0A1113.064029%0A1200.000000&freq=440&midi=69&vert=1&horiz=3&colors=&waveform=triangle&ampenv=organ 17-note well temperament] (at least if i understand this enough).


== The Scale ==
== The Scale ==
{| class="wikitable"
{| class="wikitable"
!Degree
!Degree
!
!Cents
!Cents
!Difference
!Difference
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|-
|-
|0
|0
|
|0
|0
|<u>+</u>0c
|<u>+</u>0c
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|-
|-
|1
|1
|
|69.57
|69.57
| -1.1c
| -1.1c
|
|
|25/24
|[[25/24]]
|
|
|70.59
|70.59
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|-
|-
|2
|2
|
|139.13
|139.13
| +0.5c
| +0.5c
|
|
|13/12
|[[13/12]]
|
|
|141.18
|141.18
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|-
|-
|3
|3
|
|208.69
|208.69
| +4.8c
| +4.8c
|
|
|9/8
|[[9/8]]
|
|
|211.76
|211.76
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|-
|-
|4
|4
|
|278.27
|278.27
| -10.9c, -3.1c
| -10.9c, -3.1c
|
|
|13/11, 20/17
|[[13/11]], [[20/17]]
|
|
|282.35
|282.35
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|-
|-
|5
|5
|
|347.83
|347.83
| +0.8c
| +0.8c
|
|
|11/9
|[[11/9]]
|
|
|352.94
|352.94
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|-
|-
|6
|6
|
|417.39
|417.39
| -0.1c
| -0.1c
|
|
|14/11
|[[14/11]]
|
|
|423.53
|423.53
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|-
|-
|7
|7
|
|486.97
|486.97
| -11c
| -11c
|
|
|4/3
|[[4/3]]
|
|
|494.12
|494.12
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|-
|-
|8
|8
|
|556.53
|556.53
| +5.2c
| +5.2c
|
|
|11/8
|[[11/8]]
|
|
|564.71
|564.71
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|-
|-
|9
|9
|
|626.09
|626.09
| -2.2c
| -2.2c
|
|
|23/16
|[[23/16]]
|
|
|635.29
|635.29
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|-
|-
|10
|10
|
|695.67
|695.67
| -6.3c
| -6.3c
|
|
|3/2
|[[3/2]]
|
|
|705.88
|705.88
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|-
|-
|11
|11
|
|765.23
|765.23
| +0.3c
| +0.3c
|
|
|14/9
|[[14/9]]
|
|
|776.47
|776.47
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|-
|-
|12
|12
|
|834.79
|834.79
| -5.7c
| -5.7c
|
|
|13/8
|[[13/8]]
|
|
|847.06
|847.06
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|-
|-
|13
|13
|
|904.36
|904.36
| +1.5c
| +1.5c
|
|
|27/16
|[[27/16]]
|
|
|917.65
|917.65
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|-
|-
|14
|14
|
|973.93
|973.93
| +5.1c
| +5.1c
|
|
|7/4
|[[7/4]]
|
|
|988.24
|988.24
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|-
|-
|15
|15
|
|1043.49
|1043.49
| -5.9c
| -5.9c
|
|
|11/6
|[[11/6]]
|
|
|1058.82
|1058.82
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|-
|-
|16
|16
|
|1113.06
|1113.06
| +12c, +1.9c
| +12c, +1.9c
|
|
|17/9, 19/10
|[[17/9]], [[19/10]]
|
|
|1129.41
|1129.41
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|-
|-
|17
|17
|
|1200
|1200
|<u>+</u>0c
|<u>+</u>0c
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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.
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 comma this tempers out is ''[[User:CritDeathX/Sam's Neo-Gothic Comma|((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.
 
[[File:Thirteen Preludes No. 1, tuned to sam's 17-tone temperament.wav]]


The best limit I could think of is ''2.3.6.7.11.13''. I imagine there's probably a better way to demonstrate this limit, though.
== ''Closed Up'' ==
This was a piece I decided to whip up cause [[User:Xenwolf|Xenwolf]] mentioned that [https://en.xen.wiki/w/User_talk:CritDeathX#Retuned_piece 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.
[[File:Closed Up.mp3|left|frameless]]
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