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some more harmonics in EDOs
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= User:Overthink/Sandbox (size comparison) =
{{Infobox interval|Monzo= -702 400 0 -99 100|debug=1}}
{{Harmonics in equal|10257|columns=18}}
The difference between 100 [[896/891]]'s and [[7/4]]
{{Harmonics in equal|20256|columns=18}}
{{Clear}}
{{Harmonics in equal|9999|columns=18}}
== Rank-3 tuning spectrum ==
--[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 04:41, 29 September 2025 (UTC)
9-odd-limit: 1/1, 10/9, 9/8, 8/7, 7/6, 6/5, 5/4, 9/7, 4/3, 7/5, and complements
= How =
 
== far ==
Single tunings fully defined by eigenmonzos (+ the octave):
=== can ===
 
==== this ====
{10/9, 9/8, 6/5, 5/4, 4/3}, {10/9, 8/7}, {10/9, 7/6}, {10/9, 9/7, 7/5}, {9/8, 8/7, 7/6, 9/7, 4/3}, {9/8, 4/3, 7/5}, {8/7, 6/5}, {8/7, 5/4, 7/5}, {7/6, 6/5, 7/5}, {7/6, 5/4}, {6/5, 9/7}, {5/4, 9/7}
===== even =====
 
====== go ======
=== Marvel ===
======= ? =======
 
What would happen
{10/9, 9/8, 6/5, 5/4, 4/3}:
 
~3/2: 701.955{{C}}, ~5/4: 386.314{{C}}
 
{10/9, 8/7}:
 
~3/2: 700.670{{C}}, ~5/4: 383.743{{C}}
 
{10/9, 7/6}:
 
~3/2: 700.413{{C}}, ~5/4: 383.229 {{C}}
 
{10/9, 9/7, 7/5}:
 
~3/2: 700.027{{C}}, ~5/4: 382.458{{C}}
 
{9/8, 8/7, 7/6, 9/7, 4/3}:
 
~3/2: 701.955{{C}}, ~5/4: 382.458{{C}}
 
{9/8, 4/3, 7/5}:
 
~3/2: 701.955{{C}}, ~5/4: 378.602{{C}}
 
{8/7, 6/5}:
 
~3/2: 700.027{{C}}, ~5/4: 384.386{{C}}
 
{8/7, 5/4, 7/5}:
 
~3/2: 698.099{{C}}, ~5/4: 386.314{{C}}
 
{7/6, 6/5, 7/5}:
 
~3/2: 699.384{{C}}, ~5/4: 383.743{{C}}
 
{7/6, 5/4}:
 
~3/2: 694.243{{C}}, ~5/4: 386.314{{C}}
 
{6/5, 9/7}:
 
~3/2: 698.099{{C}}, ~5/4: 382.458{{C}}
 
<s>{5/4, 9/7}</s> (impossible)
 
Additional eigenmonzo tunings up to 21-odd-limit (new intervals: 21/20, 16/15, 15/14, 21/16)
 
{21/20, 16/15}
 
{21/20, 5/4}
 
{21/20, 9/7}
 
<s>{16/15, 8/7}</s> (impossible)
 
{16/15, 7/6}
 
{16/15, 9/7}
 
{16/15, 7/5}
 
{15/14, ''Incomplete''
 
{| class="wikitable center-all"
! Gene-<br>rator
! colspan=17 | ~3/2
|-
! rowspan=17 | ~5/4
! colspan=3 | EDO tuning
! rowspan=3 |
|
| [[19edo|11\19]]
| [[31edo|18\31]]
|
|
| [[12edo|7\12]]
|
|
|
| [[53edo|31\53]]
|
| [[41edo|24\41]]
| [[22edo|1\22]]
|-
! rowspan=2 | (↓↓↓)
! colspan=2 | Eigenmonzo
| 112/75
|
|
| 25/14
| 25/21
|
| 63/50
| 189/100
| 567/400
|
| 3/2
|
|
|-
! (↓↓↓)
! Size (¢)
| 694.243
| 694.737
| 696.774
| 698.099
| 699.384
| 700.000
| 700.027
| 700.413
| 700.670
| 701.887
| 701.955
| 702.439
| 709.091
|-
! colspan=3 |
! style="height: 5px;" |
! colspan=13 |
|-
|
| 56/45
| 378.692
! rowspan=13 |
|
|
|
|
|
|
|
|
|
|
| {3/2, 7/5}
|
|
|-
| [[19edo|6\19]]
|
| 378.947
|
| [[19edo]]
|
|
|
|
|
|
|
|
|
|
|
|-
| [[41edo|13\41]]
|
| 380.488
|
|
|
|
|
|
|
|
|
|
|
| [[41edo]]
|
|-
| [[22edo|7\22]]
|
| 381.818
|
|
|
|
|
|
|
|
|
|
|
|
| [[22edo]]
|-
|
| 9/7
| 382.458
|
|
|
| {5/3, 9/7}
|
|
| {7/5, 9/5}
|
|
|
| {3/2, 7/4}
|
|
|-
|
| 245/162
| 383.229
|
|
|
|
|
|
|
| {7/6, 9/5}
|
|
|
|
|
|-
| [[72edo|23\72]]
|
| 383.333
|  
|
|
|
|
| [[72edo]]
|
|
|
|
|
|
|
|-
|
| 36/35
| 383.743
|
|
|
|
| {5/3, 7/6}
|
|
|
| {7/4, 9/5}
|
|
|
|
|-
|
| 175/144
| 384.386
|
|
|
|
|
|
| {5/3, 7/4}
|
|
|
|
|
|
|-
| [[53edo|17\53]]
|
| 384.906
|
|
|
|
|
|
|
|
|
| [[53edo]]
|
|
|
|-
|
| 5/4
| 386.314
| {5/4, 7/6}
|
|  
| {5/4, 7/4}
|
|
|
|
|
|
| {3/2, 5/4}
|
|
|-
| [[31edo|10\31]]
|
| 387.097
|
|
| [[31edo]]
|
|
|
|
|
|
|
|
|
|
|-
| [[12edo|4\12]]
|
| 400.000
|
|
|
|
|
| [[12edo]]
|
|
|
|
|
|
|
|}
 
11-odd-limit adds 12/11, 11/10, 11/9, 14/11, 11/8
 
15-odd-limit: 16/15, 15/14, 15/11
 
21-odd-limit: 22/21, 21/20, 21/16
 
== Stuff ==
=== Orwell as 9-form ===
(Note that 13/10 and 20/13 are mapped via tridecimal orwell)
 
{| class="wikitable center-all"
! Degree
! 0-step
! 1-step
! 2-step
! 3-step
! 4-step
! 5-step
! 6-step
! 7-step
! 8-step
! 9-step
|-
! Cents
| 43
| 200
| 314
| 471
| 586
| 743
| 857
| 1014
| 1129
|
|-
! Ratios
| style="background-color: white;" | 36/35~33/32
| style="background-color: white;" | 9/8
| style="background-color: white;" | 6/5
| style="background-color: white;" | 21/16
| style="background-color: white;" | 7/5
| style="background-color: white;" | (20/13)
| style="background-color: white;" | 18/11
| style="background-color: white;" | 9/5
| style="background-color: white;" | 27/14~48/25
| style="background-color: white;" |
|-
! Cents
| 0
| 157
| 271
| 429
| 543
| 700
| 814
| 971
| 1086
|
|-
! Ratios
| style="background-color: white;" | 1/1
| style="background-color: white;" | 11/10~12/11
| style="background-color: white;" | 7/6
| style="background-color: white;" | 9/7~14/11
| style="background-color: white;" | 11/8~15/11
| style="background-color: white;" | 3/2
| style="background-color: white;" | 8/5
| style="background-color: white;" | 7/4
| style="background-color: white;" | 15/8~28/15
| style="background-color: white;" |
|-
! Cents
|
| 114
| 229
| 386
| 500
| 657
| 771
| 929
| 1043
| 1200
|-
! Ratios
| style="background-color: white;" |
| style="background-color: white;" | 15/14~16/15
| style="background-color: white;" | 8/7
| style="background-color: white;" | 5/4
| style="background-color: white;" | 4/3
| style="background-color: white;" | 16/11~22/15
| style="background-color: white;" | 11/7~14/9
| style="background-color: white;" | 12/7
| style="background-color: white;" | 11/6~20/11
| style="background-color: white;" | 2/1
|-
! Cents
|
| 71
| 186
| 343
| 457
| 614
| 729
| 886
| 1000
| 1157
|-
! Ratios
| style="background-color: white;" |
| style="background-color: white;" | 25/24~28/27
| style="background-color: white;" | 10/9
| style="background-color: white;" | 11/9
| style="background-color: white;" | (13/10)
| style="background-color: white;" | 10/7
| style="background-color: white;" | 32/21
| style="background-color: white;" | 5/3
| style="background-color: white;" | 16/9
| style="background-color: white;" | 35/18~64/33
|}
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