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{{Interwiki
{{Infobox interval|Monzo= -702 400 0 -99 100|debug=1}}
| en = Main Page
The difference between 100 [[896/891]]'s and [[7/4]]
| de = Hauptseite
{{Clear}}
| es = Bienvenidos
== Rank-3 tuning spectrum ==
| ja = メインページ
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
| ro = Pagina Principală
| zh = 主页
}}__NOTOC__
== Welcome to the Xenharmonic Wiki! ==
The Xenharmonic Wiki is an open resource dedicated to musical [[tuning system]]s, focusing on [[xenharmonic music]] while also documenting [[historical tunings]] and tuning practices from [[world musical traditions|world traditions]]. It covers the [[theory]] and [[practice|practical applications]] of these systems.


For a lengthier introduction, see [[Xenharmonic Wiki: Introduction]].
Single tunings fully defined by eigenmonzos (+ the octave):


[[File:Pts-2-3-5-e2-twtop-tlin.jpg|thumb|right]]
{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}
== If you are new to musical tuning ==
* [[Why use alternative tunings?]]
* [[Microtonal music|What are microtonal and xenharmonic music]]?
* [[Listen]] to alternatively tuned music, in case you're wondering what it all sounds like.
* [[List of approaches to musical tuning|Discover approaches to musical tuning]]
* [[Links|Explore links]] to xenharmonic websites
* [[The Library|Browse the library]] of published works about microtonal/xenharmonic music
* Learn about the [[Xenharmonic Alliance]], a social group of xenharmonic musicians


== Popular topics ==
=== Marvel ===
* [[Just intonation]]
* [[EDO]]s and other [[equal-step tuning]]s
* [[MOS scale|Moment of Symmetry (MOS) scales]]
* [[Regular temperaments]]
* [[Historical temperaments]]


== Practical xenharmonics ==
{10/9, 9/8, 6/5, 5/4, 4/3}:
* [[List of music software]]
* [[List of microtonal software plugins]]
* [[Instruments]]
* [[Guides]]


== Contributing to the Xenharmonic Wiki ==
~3/2: 701.955{{C}}, ~5/4: 386.314{{C}}
This wiki is created by volunteers. It is a perpetual work in progress, depending on members of the community to help us develop it. We welcome relevant new content and constructive updates to existing pages, so please feel free to [[Help: How to get your Xenharmonic Wiki account|sign up and contribute]]!
 
{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
|}

Latest revision as of 01:49, 24 May 2026

Interval information
Factorization 2-702 × 3400 × 7-99 × 11100
Monzo [-702 400 0 -99 100⟩
Size in cents 0.02984219¢
Name(s) missing ? 
Special properties reduced
Tenney norm (log2 nd) 1959.86
Weil norm (log2 max(n, d)) 1959.86
Wilson norm (sopfr(nd)) 4397

Audio demonstration

Open this interval in xen-calc

The difference between 100 896/891's and 7/4

Rank-3 tuning spectrum

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

Single tunings fully defined by eigenmonzos (+ the octave):

{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}

Marvel

{10/9, 9/8, 6/5, 5/4, 4/3}:

~3/2: 701.955 ¢, ~5/4: 386.314 ¢

{10/9, 8/7}:

~3/2: 700.670 ¢, ~5/4: 383.743 ¢

{10/9, 7/6}:

~3/2: 700.413 ¢, ~5/4: 383.229  ¢

{10/9, 9/7, 7/5}:

~3/2: 700.027 ¢, ~5/4: 382.458 ¢

{9/8, 8/7, 7/6, 9/7, 4/3}:

~3/2: 701.955 ¢, ~5/4: 382.458 ¢

{9/8, 4/3, 7/5}:

~3/2: 701.955 ¢, ~5/4: 378.602 ¢

{8/7, 6/5}:

~3/2: 700.027 ¢, ~5/4: 384.386 ¢

{8/7, 5/4, 7/5}:

~3/2: 698.099 ¢, ~5/4: 386.314 ¢

{7/6, 6/5, 7/5}:

~3/2: 699.384 ¢, ~5/4: 383.743 ¢

{7/6, 5/4}:

~3/2: 694.243 ¢, ~5/4: 386.314 ¢

{6/5, 9/7}:

~3/2: 698.099 ¢, ~5/4: 382.458 ¢

{5/4, 9/7} (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}

{16/15, 8/7} (impossible)

{16/15, 7/6}

{16/15, 9/7}

{16/15, 7/5}

{15/14, Incomplete

Gene-
rator 		~3/2
~5/4 EDO tuning 11\19 18\31 7\12 31\53 24\41 1\22
(↓↓↓) 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
56/45 378.692 {3/2, 7/5}
6\19 378.947 19edo
13\41 380.488 41edo
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}
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}
17\53 384.906 53edo
5/4 386.314 {5/4, 7/6} {5/4, 7/4} {3/2, 5/4}
10\31 387.097 31edo
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)

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 36/35~33/32 9/8 6/5 21/16 7/5 (20/13) 18/11 9/5 27/14~48/25
Cents 0 157 271 429 543 700 814 971 1086
Ratios 1/1 11/10~12/11 7/6 9/7~14/11 11/8~15/11 3/2 8/5 7/4 15/8~28/15
Cents 114 229 386 500 657 771 929 1043 1200
Ratios 15/14~16/15 8/7 5/4 4/3 16/11~22/15 11/7~14/9 12/7 11/6~20/11 2/1
Cents 71 186 343 457 614 729 886 1000 1157
Ratios 25/24~28/27 10/9 11/9 (13/10) 10/7 32/21 5/3 16/9 35/18~64/33
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