Xenharmonic Wiki

Tetrahanson: Difference between revisions

← Older edit
VisualWikitext
Dummy index (talk | contribs)
autopatrolled, emailconfirmed
561 edits
m Text replacement - "(unchanged-interval)" to "(unchanged interval)"
 
(12 intermediate revisions by 4 users not shown)
Line 5: Line 5:
== Interval chain ==
== Interval chain ==
{| class="wikitable"
{| class="wikitable"
|-
! Generators
! Generators
! Cents (CTE)
! Cents (CTE)
! Approximate ratios
! Approximate ratios
|-
|-
| -7
| −7
| 1019.413
| 1019.413
| [[9/5]]
| [[9/5]]
|-  
|-  
| -6
| −6
| 1902.354
| 1902.354
| [[3/1]]
| [[3/1]]
|-
|-
| -5
| −5
| 385.295
| 385.295
| [[5/4]]
| [[5/4]]
|-
|-
| -4
| −4
| 1268.236
| 1268.236
| 25/12
| 25/12
|-
|-
| -3
| −3
| 2151.177
| 2151.177
| 125/36
| 125/36
|-
|-
| -2
| −2
| 634.118
| 634.118
| [[36/25]]
| [[36/25]]
|-
|-
| -1
| −1
| 1517.059
| 1517.059
| [[12/5]]
| [[12/5]]
Line 73: Line 74:
In tritave-repeating tetrahanson (3.4.5 subgroup), 36/25 actually represents 1\3edt, which makes the 3rd-tritave period.
In tritave-repeating tetrahanson (3.4.5 subgroup), 36/25 actually represents 1\3edt, which makes the 3rd-tritave period.


=== b39 & b15 ===
=== {{nowrap|b39 & b15}} ===
This is restriction of [[catalan]] extension. This can maintain the structure of the 3rd-octave period in 3.4.5, 3.5.11, and 3.5.13. Well, 3.4.5 and 3.5.13, which do not include 7 or 11 in their basis, should simply be called tetrahanson (and no-twos defactored cata, respectively).
This is restriction of [[catalan]] extension. This can maintain the structure of the 3rd-octave period in 3.4.5, 3.5.11, and 3.5.13. Well, 3.4.5 and 3.5.13, which do not include 7 nor 11 in their basis, should simply be called tetrahanson (and no-twos cata, respectively). Another point that must be noted is the tetracatakleismic on 3.4.5.14.


{| class="wikitable"
{| class="wikitable"
|-
|-
! !! 3.4.5 !! 3.5.11 !! 3.5.13
! !! 3.4.5 !! 3.5.11 !! 3.5.13
|-
|-
| CWE || 883.071 || 879.416 || 883.808
| CWE || 883.071 || 879.416 || 883.808
Line 92: Line 93:


{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
|+ Interval chain
|+ style="font-size: 105%;" | Interval chain
|-
|-
! rowspan="2" | #<br>(mingen)
! rowspan="2" | #
! colspan="2" | Period 0
! colspan="2" | Period 0
! colspan="2" | Period 1
! colspan="2" | Period 1
Line 106: Line 107:
! Approximate ratios
! Approximate ratios
|-
|-
| -1
| −1
| 1652.9
| 1656.6
| [[13/5]]
| 125/48, [[13/5]]
| 384.9
| 388.6
| 5/4
| 5/4
| 1018.9
| 1022.6
| 9/5
| 9/5
|-
|-
Line 120: Line 121:
| 36/25, [[13/9]]
| 36/25, [[13/9]]
| 1268.0
| 1268.0
| 25/12, 27/13
| 25/12, 27/13, 52/25
|-
|-
| 1
| 1
| 249.1
| 245.3
| [[125/108]], [[15/13]]
| [[125/108]], [[15/13]]
| 883.1
| 879.3
| 5/3
| 5/3, [[33/20]]
| 1517.1
| 1513.3
| 12/5
| 12/5
|-
|-
| 2
| 2
| 498.2
| 490.7
| 4/3, [[33/25]]
| 4/3, [[33/25]]
| 1132.2
| 1124.7
| 48/25, [[25/13]]
| 48/25, [[25/13]]
| 1766.1
| 1758.7
| 25/9, 36/13, 11/4
| 25/9, 36/13, 11/4
|-
|-
| 3
| 3
| 747.3
| 736.0
| [[20/13]], [[55/36]]
| [[20/13]], [[55/36]]
| 1381.2
| 1370.0
| 20/9, 11/5
| 20/9, 11/5
| 113.3
| 102.1
| [[16/15]], [[55/52]]
| [[16/15]], [[55/52]]
|-
|-
| 4
| 4
| 996.3
| 981.4
| [[16/9]], [[44/25]]
| [[16/9]], [[44/25]]
| 1630.3
| 1615.4
| 64/25,33/13
| 64/25, 33/13
| 362.4
| 347.4
| [[100/81]], [[16/13]], [[11/9]]
| [[100/81]], [[16/13]], [[11/9]]
|}
|}
<nowiki />* In 3.4.5-subgroup CWE tuning  
<nowiki />* In 3.5.11.13-subgroup 13-throdd-limit minimax tuning  
 
* [https://scaleworkshop.plainsound.org/scale/KXnc-_xfK] – 15-note scale ({{mos scalesig|9L 6s<3/1>|link=1}}, sLsLLsLsLLsLsLL)
* 2\3edt-equivalent [[Muddle]]s:
** sLsLLsLsLL parent with 2112121 target gives LsmLmLm minor? heptatonic
** sLLsLsLLsL parent with 2112121 target gives LmsLmLm major? heptatonic
 
{| class="wikitable"
|+ style="font-size: 105%;" | Chords
|-
! Mossteps !! Tonic and then every +2 mossteps
|-
| 0-3-6 || 36:44:55 → 36:45:55 → 16:20:25 &#x21a9;&#x21a9;
|-
| 0-3-7 || 9:11:15 → 12:15:20 &#x21a9;&#x21a9; → 16:20:27
|-
| 0-4-7 || 3:4:5 &#x21a9;&#x21a9; → 20:27:33 → 25:44:60
|-
| 0-5-7 || 9:13:15 &#x21a9;&#x21a9;&#x21a9; → 44:64:75
|-
| 0-6-10 || 13:20:27 &#x21a9; → 16:25:33 &#x21a9;&#x21a9;
|}


{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
|+ Tuning spectrum
|+ style="font-size: 105%;" | Tuning spectrum
|-
|-
! ET<br />generator
! ET<br />generator
! [[Eigenmonzo|Eigenmonzo<br />(unchanged-interval)]]
! [[Eigenmonzo|Eigenmonzo<br />(unchanged interval)]]
! Generator (¢)
! Generator (¢)
! Comments
! Comments
|-
|
| 11/4
| 875.659
|
|-
|
| 11/5
| 877.658
|
|-
|-
| 18\39edt
| 18\39edt
|  
|  
| 877.825
| 877.825
|
|-
|
| 33/13
| 878.675
|
|-
|
| 144/125
| 878.954
|  
|  
|-
|-
Line 172: Line 214:
| 11/9
| 11/9
| 879.333
| 879.333
| 3.5.11.13-subgroup 13-odd-limit minimax
| 3.5.11.13-subgroup 13-throdd-limit minimax
|-
|-
| 43\93edt
| 43\93edt
Line 192: Line 234:
| 5/4
| 5/4
| 881.656
| 881.656
|
|-
|
| 36/13
| 881.691
|
|-
|
| 15/13
| 881.726
|
|-
| 32\69edt
|
| 882.066
|
|-
|
| 16/13
| 882.349
|
|-
|
| 16/15
| 882.557
|  
|  
|-
|-
Line 197: Line 264:
| 4/3
| 4/3
| 883.007
| 883.007
|  
| 3.4.5-subgroup 5-throdd-limit minimax
|-
|-
|  
|  
| 5/3
| 5/3
| 884.359
| 884.359
|
|-
|
| 125/108
| 887.061
|  
|  
|-
|-
Line 210: Line 282:
|}
|}


[[Category:Temperaments]]
[[Category:Tetrahanson| ]] <!-- main article -->
[[Category:Nonoctave]]
[[Category:Rank-2 temperaments]]
[[Category:Non-octave temperaments]]
Retrieved from "https://en.xen.wiki/w/Tetrahanson"