Tetrahanson: Difference between revisions

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== 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]]
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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 nor 11 in their basis, should simply be called tetrahanson (and no-twos 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
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! Approximate ratios
! Approximate ratios
|-
|-
| -1
| −1
| 1656.6
| 1656.6
| [[13/5]]
| 125/48, [[13/5]]
| 388.6
| 388.6
| 5/4
| 5/4
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| 36/25, [[13/9]]
| 36/25, [[13/9]]
| 1268.0
| 1268.0
| 25/12, 27/13
| 25/12, 27/13, 52/25
|-
|-
| 1
| 1
Line 126: Line 127:
| [[125/108]], [[15/13]]
| [[125/108]], [[15/13]]
| 879.3
| 879.3
| 5/3
| 5/3, [[33/20]]
| 1513.3
| 1513.3
| 12/5
| 12/5
Line 150: Line 151:
| [[16/9]], [[44/25]]
| [[16/9]], [[44/25]]
| 1615.4
| 1615.4
| 64/25,33/13
| 64/25, 33/13
| 347.4
| 347.4
| [[100/81]], [[16/13]], [[11/9]]
| [[100/81]], [[16/13]], [[11/9]]
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<nowiki />* In 3.5.11.13-subgroup 13-throdd-limit minimax tuning  
<nowiki />* In 3.5.11.13-subgroup 13-throdd-limit minimax tuning  


* [https://scaleworkshop.plainsound.org/scale/KXnc-_xfK] – 15-note scale
* [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
|
|-
|-
|  
|  
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| 33/13
| 33/13
| 878.675
| 878.675
|
|-
|
| 144/125
| 878.954
|  
|  
|-
|-
Line 209: Line 239:
| 36/13
| 36/13
| 881.691
| 881.691
|
|-
|
| 15/13
| 881.726
|  
|  
|-
|-
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| 16/13
| 16/13
| 882.349
| 882.349
|
|-
|
| 16/15
| 882.557
|  
|  
|-
|-
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| 5/3
| 5/3
| 884.359
| 884.359
|
|-
|
| 125/108
| 887.061
|  
|  
|-
|-
Line 237: Line 282:
|}
|}


[[Category:Temperaments]]
[[Category:Tetrahanson| ]] <!-- main article -->
[[Category:Nonoctave]]
[[Category:Rank-2 temperaments]]
[[Category:Non-octave temperaments]]

Latest revision as of 06:59, 21 June 2025

The tetrahanson temperament is a nonoctave kleismic temperament, tempering out the kleisma in the 4.3.5 subgroup and repeating at the double octave 4/1. It is generated by 5/3 and, like in normal hanson temperament, 6 of them make a 4/3. Tetrahanson does not contain any 5-limit major or minor triads, but it does have different voicings of them (3:4:5 and 12:15:20), which, to a 12edo-accustomed listener, can make it sound like the root is the real root and the perfect fifth above it at the same time.

For technical information see Subgroup temperaments#Tetrahanson.

Interval chain

Generators Cents (CTE) Approximate ratios
−7 1019.413 9/5
−6 1902.354 3/1
−5 385.295 5/4
−4 1268.236 25/12
−3 2151.177 125/36
−2 634.118 36/25
−1 1517.059 12/5
0 0.000 1/1
1 882.941 5/3
2 1765.882 25/9
3 248.823 144/125
4 1131.764 48/25
5 2014.705 16/5
6 497.646 4/3
7 1380.587 20/9

Tetrahanson on tritave

In tritave-repeating tetrahanson (3.4.5 subgroup), 36/25 actually represents 1\3edt, which makes the 3rd-tritave period.

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 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.

3.4.5 3.5.11 3.5.13
CWE 883.071 879.416 883.808
Badness (Dirichlet) 0.155 2.708 0.069
3.4.5.11 3.4.5.13 3.5.11.13
CWE 880.672 882.854 879.352
Badness (Dirichlet) 0.384 0.066 0.44
Interval chain
# Period 0 Period 1 Period 2
Cents* Approximate ratios Cents* Approximate ratios Cents* Approximate ratios
−1 1656.6 125/48, 13/5 388.6 5/4 1022.6 9/5
0 0.0 1/1 634.0 36/25, 13/9 1268.0 25/12, 27/13, 52/25
1 245.3 125/108, 15/13 879.3 5/3, 33/20 1513.3 12/5
2 490.7 4/3, 33/25 1124.7 48/25, 25/13 1758.7 25/9, 36/13, 11/4
3 736.0 20/13, 55/36 1370.0 20/9, 11/5 102.1 16/15, 55/52
4 981.4 16/9, 44/25 1615.4 64/25, 33/13 347.4 100/81, 16/13, 11/9

* In 3.5.11.13-subgroup 13-throdd-limit minimax tuning

  • [1] – 15-note scale (9L 6s⟨3/1⟩, sLsLLsLsLLsLsLL)
  • 2\3edt-equivalent Muddles:
    • sLsLLsLsLL parent with 2112121 target gives LsmLmLm minor? heptatonic
    • sLLsLsLLsL parent with 2112121 target gives LmsLmLm major? heptatonic
Chords
Mossteps Tonic and then every +2 mossteps
0-3-6 36:44:55 → 36:45:55 → 16:20:25 ↩↩
0-3-7 9:11:15 → 12:15:20 ↩↩ → 16:20:27
0-4-7 3:4:5 ↩↩ → 20:27:33 → 25:44:60
0-5-7 9:13:15 ↩↩↩ → 44:64:75
0-6-10 13:20:27 ↩ → 16:25:33 ↩↩
Tuning spectrum
ET
generator 		Eigenmonzo
(unchanged interval) 		Generator (¢) Comments
11/4 875.659
11/5 877.658
18\39edt 877.825
33/13 878.675
144/125 878.954
11/9 879.333 3.5.11.13-subgroup 13-throdd-limit minimax
43\93edt 879.399
44/15 879.798
25\54edt 880.534
5/4 881.656
36/13 881.691
15/13 881.726
32\69edt 882.066
16/13 882.349
16/15 882.557
4/3 883.007 3.4.5-subgroup 5-throdd-limit minimax
5/3 884.359
125/108 887.061
7\15edt 887.579
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