View source for Tetrahanson

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 ==
{| class="wikitable"
|-
! 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.

=== {{nowrap|b39 &amp; 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.

{| class="wikitable"
|-
! !! 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
|}

{| class="wikitable center-1 right-2"
|+ style="font-size: 105%;" | Interval chain
|-
! rowspan="2" | #
! colspan="2" | Period 0
! colspan="2" | Period 1
! colspan="2" | 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]]
|}
<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"
|+ style="font-size: 105%;" | Tuning spectrum
|-
! ET<br />generator
! [[Eigenmonzo|Eigenmonzo<br />(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
| 
|}

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