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]]
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 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).


{| 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
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{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
|+ Interval chain
|+ style="font-size: 105%;" | Interval chain
|-
|-
! rowspan="2" | #
! rowspan="2" | #
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! Approximate ratios
! Approximate ratios
|-
|-
| -1
| −1
| 1656.6
| 1656.6
| 125/48, [[13/5]]
| 125/48, [[13/5]]
Line 159: Line 160:


{| class="wikitable"
{| class="wikitable"
|+ Chords
|+ style="font-size: 105%;" | Chords
|-
|-
! Mossteps !! Tonic and then every +2 mossteps
! Mossteps !! Tonic and then every +2 mossteps
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{| 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

Revision as of 14:40, 26 February 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).

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

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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