Canopus: Difference between revisions

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m Text replacement - "Category:Mirkwai clan" to "Category:Canopic clan"
- high prime and octaves from the interval table; standardize to CWE tuning; improve listing of mos scales; misc. cleanup
 
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| Subgroups = 3.5.7
| Subgroups = 3.5.7
| Comma basis = [[16875/16807]]
| Comma basis = [[16875/16807]]
| Edo join 1 = b13 | Edo join 2 = b88
| Edo join 1 = b10 | Edo join 2 = b13
| Mapping = 1; -5 -4
| Mapping = 1; 5 4
| Generators = 7/5
| Generators = 15/7
| Generators tuning = 583.986
| Generators tuning = 1318.0
| Optimization method = CWE
| Optimization method = CWE
| MOS scales = [[3L 1s (3/1-equivalent)|3L 1s <3/1>]], [[3L 4s (3/1-equivalent)|3L 4s <3/1>]], [[3L 7s (3/1-equivalent)|3L 7s <3/1>]]
| MOS scales = [[3L 4s (3/1-equivalent)|3L 4s <3/1>]], [[3L 7s (3/1-equivalent)|3L 7s <3/1>]], [[10L 3s (3/1-equivalent)|10L 3s <3/1>]]
| Color name =
| Odd limit 1 = 3.5.7 7 | Mistuning 1 = 1.40 | Complexity 1 = 7
| Odd limit 1 = 3.5.7 7 | Mistuning 1 = 1.40 | Complexity 1 = 7
| Odd limit 2 = 3.5.7 49 | Mistuning 2 = 2.80 | Complexity 2 = 13
| Odd limit 2 = 3.5.7 49 | Mistuning 2 = 2.80 | Complexity 2 = 13
}}
}}
'''Canopus''' is the [[Rank-2 temperament|rank two]] [[3.5.7 subgroup]] temperament [[Temper out|tempering out]] [[16875/16807]], the amount by which [[27/7]] exceeds four 7/5s. Having a generator of [[~]][[7/5|7:5]], it possesses non-trivial [[MOS]] of the families [[1L 2s (3/1-equivalent)|1L 2s]], [[3L 1s (3/1-equivalent)|3L 1s]], [[3L 4s (3/1-equivalent)|3L 4s]], [[3L 7s (3/1-equivalent)|3L 7s]], and (in most cases) [[10L 3s (3/1-equivalent)|10L 3s]]. As 16875/16807 = ([[540/539]])<sup>2</sup>*([[3025/3024]]), Canopus can be extended to the 3.5.7.11/4 subgroup extremely naturally by tempering out these two commas; prime 53 can additionally be incorporated by means of tempering out [[1325/1323]], equating (7/5)<sup>2</sup> = [[49/25]] to 53/27.
'''Canopus''' is a [[non-octave]] [[regular temperament|temperament]] of the [[3.5.7 subgroup]]. It [[tempering out|tempers out]] [[16875/16807]], the amount by which [[27/7]] exceeds four [[7/5]]'s. Having a generator of ~7/5 (or ~[[15/7]]), it possesses non-trivial [[mos]] of the families [[3L 4s (3/1-equivalent)|3L 4s]], [[3L 7s (3/1-equivalent)|3L 7s]], and [[10L 3s (3/1-equivalent)|10L 3s]].  


For technical information see [[No-twos subgroup temperaments #Canopus]].  
As 16875/16807 = ([[540/539]])<sup>2</sup>⋅([[3025/3024]]), canopus can be extended to the 3.5.7.11/4 subgroup extremely naturally by tempering out these two commas; [[prime harmonic|prime]] [[53/1|53]] can additionally be incorporated by means of tempering out [[1325/1323]], equating (7/5)<sup>2</sup> = [[49/25]] to [[53/27]].
 
For technical data, see [[No-twos subgroup temperaments #Canopus]].  


== Interval table ==
== Interval table ==
In the below, tritave-reduced harmonics and subharmonics are indicated in '''bold'''.
In the below, tritave-reduced harmonics and subharmonics 1–7 are indicated in '''bold'''.


<div style="display: inline-grid; margin-right: 25px;">
{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
|+ style="font-size: 105%;" | Canopus
|-
|-
! rowspan="2" | &#35; !! rowspan="2" | Cents* !! colspan="1" | Approximate Ratios
! # !! Cents* !! Approximate ratios
|-
|-
! 3.5.7.11/4.53 subgroup
| −3 || 150.0 || 49/45
|-
|-
| &minus;3 || 149.9 || 12/11, 49/45
| −2 || 734.0 || 75/49
|-
|-
| &minus;2 || 733.9 || 55/36, 75/49, '''81/53''', 84/55
| −1 || 1318.0 || 15/7
|-
|-
| &minus;1 || 1318.0 || 15/7
| 0 || 0.0 || '''1/1'''
|-
| 0 || 0.0 || 1/1
|-
|-
| 1 || 584.0 || 7/5
| 1 || 584.0 || 7/5
|-
|-
| 2 || 1168.0 || 49/25, '''53/27''', 55/28, 108/55
| 2 || 1168.0 || 49/25
|-
|-
| 3 || 1752.0 || 11/4, 135/49
| 3 || 1752.0 || 135/49
|-
|-
| 4 || 434.1 || '''9/7'''
| 4 || 434.0 || '''9/7'''
|-
|-
| 5 || 1018.1 || '''9/5'''
| 5 || 1018.0 || '''9/5'''
|-
|-
| 6 || 1602.1 || 53/21, 63/25
| 6 || 1602.0 || 63/25
|-
|-
| 7 || 284.1 || 33/28, 53/45
| 7 || 284.0 || 147/125
|-
|-
| 8 || 868.1 || 33/20, '''81/49'''
| 8 || 868.0 || 81/49
|-
|-
| 9 || 1452.1 || '''81/35'''
| 9 || 1452.0 || 81/35
|-
|-
| 10 || 134.2 || '''27/25''', 53/49
| 10 || 134.0 || 27/25
|}
|}
<nowiki />* In 3.5.7-targeted [[DKW theory|DKW]] tuning
<nowiki/>* In 3.5.7-subgroup CWE tuning
</div>
 


{{Todo|inline=1|expand|comment=add tuning spectrum}}
{{Todo|inline=1|expand|comment=add tuning spectrum}}

Latest revision as of 15:29, 5 November 2026

Canopus
Subgroups 3.5.7
Comma basis 16875/16807
Reduced mapping ⟨1; 5 4]
ET join b10 & b13
Generators (CWE) ~15/7 = 1318.0 ¢
MOS scales 3L 4s <3/1>, 3L 7s <3/1>, 10L 3s <3/1>
Ploidacot beta-tetragem
Minimax error 3.5.7 7-throdd-limit: 1.40 ¢;
3.5.7 49-throdd-limit: 2.80 ¢
Target scale size 3.5.7 7-throdd-limit: 7 notes;
3.5.7 49-throdd-limit: 13 notes

Canopus is a non-octave temperament of the 3.5.7 subgroup. It tempers out 16875/16807, the amount by which 27/7 exceeds four 7/5's. Having a generator of ~7/5 (or ~15/7), it possesses non-trivial mos of the families 3L 4s, 3L 7s, and 10L 3s.

As 16875/16807 = (540/539)2⋅(3025/3024), canopus can be extended to the 3.5.7.11/4 subgroup extremely naturally by tempering out these two commas; prime 53 can additionally be incorporated by means of tempering out 1325/1323, equating (7/5)2 = 49/25 to 53/27.

For technical data, see No-twos subgroup temperaments #Canopus.

Interval table

In the below, tritave-reduced harmonics and subharmonics 1–7 are indicated in bold.

# Cents* Approximate ratios
−3 150.0 49/45
−2 734.0 75/49
−1 1318.0 15/7
0 0.0 1/1
1 584.0 7/5
2 1168.0 49/25
3 1752.0 135/49
4 434.0 9/7
5 1018.0 9/5
6 1602.0 63/25
7 284.0 147/125
8 868.0 81/49
9 1452.0 81/35
10 134.0 27/25

* In 3.5.7-subgroup CWE tuning


Todo: expand

add tuning spectrum

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