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'''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 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. | ||
== Interval table == | |||
In the below, tritave-reduced harmonics and subharmonics are indicated in '''bold'''. | |||
<div><div style="display: inline-grid; margin-right: 25px;"> | |||
{| class="wikitable center-1 right-2" | |||
|+ style="font-size: 105%;" | Canopus | |||
|- | |||
! rowspan="2" | # !! rowspan="2" | Cents* !! colspan="1" | Approximate Ratios | |||
|- | |||
! 3.5.7.11/4.53 subgroup | |||
|- | |||
| −3 || 149.9 || 12/11, 49/45 | |||
|- | |||
| −2 || 733.9 || 55/36, 75/49, '''81/53''', 84/55 | |||
|- | |||
| −1 || 1318.0 || 15/7 | |||
|- | |||
| 0 || 0.0 || 1/1 | |||
|- | |||
| 1 || 584.0 || 7/5 | |||
|- | |||
| 2 || 1168.0 || 49/25, '''53/27''', 55/28, 108/55 | |||
|- | |||
| 3 || 1752.0 || 11/4, 135/49 | |||
|- | |||
| 4 || 434.1 || '''9/7''' | |||
|- | |||
| 5 || 1018.1 || '''9/5''' | |||
|- | |||
| 6 || 1602.1 || 53/21, 63/25 | |||
|- | |||
| 7 || 284.1 || 33/28, 53/45 | |||
|- | |||
| 8 || 868.1 || 33/20, '''81/49''' | |||
|- | |||
| 9 || 1452.1 || '''81/35''' | |||
|- | |||
| 10 || 134.2 || '''27/25''', 53/49 | |||
|} | |||
<nowiki />* In 3.5.7-targeted [[DKW theory|DKW]] tuning | |||
</div> | |||
{{todo|add interval table, tuning spectrum, etc.|inline=1}} | {{todo|add interval table, tuning spectrum, etc.|inline=1}} | ||
Revision as of 18:58, 10 December 2024
For technical information see No-twos subgroup temperaments#Canopus
Canopus is the rank two 3.5.7 subgroup temperament tempering out 16875/16807, the amount by which 27/7 exceeds four 7/5s. Having a generator of ~7:5, it possesses non-trivial MOS of the families 1L 2s, 3L 1s, 3L 4s, 3L 7s, and (in most cases) 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.
Interval table
In the below, tritave-reduced harmonics and subharmonics are indicated in bold.
| # | Cents* | Approximate Ratios |
|---|---|---|
| 3.5.7.11/4.53 subgroup | ||
| −3 | 149.9 | 12/11, 49/45 |
| −2 | 733.9 | 55/36, 75/49, 81/53, 84/55 |
| −1 | 1318.0 | 15/7 |
| 0 | 0.0 | 1/1 |
| 1 | 584.0 | 7/5 |
| 2 | 1168.0 | 49/25, 53/27, 55/28, 108/55 |
| 3 | 1752.0 | 11/4, 135/49 |
| 4 | 434.1 | 9/7 |
| 5 | 1018.1 | 9/5 |
| 6 | 1602.1 | 53/21, 63/25 |
| 7 | 284.1 | 33/28, 53/45 |
| 8 | 868.1 | 33/20, 81/49 |
| 9 | 1452.1 | 81/35 |
| 10 | 134.2 | 27/25, 53/49 |
* In 3.5.7-targeted DKW tuning
|
Todo: add interval table, tuning spectrum, etc. |