Canopus: Difference between revisions

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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.

== Interval table ==

In the below, tritave-reduced harmonics and subharmonics are indicated in '''bold'''.

{| 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>

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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.
+== 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" | &#35; !! rowspan="2" | Cents* !! colspan="1" | Approximate Ratios
+|-
+! 3.5.7.11/4.53 subgroup
+|-
+| &minus;3 || 149.9 || 12/11, 49/45
+|-
+| &minus;2 || 733.9 || 55/36, 75/49, '''81/53''', 84/55
+|-
+| &minus;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}}