Gamelismic clan: Difference between revisions

* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~8/7 = {{monzo| 0 0 1/12 }}

: [{{monzo| 1 0 0 0 }}, {{monzo| 1 0 1/4 0 }}, {{monzo| 0 0 1 0 }}, {{monzo| 3 0 -1/12 0 }}]

{{Optimal ET sequence|legend=1| 5, 21c, 26, 31 }}

* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~8/7 = {{monzo| 2/9 0 1/18 -1/18 }}

: {{monzo list| 1 0 0 0 | 5/3 0 1/6 -1/6 | 25/9 0 17/18 -17/18 | 25/9 0 -1/18 1/18 }}

[[Algebraic generator]]: larger root of 20''x''² - 36''x'' + 15, or (9 + √6)/10.

* 11-odd-limit: ~8/7 = {{monzo| 4/19 2/19 0 0 -1/19 }}

: [{{monzo| 1 0 0 0 0 }}, {{monzo| 31/19 6/19 0 0 -3/19 }}, {{monzo| 49/19 34/19 0 0 -17/19 }}, {{monzo| 53/19 -2/19 0 0 1/19 }}, {{monzo| 62/19 -26/19 0 0 13/19 }}]

Algebraic generator: positive root of ''x''² + 16''x'' - 31, or √95 - 8.

Minimax tuning:  

* 13- and 15-odd-limit: ~8/7 = {{monzo| 3/14 1/14 0 0 0 -1/28 }}

Algebraic generator: Gatetone, positive root of 4''x''⁶ - 7''x'' - 1. Recurrence converges slowly.

Minimax tuning:

* 17-odd-limit: ~8/7 = {{monzo| 3/13 1/13 0 0 0 0 -1/26 }}

{{Optimal ET sequence|legend=0| 41, 46, 87 }}

* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~8/7 = {{monzo| 7/24 0 -1/24 }}

: {{monzo list| 1 0 0 0 | 15/8 0 -1/8 0 | 0 0 1 0 | 65/24 0 1/24 0 }}

{{Optimal ET sequence|legend=1| 36, 41, 77, 118, 277d }}

* 11-odd-limit: ~8/7 = {{monzo| 7/24 0 -1/24 }}

: [{{monzo| 1 0 0 0 0 }}, {{monzo| 15/8 0 -1/8 0 0 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 65/24 0 1/24 0 0 }}, {{monzo| 37/6 0 -7/6 0 0 }}]

{{Optimal ET sequence|legend=0| 36e, 41, 77, 118, 159, 277d }}

* [[7-odd-limit]]: ~15/14 = {{monzo| 2/13 1/13 -1/13 }}

: {{monzo list| 1 0 0 0 | 25/13 6/13 -6/13 0 | 25/13 -7/13 7/13 0 | 35/13 -2/13 2/13 0 }}

* [[9-odd-limit]]: ~15/14 = {{monzo| 1/19 2/19 -1/19 }}

: {{monzo list| 1 0 0 0 | 25/19 12/19 -6/19 0 | 50/19 -14/19 7/19 0 | 55/19 -4/19 2/19 0 }}

[[Tuning ranges]]:

* 11-odd-limit: ~15/14 = {{monzo| 1/19 2/19 -1/19 }}

: [{{monzo| 1 0 0 0 0 }}, {{monzo| 25/19 12/19 -6/19 0 0 }}, {{monzo| 50/19 -14/19 7/19 0 0 }}, {{monzo| 55/19 -4/19 2/19 0 0 }}, {{monzo| 53/19 30/19 -15/19 0 0 }}]

Tuning ranges:

* [[7-odd-limit]]: ~21/20 = {{monzo| 1/6 1/12 0 -1/12 }}

: {{monzo list| 1 0 0 0 | 5/2 3/4 0 -3/4 | 17/6 5/12 0 -5/12 | 5/2 -1/4 0 1/4 }}

* [[9-odd-limit]]: ~21/20 = {{monzo| 1/21 2/21 0 -1/21}}

: {{monzo list| 1 0 0 0 | 10/7 6/7 0 -3/7 | 47/21 10/21 0 -5/21 | 20/7 -2/7 0 1/7 }}

[[Algebraic generator]]: smaller root of ''x''² - 89''x'' + 92, or (89 - sqrt (7553))/2, at 77.8616 cents.  

* 11-odd-limit: ~21/20 = {{monzo| 0 0 0 -1/10 1/10 }}

: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1 0 0 -9/10 9/10 }}, {{monzo| 2 0 0 -1/2 1/2 }}, {{monzo| 3 0 0 3/10 -3/10 }}, {{monzo| 3 0 0 -7/10 7/10 }}]

Algebraic generator: positive root of 4''x''³ + 15''x''² - 21, or else Gontrand2, the smallest positive root of 4''x''⁷ - 8''x''⁶ + 5.

* [[7-odd-limit]]: ~10/9 = {{monzo| 3/26 0 -1/13 1/13 }}

: {{monzo list| 1 0 0 0 | 47/26 0 6/13 -6/13 | 71/26 0 11/13 -11/13 | 71/26 0 -2/13 2/13 }}

* [[9-odd-limit]]: ~10/9 = {{monzo| 5/28 -1/7 0 1/14 }}

: {{Monzo list| 1 0 0 0 | 10/7 6/7 0 -3/7 | 57/28 11/7 0 -11/14 | 20/7 -2/7 0 1/7 }}

{{Optimal ET sequence|legend=1| 26, 46, 72, 118, 190 }}

* [[11-odd-limit]]: ~10/9 = {{monzo| 5/28 -1/7 0 1/14 }}

: [{{monzo| 1 0 0 0 0 }}, {{monzo| 10/7 6/7 0 -3/7 0 }}, {{monzo| 57/28 11/7 0 -11/14 0 }}, {{monzo| 20/7 -2/7 0 1/7 0 }}, {{monzo| 99/28 -3/7 0 3/14 0 }}]

{{Optimal ET sequence|legend=0| 26, 46, 72, 118, 190 }}