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== Distincly tempered out commas ==
12edo is distincly consistent in the 5-limit. There are precisely 14 distinct tempered out commas, which are ratios formed by sequences of intervals where every note is unique, except for the starting and ending notes, which remain the same.
{| class="wikitable center-all left-2 right-3 right-5 left-6"
|+style=white-space:nowrap|All commas tempered out in 12-tet throughout series of 5-odd-limit intervals with all notes distinct
! Ratio
! Factorization
! Cents
! Limit
! - Cents
! 1 / Factorization
! 1 / Ratio
|-
|[[81/80]]
| 2<sup>-4</sup> • 3<sup>4</sup> • 5<sup>-1</sup>
| 21.506
| 5
|  -21.506
| 2<sup>4</sup> • 3<sup>-4</sup> • 5<sup>1</sup>
|[[80/81]]
|-
|[[128/125]]
| 2<sup>7</sup> • 5<sup>-3</sup>
| 41.059
| 5
|  -41.059
| 2<sup>-7</sup> • 5<sup>3</sup>
|[[125/128]]
|-
|[[648/625]]
| 2<sup>3</sup> • 3<sup>4</sup> • 5<sup>-4</sup>
| 62.565
| 5
|  -62.565
| 2<sup>-3</sup> • 3<sup>-4</sup> • 5<sup>4</sup>
|[[625/648]]
|-
|[[2048/2025]]
| 2<sup>11</sup> • 3<sup>-4</sup> • 5<sup>-2</sup>
| 19.553
| 5
|  -19.553
| 2<sup>-11</sup> • 3<sup>4</sup> • 5<sup>2</sup>
|[[2025/2048]]
|-
|[[6561/6250]]
| 2<sup>-1</sup> • 3<sup>8</sup> • 5<sup>-5</sup>
| 84.071
| 5
|  -84.071
| 2<sup>1</sup> • 3<sup>-8</sup> • 5<sup>5</sup>
|[[6250/6561]]
|-
|[[32805/32768]]
| 2<sup>-15</sup> • 3<sup>8</sup> • 5<sup>1</sup>
| 1.954
| 5
|  -1.954
| 2<sup>15</sup> • 3<sup>-8</sup> • 5<sup>-1</sup>
|[[32768/32805]]
|-
|[[82944/78125]]
| 2<sup>10</sup> • 3<sup>4</sup> • 5<sup>-7</sup>
| 103.624
| 5
|  -103.624
| 2<sup>-10</sup> • 3<sup>-4</sup> • 5<sup>7</sup>
|[[78125/82944]]
|-
|[[262144/253125]]
| 2<sup>18</sup> • 3<sup>-4</sup> • 5<sup>-5</sup>
| 60.611
| 5
|  -60.611
| 2<sup>-18</sup> • 3<sup>4</sup> • 5<sup>5</sup>
|[[253125/262144]]
|-
|[[531441/500000]]
| 2<sup>-5</sup> • 3<sup>12</sup> • 5<sup>-6</sup>
| 105.578
| 5
|  -105.578
| 2<sup>5</sup> • 3<sup>-12</sup> • 5<sup>6</sup>
|[[500000/531441]]
|-
| [[531441/524288]]
| 2<sup>-19</sup> • 3<sup>12</sup>
| 23.460
| 3
| -23.460
| 2<sup>19</sup> • 3<sup>-12</sup>
| [[524288/531441]]
|-
| [[2125764/1953125]]
| 2<sup>2</sup> • 3<sup>12</sup> • 5<sup>-9</sup>
| 146.637
| 5
| -146.637
| 2<sup>-2</sup> • 3<sup>-12</sup> • 5<sup>9</sup>
| [[1953125/2125764]]
|-
| [[10616832/9765625]]
| 2<sup>17</sup> • 3<sup>4</sup> • 5<sup>-10</sup>
| 144.683
| 5
| -144.683
| 2<sup>-17</sup> • 3<sup>-4</sup> • 5<sup>10</sup>
| [[9765625/10616832]]
|-
| [[33554432/31640625]]
| 2<sup>25</sup> • 3<sup>-4</sup> • 5<sup>-8</sup>
| 101.670
| 5
| -101.670
| 2<sup>-25</sup> • 3<sup>4</sup> • 5<sup>8</sup>
| [[31640625/33554432]]
|-
| [[53747712/48828125]]
| 2<sup>13</sup> • 3<sup>8</sup> • 5<sup>-11</sup>
| 166.189
| 5
| -166.189
| 2<sup>-13</sup> • 3<sup>-8</sup> • 5<sup>11</sup>
| [[48828125/53747712]]
|}
{| class="wikitable center-all left-2 right-3 right-5 left-6"
|+style=white-space:nowrap| 7-limit commas tempered out in 12-tet with Benedetti height < 2**16
! Ratio
! Factorization
! Cents
! Limit
! - Cents
! 1 / Factorization
! 1 / Ratio
|-
| [[36/35]]
| 2<sup>2</sup> • 3<sup>2</sup> • 5<sup>-1</sup> • 7<sup>-1</sup>
| 48.770
| 7
| -48.770
| 2<sup>-2</sup> • 3<sup>-2</sup> • 5<sup>1</sup> • 7<sup>1</sup>
| [[35/36]]
|-
| [[50/49]]
| 2<sup>1</sup> • 5<sup>2</sup> • 7<sup>-2</sup>
| 34.976
| 7
| -34.976
| 2<sup>-1</sup> • 5<sup>-2</sup> • 7<sup>2</sup>
| [[49/50]]
|-
| [[64/63]]
| 2<sup>6</sup> • 3<sup>-2</sup> • 7<sup>-1</sup>
| 27.264
| 7
| -27.264
| 2<sup>-6</sup> • 3<sup>2</sup> • 7<sup>1</sup>
| [[63/64]]
|-
| [[126/125]]
| 2<sup>1</sup> • 3<sup>2</sup> • 5<sup>-3</sup> • 7<sup>1</sup>
| 13.795
| 7
| -13.795
| 2<sup>-1</sup> • 3<sup>-2</sup> • 5<sup>3</sup> • 7<sup>-1</sup>
| [[125/126]]
|-
| [[225/224]]
| 2<sup>-5</sup> • 3<sup>2</sup> • 5<sup>2</sup> • 7<sup>-1</sup>
| 7.712
| 7
| -7.712
| 2<sup>5</sup> • 3<sup>-2</sup> • 5<sup>-2</sup> • 7<sup>1</sup>
| [[224/225]]
|-
| [[256/245]]
| 2<sup>8</sup> • 5<sup>-1</sup> • 7<sup>-2</sup>
| 76.034
| 7
| -76.034
| 2<sup>-8</sup> • 5<sup>1</sup> • 7<sup>2</sup>
| [[245/256]]
|}
== Modes ==
== Modes ==


Line 743: Line 930:
|3 8 3 3 3 8 3 3 3 8 3
|3 8 3 3 3 8 3 3 3 8 3
|3<sup>8</sup> • 5<sup>-11</sup>
|3<sup>8</sup> • 5<sup>-11</sup>
|}
== Tempered commas ==
{| class="wikitable center-all left-2 right-3 right-5 left-6"
|+style=white-space:nowrap|All commas tempered out in 12-tet throughout series of 5-odd-limit intervals with all notes distinct
! Ratio
! Factorization
! Cents
! Limit
! - Cents
! 1 / Factorization
! 1 / Ratio
|-
|[[81/80]]
| 2<sup>-4</sup> • 3<sup>4</sup> • 5<sup>-1</sup>
| 21.506
| 5
|  -21.506
| 2<sup>4</sup> • 3<sup>-4</sup> • 5<sup>1</sup>
|[[80/81]]
|-
|[[128/125]]
| 2<sup>7</sup> • 5<sup>-3</sup>
| 41.059
| 5
|  -41.059
| 2<sup>-7</sup> • 5<sup>3</sup>
|[[125/128]]
|-
|[[648/625]]
| 2<sup>3</sup> • 3<sup>4</sup> • 5<sup>-4</sup>
| 62.565
| 5
|  -62.565
| 2<sup>-3</sup> • 3<sup>-4</sup> • 5<sup>4</sup>
|[[625/648]]
|-
|[[2048/2025]]
| 2<sup>11</sup> • 3<sup>-4</sup> • 5<sup>-2</sup>
| 19.553
| 5
|  -19.553
| 2<sup>-11</sup> • 3<sup>4</sup> • 5<sup>2</sup>
|[[2025/2048]]
|-
|[[6561/6250]]
| 2<sup>-1</sup> • 3<sup>8</sup> • 5<sup>-5</sup>
| 84.071
| 5
|  -84.071
| 2<sup>1</sup> • 3<sup>-8</sup> • 5<sup>5</sup>
|[[6250/6561]]
|-
|[[32805/32768]]
| 2<sup>-15</sup> • 3<sup>8</sup> • 5<sup>1</sup>
| 1.954
| 5
|  -1.954
| 2<sup>15</sup> • 3<sup>-8</sup> • 5<sup>-1</sup>
|[[32768/32805]]
|-
|[[82944/78125]]
| 2<sup>10</sup> • 3<sup>4</sup> • 5<sup>-7</sup>
| 103.624
| 5
|  -103.624
| 2<sup>-10</sup> • 3<sup>-4</sup> • 5<sup>7</sup>
|[[78125/82944]]
|-
|[[262144/253125]]
| 2<sup>18</sup> • 3<sup>-4</sup> • 5<sup>-5</sup>
| 60.611
| 5
|  -60.611
| 2<sup>-18</sup> • 3<sup>4</sup> • 5<sup>5</sup>
|[[253125/262144]]
|-
|[[531441/500000]]
| 2<sup>-5</sup> • 3<sup>12</sup> • 5<sup>-6</sup>
| 105.578
| 5
|  -105.578
| 2<sup>5</sup> • 3<sup>-12</sup> • 5<sup>6</sup>
|[[500000/531441]]
|-
| [[531441/524288]]
| 2<sup>-19</sup> • 3<sup>12</sup>
| 23.460
| 3
| -23.460
| 2<sup>19</sup> • 3<sup>-12</sup>
| [[524288/531441]]
|-
| [[2125764/1953125]]
| 2<sup>2</sup> • 3<sup>12</sup> • 5<sup>-9</sup>
| 146.637
| 5
| -146.637
| 2<sup>-2</sup> • 3<sup>-12</sup> • 5<sup>9</sup>
| [[1953125/2125764]]
|-
| [[10616832/9765625]]
| 2<sup>17</sup> • 3<sup>4</sup> • 5<sup>-10</sup>
| 144.683
| 5
| -144.683
| 2<sup>-17</sup> • 3<sup>-4</sup> • 5<sup>10</sup>
| [[9765625/10616832]]
|-
| [[33554432/31640625]]
| 2<sup>25</sup> • 3<sup>-4</sup> • 5<sup>-8</sup>
| 101.670
| 5
| -101.670
| 2<sup>-25</sup> • 3<sup>4</sup> • 5<sup>8</sup>
| [[31640625/33554432]]
|-
| [[53747712/48828125]]
| 2<sup>13</sup> • 3<sup>8</sup> • 5<sup>-11</sup>
| 166.189
| 5
| -166.189
| 2<sup>-13</sup> • 3<sup>-8</sup> • 5<sup>11</sup>
| [[48828125/53747712]]
|}
{| class="wikitable center-all left-2 right-3 right-5 left-6"
|+style=white-space:nowrap| 7-limit commas tempered out in 12-tet with Benedetti height < 2**16
! Ratio
! Factorization
! Cents
! Limit
! - Cents
! 1 / Factorization
! 1 / Ratio
|-
| [[36/35]]
| 2<sup>2</sup> • 3<sup>2</sup> • 5<sup>-1</sup> • 7<sup>-1</sup>
| 48.770
| 7
| -48.770
| 2<sup>-2</sup> • 3<sup>-2</sup> • 5<sup>1</sup> • 7<sup>1</sup>
| [[35/36]]
|-
| [[50/49]]
| 2<sup>1</sup> • 5<sup>2</sup> • 7<sup>-2</sup>
| 34.976
| 7
| -34.976
| 2<sup>-1</sup> • 5<sup>-2</sup> • 7<sup>2</sup>
| [[49/50]]
|-
| [[64/63]]
| 2<sup>6</sup> • 3<sup>-2</sup> • 7<sup>-1</sup>
| 27.264
| 7
| -27.264
| 2<sup>-6</sup> • 3<sup>2</sup> • 7<sup>1</sup>
| [[63/64]]
|-
| [[126/125]]
| 2<sup>1</sup> • 3<sup>2</sup> • 5<sup>-3</sup> • 7<sup>1</sup>
| 13.795
| 7
| -13.795
| 2<sup>-1</sup> • 3<sup>-2</sup> • 5<sup>3</sup> • 7<sup>-1</sup>
| [[125/126]]
|-
| [[225/224]]
| 2<sup>-5</sup> • 3<sup>2</sup> • 5<sup>2</sup> • 7<sup>-1</sup>
| 7.712
| 7
| -7.712
| 2<sup>5</sup> • 3<sup>-2</sup> • 5<sup>-2</sup> • 7<sup>1</sup>
| [[224/225]]
|-
| [[256/245]]
| 2<sup>8</sup> • 5<sup>-1</sup> • 7<sup>-2</sup>
| 76.034
| 7
| -76.034
| 2<sup>-8</sup> • 5<sup>1</sup> • 7<sup>2</sup>
| [[245/256]]
|}
|}
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