Miracle extensions
The basic 7-limit miracle temperament has various extensions to the 11- and 13-limit. The following temperaments are discussed in this article:
- Miraculous (31 & 41) – tempering out 105/104, 144/143, 196/195, and 243/242;
- Benediction (31 & 41f) – tempering out 225/224, 243/242, 351/350, and 385/384;
- Manna (31f & 41f) – tempering out 225/224, 243/242, 325/324, and 385/384;
In addition, we also consider the only alternative 11-limit extension:
- Revelation (21 & 31) – tempering out 66/65, 99/98, 105/104, and 512/507.
For technical information see Gamelismic clan #Miracle.
Tuning spectra
Miraculous
| Edo generator |
Eigenmonzo (unchanged-interval) |
Generator (¢) | Comments |
|---|---|---|---|
| 16/15 | 111.731 | ||
| 13/10 | 113.553 | ||
| 8/7 | 115.587 | ||
| 11/9 | 115.803 | ||
| 3\31 | 116.129 | ||
| 5/4 | 116.241 | ||
| 15/11 | 116.441 | ||
| 7/5 | 116.502 | ||
| 10\103 | 116.505 | ||
| 17\175 | 116.571 | ||
| [0 -27 25 5⟩ | 116.573 | 7 limit least squares | |
| [0 -19 20⟩ | 116.578 | 5 limit least squares | |
| 6/5 | 116.588 | 5 and 7 limit minimax | |
| 11/10 | 116.591 | ||
| 12/11 | 116.596 | ||
| 14/11 | 116.617 | ||
| 116.633 | POTE11 | ||
| 7/6 | 116.641 | ||
| 7\72 | 116.667 | ||
| [0 17 -11 -6 11⟩ | 116.672 | 11 limit least squares | |
| 116.673 | POTE5 | ||
| 116.675 | POTE7 | ||
| 10/9 | 116.716 | 9, 11 and 13 limit minimax | |
| [0 117 -44 -19⟩ | 116.721 | 9 limit least squares | |
| 116.747 | POTE13 | ||
| 11/8 | 116.755 | ||
| 9/7 | 116.792 | ||
| 11\113 | 116.814 | ||
| [0 127 -84 -36 100 -44⟩ | 116.820 | 15 limit least squares | |
| [0 141 -70 -35 84 -42⟩ | 116.846 | 13 limit least squares | |
| 4/3 | 116.993 | 15 limit minimax | |
| 4\41 | 117.073 | ||
| 13/11 | 117.266 | ||
| 18/13 | 117.559 | ||
| 13/12 | 117.936 | ||
| 15/14 | 119.443 | ||
| 16/13 | 119.824 | ||
| 15/13 | 123.871 | ||
| 14/13 | 128.298 |
Benediction
| Edo generator |
Eigenmonzo (unchanged-interval) |
Generator (¢) | Comments |
|---|---|---|---|
| 16/15 | 111.731 | ||
| 8/7 | 115.587 | ||
| 11/9 | 115.803 | ||
| 3\31 | 116.129 | ||
| 5/4 | 116.241 | ||
| 15/11 | 116.441 | ||
| 16/13 | 116.455 | ||
| 7/5 | 116.502 | ||
| 10\103 | 116.505 | ||
| 14/13 | 116.509 | ||
| 13/10 | 116.511 | ||
| 13/12 | 116.536 | ||
| 13/11 | 116.547 | ||
| [0 -234 39 4 -115 228⟩ | 116.56309 | 13 limit least squares | |
| [0 -251 22 5 -131 261⟩ | 116.56348 | 15 limit least squares | |
| 17\175 | 116.571 | ||
| [0 -27 25 5⟩ | 116.573 | 7 limit least squares | |
| 116.574 | POTE13 | ||
| [0 -19 20⟩ | 116.578 | 5 limit least squares | |
| 6/5 | 116.588 | 5, 7 and 15 limit minimax | |
| 11/10 | 116.591 | ||
| 18/13 | 116.595 | 13 limit minimax | |
| 12/11 | 116.596 | ||
| 15/13 | 116.598 | ||
| 14/11 | 116.617 | ||
| 116.633 | POTE11 | ||
| 7/6 | 116.641 | ||
| 7\72 | 116.667 | ||
| [0 17 -11 -6 11⟩ | 116.672 | 11 limit least squares | |
| 116.673 | POTE5 | ||
| 116.675 | POTE7 | ||
| 10/9 | 116.716 | 9 and 11 limit minimax | |
| [0 117 -44 -19⟩ | 116.721 | 9 limit least squares | |
| 11/8 | 116.755 | ||
| 9/7 | 116.792 | ||
| 11\113 | 116.814 | ||
| 4/3 | 116.993 | ||
| 4\41 | 117.073 | ||
| 15/14 | 119.443 |
Manna
| Edo generator |
Eigenmonzo (unchanged-interval) |
Generator (¢) | Comments |
|---|---|---|---|
| 16/15 | 111.731 | ||
| 8/7 | 115.587 | ||
| 11/9 | 115.803 | ||
| 3\31 | 116.129 | ||
| 5/4 | 116.241 | ||
| 15/11 | 116.441 | ||
| 7/5 | 116.502 | ||
| 10\103 | 116.505 | ||
| 17\175 | 116.571 | ||
| [0 -27 25 5⟩ | 116.573 | 7 limit least squares | |
| [0 -19 20⟩ | 116.578 | 5 limit least squares | |
| 6/5 | 116.588 | 5 and 7 limit minimax | |
| 11/10 | 116.591 | ||
| 12/11 | 116.596 | ||
| 14/11 | 116.617 | ||
| 116.633 | POTE11 | ||
| 7/6 | 116.641 | ||
| 7\72 | 116.667 | ||
| [0 17 -11 -6 11⟩ | 116.672 | 11 limit least squares | |
| 116.673 | POTE5 | ||
| 116.675 | POTE7 | ||
| 10/9 | 116.716 | 9 and 11 limit minimax | |
| [0 117 -44 -19⟩ | 116.721 | 9 limit least squares | |
| 15/13 | 116.725 | 15 limit minimax | |
| 116.739 | POTE13 | ||
| 11/8 | 116.755 | ||
| 13/10 | 116.760 | 13 limit minimax | |
| [0 -37 -166 -77 59 243⟩ | 116.764 | 15 limit least squares | |
| [0 18 -111 -76 43 204⟩ | 116.780 | 13 limit least squares | |
| 9/7 | 116.792 | ||
| 14/13 | 116.79254 | ||
| 18/13 | 116.79299 | ||
| 11\113 | 116.814 | ||
| 13/12 | 116.830 | ||
| 16/13 | 116.856 | ||
| 13/11 | 116.922 | ||
| 4/3 | 116.993 | ||
| 4\41 | 117.073 | ||
| 15/14 | 119.443 |
Revelation
| Edo generator |
Eigenmonzo (unchanged-interval) |
Generator (¢) | Comments |
|---|---|---|---|
| 16/15 | 111.731 | ||
| 13/10 | 113.553 | ||
| 13/11 | 114.555 | ||
| 11/10 | 115.000 | ||
| 14/11 | 115.536 | ||
| 11/8 | 115.543 | ||
| 8/7 | 115.587 | ||
| 15/11 | 115.797 | ||
| 12/11 | 115.938 | ||
| 3\31 | 116.129 | ||
| 11/9 | 116.164 | 11, 13 and 15 limit minimax | |
| [0 -195 35 5 89⟩ | 116.198 | 11 limit least squares | |
| [0 -251 22 5 117 13⟩ | 116.229 | 15 limit least squares | |
| 5/4 | 116.241 | ||
| [0 -234 39 4 102 11⟩ | 116.249 | 13 limit least squares | |
| 116.268 | POTE13 | ||
| 116.277 | POTE11 | ||
| 7/5 | 116.502 | ||
| 10\103 | 116.505 | ||
| 17\175 | 116.571 | ||
| [0 -27 25 5⟩ | 116.573 | 7 limit least squares | |
| [0 -19 20⟩ | 116.578 | 5 limit least squares | |
| 6/5 | 116.588 | 5 and 7 limit minimax | |
| 7/6 | 116.641 | ||
| 7\72 | 116.667 | ||
| 116.673 | POTE5 | ||
| 116.675 | POTE7 | ||
| 10/9 | 116.716 | 9 limit minimax | |
| [0 117 -44 -19⟩ | 116.721 | 9 limit least squares | |
| 9/7 | 116.792 | ||
| 11\113 | 116.814 | ||
| 4/3 | 116.993 | ||
| 4\41 | 117.073 | ||
| 18/13 | 117.559 | ||
| 13/12 | 117.936 | ||
| 15/14 | 119.443 | ||
| 16/13 | 119.824 | ||
| 15/13 | 123.871 | ||
| 14/13 | 128.298 |