Xenness
Xenness is a subjective measure of how xenharmonic an interval or chord is. There are many ways to quantify it, but an understanding can be helpful when making xenharmonic music.
| Range ¢ | Closer to |
|---|---|
| -25 ~ 25 | Unison* |
| 25 ~ 75 | Quartertone |
| 75 ~ 125 | Semitone |
| 125 ~ 175 | Neutral second |
| 175 ~ 225 | Whole Tone |
| 225 ~ 275 | Semifourth |
| 275 ~ 325 | Minor Third |
| 325 ~ 375 | Neutral third |
| 375 ~ 425 | Major third |
| 425 ~ 475 | Semisixth |
| 475 ~ 525 | Perfect fourth |
| 525 ~ 575 | Superfourth |
| 575 ~ 625 | Tritone |
| 625 ~ 675 | Subfifth |
| 675 ~ 725 | Perfect fifth |
| 725 ~ 775 | Semitenth |
| 775 ~ 825 | Minor sixth |
| 825 ~ 875 | Neutral sixth |
| 875 ~ 925 | Major sixth |
| 925 ~ 975 | Semitwelfth |
| 975 ~ 1025 | Minor seventh |
| 1025 ~ 1075 | Neutral seventh |
| 1075 ~ 1125 | Major seventh |
| 1125 ~ 1175 | Semifourteenth |
| 1175 ~ 1225 | Octave |
Distance from 12edo
Being the dominant tuning system worldwide, 12edo is the standard against which xenharmonic music is judged. By this metric, all intervals 50¢ from the nearest 12edo step (alternating steps of 24edo) are considered the most xenharmonic.
Intervals can be categorized by whether they are closer to a 12edo interval or an alternating 24edo interval. For example, 75¢ to 125¢ is closer to a minor second and 125¢ to 175¢ is closer to a neutral second. This category could be called "middle second" to avoid ambiguity.
* Note that 25¢ is Ground's recommendation for the lower bound of an Aberrisma.
Distance from LCJI
A related metric is to calculate the distance from nearby low-complexity just intonation ratios, such as 4/3 or 9/7. This is not true xenness as there are many LCJI intervals that are simple but very far from 12edo, such as 11/8. This method is also more mathematically complicated, as it may require an infinite sum of distances from various intervals divided by some complexity measure, such as N*D.
Xenness of Tuning Systems
Tuning systems as a whole can also be evaluated for xenness. This could either be based on a lack of "unxen" intervals, the presence of "xen" intervals, or certain structural properties like the stacking of approximations of 3/2.
