Mediant (operation): Difference between revisions
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== Examples == | == Examples == | ||
The following table shows the mediant ''m'' of some fraction pairs ''f1'', ''f2''. | |||
{| class="wikitable" | |||
! ''f1'' | |||
! ''f2'' | |||
! ''m'' | |||
! intermediate step(s) | |||
|- | |||
| [[3/2]] || [[5/4]] || [[4/3]] || (3+5)/(2+4) = 8/6 | |||
|- | |||
| [[3/2]] || [[4/3]] || [[7/5]] || (3+4)/(2+3) | |||
|- | |||
| [[5/4]] || [[6/5]] || [[11/9]] || (5+6)/(4+5) | |||
|- | |||
| [[9/8]] || [[10/9]] || [[19/17]] || (9+10)/(8+9) | |||
|- | |||
| 9/8 || 19/17 || [[28/25]] || (9+19)/(8+17) | |||
|- | |||
| 19/17 || 10/9 || [[29/26]] || (19+10)/(17+9) | |||
|} | |||
== Generalization == | == Generalization == | ||
Revision as of 06:02, 1 July 2021
In the context of JI ratios, the mediant M of two ratios a/c and b/d in lowest terms is M=(a+b)/(c+d). It will always be between the two ratios (a/c < M < b/d, assuming a/c < b/d).
Examples
The following table shows the mediant m of some fraction pairs f1, f2.
| f1 | f2 | m | intermediate step(s) |
|---|---|---|---|
| 3/2 | 5/4 | 4/3 | (3+5)/(2+4) = 8/6 |
| 3/2 | 4/3 | 7/5 | (3+4)/(2+3) |
| 5/4 | 6/5 | 11/9 | (5+6)/(4+5) |
| 9/8 | 10/9 | 19/17 | (9+10)/(8+9) |
| 9/8 | 19/17 | 28/25 | (9+19)/(8+17) |
| 19/17 | 10/9 | 29/26 | (19+10)/(17+9) |
Generalization
The mediant operation can also be used to find generators and scales in edos representing temperaments. For example, the perfect fifth (3/2) in 12edo which supports meantone is 7 steps out of 12, and the fifth in 19edo, another meantone tuning, is 11 steps out of 19. Hence the perfect fifth in 31edo (which is a meantone tuning because 31 = 12 + 19; more precisely, the 5-limit val of 31edo is the sum of the 5-limit vals of 12edo and 19edo) is (7+11)\(12+19) = 18\31, which is in between the sizes of the 12edo fifth and the 19edo one.