575/574
| Interval information |
fowutwethoruyoyo unison
reduced
575/574, Tutte's comma (or the tutte comma), is a 41-limit superparticular interval superparticular comma of about 3.0 cents.
Commatic relationships
This comma is the difference between the following superparticular pairs:
It can be factored into the following superparticular ratios:
- 576/575 and 330625/330624
- 589/588 and 24150/24149
- 616/615 and 8625/8624
- 621/620 and 7750/7749
- 625/624 and 7176/7175
- 736/735 and 2625/2624
- 780/779 and 2185/2184
- 897/896 and 1600/1599
- 925/924 and 1518/1517
Temperaments
Tempering out this comma in the full 41-limit leads to the rank-12 temperament tuttsmic. Using the 2.5.7.23.41 subgroup leads to the rank-4 temperament tutte.
Tutte temperament
Subgroup: 2.5.7.23.41
Comma list: 575/574
| [⟨ | 1 | 2 | 2 | 4 | 5 | ], |
| ⟨ | 0 | 1 | 0 | 0 | 2 | ], |
| ⟨ | 0 | 0 | 1 | 0 | -1 | ], |
| ⟨ | 0 | 0 | 0 | 1 | 1 | ]] |
Optimal tuning (CTE): ~ 2 = 1\1, ~5/4 = 385.900, ~7/4 = 969.128, ~23/16 = 627.490
Optimal ET sequence: 4, 6, 10m, 15m, 18im, 19, 21, 25, 31, 56, 74m, 78, 84, 109, 249, 358, 432m, 541m, 650m
Badness (Sintel): 0.045
Etymology
It is called Tutte's comma because it allowed British mathematician Bill Tutte, at Bletchley Park in 1941, to achieve a major breakthrough in decoding the Lorenz cipher used by the German High Command. Every encrypted message was preceded by a string of twelve characters. It was assumed that these related to settings on the machine. 25 different letters had been seen in all positions except for one in which only 23 were seen. So Tutte took the 4000 character key stream that had been extracted by John Tiltman, after a procedural error by a German operator, and placed it on a grid with 575 columns to look for repetitions (575 = 25 x 23). He did not see the hoped-for vertical repetitions, as the number 25 had been used to mislead. However he did notice patterns repeating along diagonals. This implied an actual repetition at 574 columns, one less. Because 574 factors into 2 × 7 × 41, Tutte deduced that one of the encryption wheels inside the German machine had 41 pins. This critical breakthrough encouraged the British codebreakers to extract another eleven relatively-prime numbers and map the entire structure of the Lorenz machine without ever having seen one. This directly led to the creation of Colossus, the world's first programmable digital electronic computer, which was built to automate Tutte's codebreaking algorithms and intercept top-secret communications from Hitler and his High Command. Most significantly, it allowed the Allies to confirm that their deceptions about the location of the D-day landings had been successful.
When a suffix is added to create a short comma name or a temperament name, the silent "e" is dropped to maintain correct pronunciation as in tuttsma.