575/574 - Revision history

Dave Keenan: /* Etymology */ Changed "allowed the British codebreakers" to "encouraged the British codebreakers". Slightly more accurate.

2026-08-28T02:26:21Z

Etymology: Changed "allowed the British codebreakers" to "encouraged the British codebreakers". Slightly more accurate.

← Older revision		Revision as of 02:26, 28 August 2026
Line 53:		Line 53:

	== Etymology ==		== Etymology ==
	It is called ''Tutte's comma'' because it allowed British mathematician [https://en.wikipedia.org/wiki/W._T._Tutte Bill Tutte], at Bletchley Park in 1941, to achieve a major breakthrough in decoding the [https://en.wikipedia.org/wiki/Lorenz_cipher 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 [https://en.wikipedia.org/wiki/John_Tiltman 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 ~~allowed~~ 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 [https://en.wikipedia.org/wiki/Colossus_computer 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 [https://en.wikipedia.org/wiki/Normandy_landings D-day] landings had been successful.		It is called ''Tutte's comma'' because it allowed British mathematician [https://en.wikipedia.org/wiki/W._T._Tutte Bill Tutte], at Bletchley Park in 1941, to achieve a major breakthrough in decoding the [https://en.wikipedia.org/wiki/Lorenz_cipher 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 [https://en.wikipedia.org/wiki/John_Tiltman 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 [https://en.wikipedia.org/wiki/Colossus_computer 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 [https://en.wikipedia.org/wiki/Normandy_landings 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'''.		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'''.

Dave Keenan: /* Etymology */ Corrected "subgroup temperament name" to "temperament name".

2026-08-28T02:07:39Z

Etymology: Corrected "subgroup temperament name" to "temperament name".

← Older revision		Revision as of 02:07, 28 August 2026
Line 55:		Line 55:
	It is called ''Tutte's comma'' because it allowed British mathematician [https://en.wikipedia.org/wiki/W._T._Tutte Bill Tutte], at Bletchley Park in 1941, to achieve a major breakthrough in decoding the [https://en.wikipedia.org/wiki/Lorenz_cipher 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 [https://en.wikipedia.org/wiki/John_Tiltman 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 allowed 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 [https://en.wikipedia.org/wiki/Colossus_computer 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 [https://en.wikipedia.org/wiki/Normandy_landings D-day] landings had been successful.		It is called ''Tutte's comma'' because it allowed British mathematician [https://en.wikipedia.org/wiki/W._T._Tutte Bill Tutte], at Bletchley Park in 1941, to achieve a major breakthrough in decoding the [https://en.wikipedia.org/wiki/Lorenz_cipher 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 [https://en.wikipedia.org/wiki/John_Tiltman 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 allowed 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 [https://en.wikipedia.org/wiki/Colossus_computer 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 [https://en.wikipedia.org/wiki/Normandy_landings D-day] landings had been successful.

	When a suffix is added to create a short comma name or a ~~subgroup~~ temperament name, the silent "e" is dropped to maintain correct pronunciation as in '''tuttsma'''.		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'''.

	== See also ==		== See also ==

Dave Keenan: Renamed the comma as per the talk page, following the temperament naming conventions and dropping the silent "e" when adding a suffix.

2026-08-28T02:05:39Z

Renamed the comma as per the talk page, following the temperament naming conventions and dropping the silent "e" when adding a suffix.

← Older revision		Revision as of 02:05, 28 August 2026
Line 1:		Line 1:
	{{Infobox Interval		{{Infobox Interval
	\| Ratio = 575/574		\| Ratio = 575/574
	\| Name = ~~renatisma~~		\| Name = tutte comma
	\| Color name = 41u23oryy1 <br> fowutwethoruyoyo unison		\| Color name = 41u23oryy1 <br> fowutwethoruyoyo unison
	\| Comma = yes		\| Comma = yes
	}}		}}

	'''575/574''', the '''~~renatisma~~''', is a [[41-limit]] [[superparticular interval]] [[superparticular comma]] of about 3.0 [[cent]]s.		'''575/574''', '''Tutte's comma''' (or the '''tutte comma'''), is a [[41-limit]] [[superparticular interval]] [[superparticular comma]] of about 3.0 [[cent]]s.

	It could be called the '''Tutte comma''' because it allowed British mathematician [https://en.wikipedia.org/wiki/W._T._Tutte Bill Tutte], at Bletchley Park in 1941, to achieve a major breakthrough in decoding the [https://en.wikipedia.org/wiki/Lorenz_cipher 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 [https://en.wikipedia.org/wiki/John_Tiltman 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 allowed 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 [https://en.wikipedia.org/wiki/Colossus_computer 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 [https://en.wikipedia.org/wiki/Normandy_landings D-day] landings had been successful.

	== Commatic relationships ==		== Commatic relationships ==
Line 28:		Line 26:

	== Temperaments ==		== Temperaments ==
	Tempering out this comma in the full 41-limit leads to the rank-12 temperament '''~~renatismic~~'''. Using the 2.5.7.23.41 subgroup leads to the rank-4 temperament '''~~renatic~~'''.		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'''.

	=== ~~Renatic~~ ===		=== Tutte temperament ===

	[[Subgroup]]: 2.5.7.23.41		[[Subgroup]]: 2.5.7.23.41
Line 55:		Line 53:

	== Etymology ==		== Etymology ==
	~~This~~ comma ~~was named by~~ [[~~Francium]~~] in ~~2025~~. ~~It refers~~ to ~~{{W\|~~575 ~~Renate}}~~, the ~~asteroid~~. ~~The name~~'s ~~origin is unknown~~, it ~~may be inspired by~~ the ~~asteroid's provisional designation~~, ~~containing~~ the ~~letters~~ "RE".		It is called ''Tutte's comma'' because it allowed British mathematician [https://en.wikipedia.org/wiki/W._T._Tutte Bill Tutte], at Bletchley Park in 1941, to achieve a major breakthrough in decoding the [https://en.wikipedia.org/wiki/Lorenz_cipher 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 [https://en.wikipedia.org/wiki/John_Tiltman 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 allowed 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 [https://en.wikipedia.org/wiki/Colossus_computer 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 [https://en.wikipedia.org/wiki/Normandy_landings D-day] landings had been successful.

			When a suffix is added to create a short comma name or a subgroup temperament name, the silent "e" is dropped to maintain correct pronunciation as in '''tuttsma'''.

	== See also ==		== See also ==

Dave Keenan: Added an anecdote about the cryptographic importance, in the Second World War, of this otherwise obscure comma.

2026-08-24T07:18:29Z

Added an anecdote about the cryptographic importance, in the Second World War, of this otherwise obscure comma.

← Older revision		Revision as of 07:18, 24 August 2026
Line 7:		Line 7:

	'''575/574''', the '''renatisma''', is a [[41-limit]] [[superparticular interval]] [[superparticular comma]] of about 3.0 [[cent]]s.		'''575/574''', the '''renatisma''', is a [[41-limit]] [[superparticular interval]] [[superparticular comma]] of about 3.0 [[cent]]s.

			It could be called the '''Tutte comma''' because it allowed British mathematician [https://en.wikipedia.org/wiki/W._T._Tutte Bill Tutte], at Bletchley Park in 1941, to achieve a major breakthrough in decoding the [https://en.wikipedia.org/wiki/Lorenz_cipher 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 [https://en.wikipedia.org/wiki/John_Tiltman 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 allowed 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 [https://en.wikipedia.org/wiki/Colossus_computer 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 [https://en.wikipedia.org/wiki/Normandy_landings D-day] landings had been successful.

	== Commatic relationships ==		== Commatic relationships ==

Francium: /* Etymology */ typo

2025-03-17T10:58:55Z

Etymology: typo

← Older revision		Revision as of 10:58, 17 March 2025
Line 53:		Line 53:

	== Etymology ==		== Etymology ==
	This comma was named by [[Francium]] in 2025. It refers to {{W\|575 Renate}}, the asteroid. The ~~name’s~~ origin is unknown, it may be inspired by the ~~asteroid’s~~ provisional designation, containing the letters ~~“RE”~~.		This comma was named by [[Francium]] in 2025. It refers to {{W\|575 Renate}}, the asteroid. The name's origin is unknown, it may be inspired by the asteroid's provisional designation, containing the letters "RE".

	== See also ==		== See also ==

Francium: Created page with "{{Infobox Interval | Ratio = 575/574 | Name = renatisma | Color name = 41u23oryy1
fowutwethoruyoyo unison | Comma = yes }} '''575/574''', the '''renatisma''', is a 41-..."

2025-03-17T10:58:15Z

Created page with "{{Infobox Interval | Ratio = 575/574 | Name = renatisma | Color name = 41u23oryy1 <br> fowutwethoruyoyo unison | Comma = yes }} '''575/574''', the '''renatisma''', is a 41-..."

New page

{{Infobox Interval
| Ratio = 575/574
| Name = renatisma
| Color name = 41u23oryy1 <br> fowutwethoruyoyo unison
| Comma = yes
}}

'''575/574''', the '''renatisma''', is a [[41-limit]] [[superparticular interval]] [[superparticular comma]] of about 3.0 [[cent]]s.

== Commatic relationships ==
This comma is the difference between the following superparticular pairs:
* [[165/164]] and [[231/230]]
* [[225/224]] and [[369/368]]
* [[253/252]] and [[451/450]]

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 '''renatismic'''. Using the 2.5.7.23.41 subgroup leads to the rank-4 temperament '''renatic'''.

=== Renatic ===

[[Subgroup]]: 2.5.7.23.41

[[Comma list]]: 575/574

[[Mapping]]: <br>
{| class="right-all"
|-
| [⟨ || 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|legend=1| 4, 6, 10m, 15m, 18im, 19, 21, 25, 31, 56, 74m, 78, 84, 109, 249, 358, 432m, 541m, 650m }}

[[Badness]] (Sintel): 0.045

== Etymology ==
This comma was named by [[Francium]] in 2025. It refers to {{W|575 Renate}}, the asteroid. The name’s origin is unknown, it may be inspired by the asteroid’s provisional designation, containing the letters “RE”.

== See also ==
* [[Unnoticeable comma]]
* [[List of superparticular intervals]]

[[Category:Commas named after asteroids]]
[[Category:Commas with unknown etymology]]