Kite's thoughts on ring number - Revision history

TallKite: TallKite moved page Ring number to Kite's thoughts on ring number: Editor VectorGraphics has repeatedly edited pages I've written about my own research and inserted misinformation. On discord he is openly hostile to me. This move is necessary to avoid a toxic work environment for me.

2025-08-20T01:52:49Z

TallKite moved page Ring number to Kite's thoughts on ring number: Editor VectorGraphics has repeatedly edited pages I've written about my own research and inserted misinformation. On discord he is openly hostile to me. This move is necessary to avoid a toxic work environment for me.

← Older revision	Revision as of 01:52, 20 August 2025
(No difference)

TallKite: Undo revision 207455 by VectorGraphics (talk)

2025-08-20T01:52:28Z

Undo revision 207455 by VectorGraphics (talk)

← Older revision		Revision as of 01:52, 20 August 2025
Line 9:		Line 9:
	[[Chain-of-fifths notation]] only works for single-ring edos.		[[Chain-of-fifths notation]] only works for single-ring edos.

	On the scale tree, multi-ring edos appear only on the ~~vertical~~ lines ~~descending from points where the tree branches~~:		On the scale tree, multi-ring edos appear only on the spines of the kites, shown here as dotted lines:

	[[File:Scale_Tree_close-up.png\|alt=Scale Tree close-up.png\|Scale Tree close-up.png\|359x359px]]		[[File:Scale_Tree_close-up.png\|alt=Scale Tree close-up.png\|Scale Tree close-up.png\|359x359px]]

VectorGraphics at 09:10, 18 August 2025

2025-08-18T09:10:09Z

← Older revision		Revision as of 09:10, 18 August 2025
Line 9:		Line 9:
	[[Chain-of-fifths notation]] only works for single-ring edos.		[[Chain-of-fifths notation]] only works for single-ring edos.

	On the scale tree, multi-ring edos appear only on the ~~spines of~~ the ~~kites, shown here as dotted lines~~:		On the scale tree, multi-ring edos appear only on the vertical lines descending from points where the tree branches:

	[[File:Scale_Tree_close-up.png\|alt=Scale Tree close-up.png\|Scale Tree close-up.png\|359x359px]]		[[File:Scale_Tree_close-up.png\|alt=Scale Tree close-up.png\|Scale Tree close-up.png\|359x359px]]

TallKite at 21:27, 11 August 2024

2024-08-11T21:27:39Z

← Older revision		Revision as of 21:27, 11 August 2024
Line 19:		Line 19:

	Like the ring number, the chain number can be generalized to other intervals besides the octave and the fifth.		Like the ring number, the chain number can be generalized to other intervals besides the octave and the fifth.

			The terms ring number and chain number were coined by [[Kite Giedraitis]].

	[[Category:Equal-step tuning]]		[[Category:Equal-step tuning]]
	[[Category:Terms]]		[[Category:Terms]]

FloraC: Rework the intro and misc. style improvements

2024-07-01T12:51:38Z

Rework the intro and misc. style improvements

← Older revision		Revision as of 12:51, 1 July 2024
Line 1:		Line 1:
	~~Every~~ edo ~~has a ring~~ number, which equals the GCD of the edostepspans of the octave and the fifth, or more generally the equave and the generator.		The '''ring number''' of an [[edo]] is the number of [[circle of fifths\|circles of fifths]] in it, which equals the {{w\|Greatest common divisor\|GCD}} of the edostepspans of the [[octave]] and the [[perfect fifth\|fifth]], or more generally the [[equave]] and the [[generator]]. Every edo has a ring number associated with its approximation to the fifth.

	==Examples==		== Examples ==
	~~12-edo~~'s best approximation of [[3/2]] is 7\12. Since 7 and 12 are co-prime, ~~12-edo~~ has only one circle of fifths. Its ring number is 1, and ~~12-edo~~ is said to be '''single-ring'''. An edo with a non-co-prime fifth is '''multi-ring,''' or "ringy". For example, [[15edo~~\|15-edo~~'s]] best approximation of 3/2 is 9\15. The ring number is the GCD of 9 and 15, which is 3. Thus ~~15-edo~~ is a triple-ring edo. Using an alternative approximation of 3/2 affects the "ringiness": ~~18-edo~~ is not multi-ring, but 18b-edo is.		12edo's best approximation of [[3/2]] is 7\12. Since 7 and 12 are co-prime, 12edo has only one circle of fifths. Its ring number is 1, and 12edo is said to be '''single-ring'''. An edo with a non-co-prime fifth is '''multi-ring,''' or "ringy". For example, [[15edo]]'s best approximation of 3/2 is 9\15. The ring number is the GCD of 9 and 15, which is 3. Thus 15edo is a triple-ring edo. Using an alternative approximation of 3/2 affects the "ringiness": 18edo is not multi-ring, but 18b-edo is.

	== Properties ==		== Properties ==
	If N is a prime number, N-edo is a single-ring edo. Note that if N is not prime, N-edo may still be single-ring.		If ''N'' is a [[prime number]], ''N''-edo is a single-ring edo. Note that if ''N'' is not prime, ''N''-edo may still be single-ring.

	[[~~Circle~~-of-fifths notation]] only works for single-ring edos.		[[Chain-of-fifths notation]] only works for single-ring edos.

	On the scale tree, multi-ring edos appear only on the spines of the kites, shown here as dotted lines:		On the scale tree, multi-ring edos appear only on the spines of the kites, shown here as dotted lines:
Line 13:		Line 13:
	[[File:Scale_Tree_close-up.png\|alt=Scale Tree close-up.png\|Scale Tree close-up.png\|359x359px]]		[[File:Scale_Tree_close-up.png\|alt=Scale Tree close-up.png\|Scale Tree close-up.png\|359x359px]]

	==Generalizations==		== Generalizations ==
	The ring number can be defined using any two intervals, not just the octave or the fifth. The two intervals are treated as [[equave]] and generator. For example, 15-edo can be thought of as generated by 2\15 ([[Porcupine\|Porcupine/Triyo]]), which makes it single-ring. And [[Bohlen-Pierce\|13ed3]] can be thought of as generated by 6\13 (a tempered 5/3), again single-ring.		The ring number can be defined using any two intervals, not just the octave or the fifth. The two intervals are treated as [[equave]] and generator. For example, 15-edo can be thought of as generated by 2\15 ([[Porcupine\|Porcupine/Triyo]]), which makes it single-ring. And [[Bohlen-Pierce\|13ed3]] can be thought of as generated by 6\13 (a tempered 5/3), again single-ring.

	Analogous to ring number, rank-2 temperaments have a '''chain number'''. For example, any rank-2 temperament with a pergen of (P8, P5) has a chain number of 1, and is single-chain. All other pergens are multi-chain. For example, Porcupine/Triyo has pergen (P8, P4/3) and is triple-chain. Diaschismatic/Sagugu has pergen (P8/2, P5) and is double-chain. A pergen (P8/m, M/n) has chain number m*n/\|f\|, where M is the multigen and f is M's [[fifthspan]]. For example (P8/2, M2/4) is quadruple-chain.		Analogous to ring number, rank-2 temperaments have a '''chain number'''. For example, any rank-2 temperament with a pergen of (P8, P5) has a chain number of 1, and is single-chain. All other pergens are multi-chain. For example, Porcupine/Triyo has pergen (P8, P4/3) and is triple-chain. Diaschismatic/Sagugu has pergen (P8/2, P5) and is double-chain. A pergen (P8/''m'', M/''n'') has chain number ''m''·''n''/\|''f''\|, where M is the multigen and ''f'' is M's [[fifthspan]]. For example (P8/2, M2/4) is quadruple-chain.

	Like the ring number, the chain number can be generalized to other intervals besides the octave and the fifth.		Like the ring number, the chain number can be generalized to other intervals besides the octave and the fifth.

	[[Category:Equal-step tuning]]		[[Category:Equal-step tuning]]
	[[Category:Terms]]		[[Category:Terms]]

BudjarnLambeth: Categorised this uncategorised page

2023-06-13T23:04:06Z

Categorised this uncategorised page

← Older revision		Revision as of 23:04, 13 June 2023
Line 19:		Line 19:

	Like the ring number, the chain number can be generalized to other intervals besides the octave and the fifth.		Like the ring number, the chain number can be generalized to other intervals besides the octave and the fifth.
			[[Category:Equal-step tuning]]
			[[Category:Terms]]

TallKite: Created page with "Every edo has a ring number, which equals the GCD of the edostepspans of the octave and the fifth, or more generally the equave and the generator. ==Examples== 12-edo's best..."

2023-06-01T08:20:45Z

Created page with "Every edo has a ring number, which equals the GCD of the edostepspans of the octave and the fifth, or more generally the equave and the generator. ==Examples== 12-edo's best..."

New page

Every edo has a ring number, which equals the GCD of the edostepspans of the octave and the fifth, or more generally the equave and the generator.

==Examples==
12-edo's best approximation of [[3/2]] is 7\12. Since 7 and 12 are co-prime, 12-edo has only one circle of fifths. Its ring number is 1, and 12-edo is said to be '''single-ring'''. An edo with a non-co-prime fifth is '''multi-ring,''' or "ringy". For example, [[15edo|15-edo's]] best approximation of 3/2 is 9\15. The ring number is the GCD of 9 and 15, which is 3. Thus 15-edo is a triple-ring edo. Using an alternative approximation of 3/2 affects the "ringiness": 18-edo is not multi-ring, but 18b-edo is.

== Properties ==
If N is a prime number, N-edo is a single-ring edo. Note that if N is not prime, N-edo may still be single-ring.

[[Circle-of-fifths notation]] only works for single-ring edos.

On the scale tree, multi-ring edos appear only on the spines of the kites, shown here as dotted lines:

[[File:Scale_Tree_close-up.png|alt=Scale Tree close-up.png|Scale Tree close-up.png|359x359px]]

==Generalizations==
The ring number can be defined using any two intervals, not just the octave or the fifth. The two intervals are treated as [[equave]] and generator. For example, 15-edo can be thought of as generated by 2\15 ([[Porcupine|Porcupine/Triyo]]), which makes it single-ring. And [[Bohlen-Pierce|13ed3]] can be thought of as generated by 6\13 (a tempered 5/3), again single-ring.

Analogous to ring number, rank-2 temperaments have a '''chain number'''. For example, any rank-2 temperament with a pergen of (P8, P5) has a chain number of 1, and is single-chain. All other pergens are multi-chain. For example, Porcupine/Triyo has pergen (P8, P4/3) and is triple-chain. Diaschismatic/Sagugu has pergen (P8/2, P5) and is double-chain. A pergen (P8/m, M/n) has chain number m*n/|f|, where M is the multigen and f is M's [[fifthspan]]. For example (P8/2, M2/4) is quadruple-chain.

Like the ring number, the chain number can be generalized to other intervals besides the octave and the fifth.