Kite's thoughts on ring number: Difference between revisions

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~~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 ==

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:

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[[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.

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.

[[Category:Equal-step tuning]]

[[Category:Terms]]

@@ 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 ==
--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.
+edo'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 ==
-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:
@@ Line 13: / Line 13: @@
 [[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.
-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.
 [[Category:Equal-step tuning]]
 [[Category:Terms]]