Dual cycle system: Difference between revisions

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The dual cycle system is a just-intonation tuning system formed from two finite Pythagorean cycles related by the syntonic comma (80~~:81~~), producing 5-limit harmonic relationships through cross-cycle interaction. The system yields an ~~Eight~~-~~Pitch Collection~~ that preserves diatonic ordering with a single duplicated degree, enabling pure 5-limit triads and practical modulation.

The '''dual cycle system''' is a [[just intonation|just-intonation]] [[tuning system]] formed from two finite [[Pythagorean]] cycles related by the [[syntonic comma]] (81/80), producing [[5-limit]] harmonic relationships through cross-cycle interaction. The system yields an eight-pitch collection that preserves diatonic ordering with a single duplicated degree, enabling pure 5-limit triads and practical modulation.

[[File:Dual Cycle System.png|thumb|right|Dual-cycle system]]

== Music ==

; [[Siavash Talebi]]

* [https://www.youtube.com/watch?v=PsVhas-P-OM ''Also Singt Harmonie''] (2026)

== References ==

* Talebi, Siavash. “Dual-Cycle System: Pure 5-Limit Harmony Within a 3-Limit Framework.” Zenodo, 2025. https://doi.org/10.5281/zenodo.17846833

~~Music:~~

~~Also Singt Harmonie — composition demonstrating the dual-cycle system.~~

~~https://youtu.be/PsVhas-P-OM?si=X0IA0ILjOuerAVoS~~

~~Reference:~~

~~Talebi, Siavash. “Dual-Cycle System: Pure 5-Limit Harmony Within a 3-Limit Framework.” Zenodo, 2025.~~

~~https://doi.org/10.5281/zenodo.17846833~~

~~[[Category:Just intonation]]~~

~~[[Category:3-limit]]~~

[[Category:5-limit]]

~~[[Category:Syntonic comma]]~~

@@ Line 1: / Line 1: @@
-The dual cycle system is a just-intonation tuning system formed from two finite Pythagorean cycles related by the syntonic comma (80:81), producing 5-limit harmonic relationships through cross-cycle interaction. The system yields an Eight-Pitch Collection that preserves diatonic ordering with a single duplicated degree, enabling pure 5-limit triads and practical modulation.
+The '''dual cycle system''' is a [[just intonation|just-intonation]] [[tuning system]] formed from two finite [[Pythagorean]] cycles related by the [[syntonic comma]] (81/80), producing [[5-limit]] harmonic relationships through cross-cycle interaction. The system yields an eight-pitch collection that preserves diatonic ordering with a single duplicated degree, enabling pure 5-limit triads and practical modulation.
 [[File:Dual Cycle System.png|thumb|right|Dual-cycle system]]
+== Music ==
+; [[Siavash Talebi]]
+* [https://www.youtube.com/watch?v=PsVhas-P-OM ''Also Singt Harmonie''] (2026)
+== References ==
+* Talebi, Siavash. “Dual-Cycle System: Pure 5-Limit Harmony Within a 3-Limit Framework.” Zenodo, 2025. https://doi.org/10.5281/zenodo.17846833
-Music:
-Also Singt Harmonie — composition demonstrating the dual-cycle system.
-https://youtu.be/PsVhas-P-OM?si=X0IA0ILjOuerAVoS
-Reference:
-Talebi, Siavash. “Dual-Cycle System: Pure 5-Limit Harmony Within a 3-Limit Framework.” Zenodo, 2025.
-https://doi.org/10.5281/zenodo.17846833
-[[Category:Just intonation]]
-[[Category:3-limit]]
 [[Category:5-limit]]
-[[Category:Syntonic comma]]