EDO: Difference between revisions

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If your interest lies in the nuanced approximation of just intonation through EDOs, then delving into EDOs characterized by a strong [[The Riemann zeta function and tuning|local zeta peak]] could be especially captivating. Such EDOs, including [[12edo|12]], [[19edo|19]], [[22edo|22]], [[27edo|27]], [[31edo|31]], [[34edo|34]], [[41edo|41]], [[46edo|46]], [[53edo|53]], [[58edo|58]], [[60edo|60]], [[65edo|65]], [[68edo|68]], [[72edo|72]], [[77edo|77]], [[80edo|80]], [[84edo|84]], [[87edo|87]], [[94edo|94]], and [[99edo|99]], offer rich avenues for exploration in the quest for harmonic purity and transparent [[temperament|temperaments]].

EDOs with a less pronounced, yet still noteworthy [[The Riemann zeta function and tuning|local zeta peak]]—specifically [[10edo|10]], [[14edo|14]], [[15edo|15]], [[17edo|17]], [[24edo|24]], [[26edo|26]], [[29edo|29]], [[36edo|36]], [[37edo|37]], [[38edo|38]], [[39edo|39]], [[43edo|43]], [[48edo|48]], [[49edo|49]], [[50edo|50]], [[56edo|56]], [[62edo|62]], [[63edo|63]], [[89edo|89]], and [[96edo|96]] EDOs— present a unique palette for harmony explorers. Although these systems may lack the harmonic precision found in EDOs with more prominent zeta peaks, they strike an intriguing balance between consonance and more distant harmonic textures.

EDOs with a less pronounced, yet still noteworthy [[The Riemann zeta function and tuning|local zeta peak]]—specifically [[10edo|10]], [[14edo|14]], [[15edo|15]], [[16edo|16]], [[17edo|17]], [[21edo|21]], [[24edo|24]], [[26edo|26]], [[29edo|29]], [[32edo|32]], [[36edo|36]], [[37edo|37]], [[38edo|38]], [[39edo|39]], [[43edo|43]], [[45edo|45]], [[48edo|48]], [[49edo|49]], [[50edo|50]], [[56edo|56]], [[62edo|62]], [[63edo|63]], [[82edo|82]], [[89edo|89]], and [[96edo|96]] EDOs— present a unique palette for harmony explorers. Although these systems may lack the harmonic precision found in EDOs with more prominent zeta peaks, they strike an intriguing balance between consonance and more distant harmonic textures.

EDOs can be further subdivided and classified according to the size of the fifth, such as with [[Margo Schulter]]'s [[gentle region]] or the distinction between negative, positive, doubly negative and doubly positive of [[R. H. M. Bosanquet]]. [[Kite Giedraitis]] has proposed these six categories, based on the size of the fifth. From narrowest to widest:

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 If your interest lies in the nuanced approximation of just intonation through EDOs, then delving into EDOs characterized by a strong [[The Riemann zeta function and tuning|local zeta peak]] could be especially captivating. Such EDOs, including [[12edo|12]], [[19edo|19]], [[22edo|22]], [[27edo|27]], [[31edo|31]], [[34edo|34]], [[41edo|41]], [[46edo|46]], [[53edo|53]], [[58edo|58]], [[60edo|60]], [[65edo|65]], [[68edo|68]], [[72edo|72]], [[77edo|77]], [[80edo|80]], [[84edo|84]], [[87edo|87]], [[94edo|94]], and [[99edo|99]], offer rich avenues for exploration in the quest for harmonic purity and transparent [[temperament|temperaments]].
-EDOs with a less pronounced, yet still noteworthy [[The Riemann zeta function and tuning|local zeta peak]]—specifically [[10edo|10]], [[14edo|14]], [[15edo|15]], [[17edo|17]], [[24edo|24]], [[26edo|26]], [[29edo|29]], [[36edo|36]], [[37edo|37]], [[38edo|38]], [[39edo|39]], [[43edo|43]], [[48edo|48]], [[49edo|49]], [[50edo|50]], [[56edo|56]], [[62edo|62]], [[63edo|63]], [[89edo|89]], and [[96edo|96]] EDOs— present a unique palette for harmony explorers. Although these systems may lack the harmonic precision found in EDOs with more prominent zeta peaks, they strike an intriguing balance between consonance and more distant harmonic textures.
+EDOs with a less pronounced, yet still noteworthy [[The Riemann zeta function and tuning|local zeta peak]]—specifically [[10edo|10]], [[14edo|14]], [[15edo|15]], [[16edo|16]], [[17edo|17]], [[21edo|21]], [[24edo|24]], [[26edo|26]], [[29edo|29]], [[32edo|32]], [[36edo|36]], [[37edo|37]], [[38edo|38]], [[39edo|39]], [[43edo|43]], [[45edo|45]], [[48edo|48]], [[49edo|49]], [[50edo|50]], [[56edo|56]], [[62edo|62]], [[63edo|63]], [[82edo|82]], [[89edo|89]], and [[96edo|96]] EDOs— present a unique palette for harmony explorers. Although these systems may lack the harmonic precision found in EDOs with more prominent zeta peaks, they strike an intriguing balance between consonance and more distant harmonic textures.
 EDOs can be further subdivided and classified according to the size of the fifth, such as with [[Margo Schulter]]'s [[gentle region]] or the distinction between negative, positive, doubly negative and doubly positive of [[R. H. M. Bosanquet]]. [[Kite Giedraitis]] has proposed these six categories, based on the size of the fifth. From narrowest to widest: