EDO vs ET: Difference between revisions

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However, it is not always true that those using EDOs are interested in approximating JI, nor is it true that describing all EDOs in terms of approximate JI is universally helpful or illuminating. As an example of the former, consider the atonalists, a loose school of 20th-century composers who sought to embrace the "equality" of equal temperament by treating every note as having equal musical importance, and thereby escape connotations of tonality that had previously defined Western classical music.

Consider also 7-EDO: there is not a single triadic sonority within the EDO that is concordant enough to plausibly be conflated with Just Intonation, and attempts to describe its harmonic structures in terms of Just ratios is often more confusing than it is illuminating. It is not impossible to treat 7-EDO as an equal temperament, but the question arises of what is being gained in the process. One possible answer to that is that 7-EDO treated as a temperament, even though it is not actually used as one, is basic to Western musical theory. One step is a tone, two steps a third, three steps a fourth, four steps a fifth, five steps a sixth, six steps a seventh, and seven steps an octave. These can be major, minor, diminished or augmented, which are all the same to 7-EDO. It is understood that the perfect octave is a 2, and the perfect fifth must approximate 3/2; if the perfect major third approximates 5/4 then we have the <7 11 16| val of 7-EDO as a temperament lying behind the terminology of Western music. While true, this is also not really relevant to the use of 7-EDO itself as a musical scale.

Consider also 9-EDO; this has nearly pure intervals of 7/6 and 12/7, so close (one fifth of a cent) that they really cannot be heard as other than JI. However that is not enough to give 9-EDO the overall character of approximate JI. Nor does the fact that it possesses the same 400 cent major thirds as 12-EDO really do it, and the attempt to hear 667 cents as a fifth is at best marginally successful. What we find is a peculiar hybrid, a chimera neither fish nor fowl.

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While the equal temperament paradigm is firmly established, with its benefits clearly understood, little has been done to establish a competitive EDO paradigm. Ivor Darreg is probably the first composer and writer to discuss EDOs without any explicit reference to their proximity to JI. Easley Blackwood (in his microtonal work) seems to have ignored or been otherwise ignorant of Just Intonation and its relationship to EDOs, instead focusing on the "recognizably diatonic" properties held by each EDO he worked with. Various other microtonal composers have, in recent years, produced compositions in various EDOs without making any conscious musical references to JI, but the question may still be asked: what is gained in this approach, that is not present in the ET paradigm?

A common JI-agnostic way of looking at edos is by looking at which [[MOS scale|mosses]] they contain. Small edos especially have limited palettes of these scales, which can help define an edo's sound. Any particular mos pattern (e.g. [[5L 3s]]) is present in multiple edos, but a mos's [[step ratio]] can be different in different edos. The step ratio of a mos has a clear effect on its sound, an effect not defined by any JI interpretation. Using the [[TAMNAMS]] system for naming mosses and their step ratios, you can list some of the characteristic mos sounds of an edo. For example, [[26edo]] contains supersoft [[5L 2s|diatonic]], basic [[5L 3s|oneirotonic]], and semihard [[4L 3s|smitonic]].

Choosing to not use RTT to interpret an edo's intervals doesn't mean you have to ignore the effect of JI entirely. For example, 5edo's perfect fifth is about 18 cents sharp of 3/2, and its proximity to this JI interval is very important to its sound. 3\5 sounds the way it does both because it's close to 3/2 and because it's quite sharp of 3/2. An RTT approach can view 3\5 as an imperfect representation of 3/2, but a more sound-based approach can view it as an interval with a unique character, whose sound is informed by its proximity to JI. You can also look at JI proximity while rejecting the linearity of RTT. You may consider 3\5 to be a tweaked version of 3/2, but not consider 1\5 to be related to 9/8. RTT would say that the interval formed by stacking two 3/2 approximations must "represent" 9/8.

One common feature discussions of EDOs outside the ET paradigm is the concept of "mood", first used in the writings of Ivor Darreg. An EDO's mood is typically described as a subjective and unquantifiable property that can only be ascertained through listening to and composing in the EDO. The moods of EDOs are often described rather prosaically, using terms such as "bright", "narcotic", "spicy", "mysterious", "strident", or "strong". In the absence of the objective quantitative dimension of the ET paradigm, such subjective qualitative approaches are really the only possible way to discuss EDOs. One of the benefits of this is that it avoids creating a hierarchy of EDOs, and thus avoids creating a cognitive bias against the use of certain EDOs.

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 However, it is not always true that those using EDOs are interested in approximating JI, nor is it true that describing all EDOs in terms of approximate JI is universally helpful or illuminating. As an example of the former, consider the atonalists, a loose school of 20th-century composers who sought to embrace the "equality" of equal temperament by treating every note as having equal musical importance, and thereby escape connotations of tonality that had previously defined Western classical music.
-Consider also 7-EDO: there is not a single triadic sonority within the EDO that is concordant enough to plausibly be conflated with Just Intonation, and attempts to describe its harmonic structures in terms of Just ratios is often more confusing than it is illuminating. It is not impossible to treat 7-EDO as an equal temperament, but the question  arises of what is being gained in the process. One possible answer to that is that 7-EDO treated as a temperament, even though it is not actually used as one, is basic to Western musical theory. One step is a tone, two steps a third, three steps a fourth, four steps a fifth, five steps a sixth, six steps a seventh, and seven steps an octave. These can be major, minor, diminished or augmented, which are all the same to 7-EDO. It is understood that the perfect octave is a 2, and the perfect fifth must approximate 3/2; if the perfect major third approximates 5/4 then we have the &lt;7 11 16| val of 7-EDO as a temperament lying behind the terminology of Western music. While true, this is also not really relevant to the use of 7-EDO itself as a musical scale.
+Consider also 7-EDO: there is not a single triadic sonority within the EDO that is concordant enough to plausibly be conflated with Just Intonation, and attempts to describe its harmonic structures in terms of Just ratios is often more confusing than it is illuminating. It is not impossible to treat 7-EDO as an equal temperament, but the question arises of what is being gained in the process. One possible answer to that is that 7-EDO treated as a temperament, even though it is not actually used as one, is basic to Western musical theory. One step is a tone, two steps a third, three steps a fourth, four steps a fifth, five steps a sixth, six steps a seventh, and seven steps an octave. These can be major, minor, diminished or augmented, which are all the same to 7-EDO. It is understood that the perfect octave is a 2, and the perfect fifth must approximate 3/2; if the perfect major third approximates 5/4 then we have the &lt;7 11 16| val of 7-EDO as a temperament lying behind the terminology of Western music. While true, this is also not really relevant to the use of 7-EDO itself as a musical scale.
 Consider also 9-EDO; this has nearly pure intervals of 7/6 and 12/7, so close (one fifth of a cent) that they really cannot be heard as other than JI. However that is not enough to give 9-EDO the overall character of approximate JI. Nor does the fact that it possesses the same 400 cent major thirds as 12-EDO really do it, and the attempt to hear 667 cents as a fifth is at best marginally successful. What we find is a peculiar hybrid, a chimera neither fish nor fowl.
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 While the equal temperament paradigm is firmly established, with its benefits clearly understood, little has been done to establish a competitive EDO paradigm. Ivor Darreg is probably the first composer and writer to discuss EDOs without any explicit reference to their proximity to JI. Easley Blackwood (in his microtonal work) seems to have ignored or been otherwise ignorant of Just Intonation and its relationship to EDOs, instead focusing on the "recognizably diatonic" properties held by each EDO he worked with. Various other microtonal composers have, in recent years, produced compositions in various EDOs without making any conscious musical references to JI, but the question may still be asked: what is gained in this approach, that is not present in the ET paradigm?
+A common JI-agnostic way of looking at edos is by looking at which [[MOS scale|mosses]] they contain. Small edos especially have limited palettes of these scales, which can help define an edo's sound. Any particular mos pattern (e.g. [[5L 3s]]) is present in multiple edos, but a mos's [[step ratio]] can be different in different edos. The step ratio of a mos has a clear effect on its sound, an effect not defined by any JI interpretation. Using the [[TAMNAMS]] system for naming mosses and their step ratios, you can list some of the characteristic mos sounds of an edo. For example, [[26edo]] contains supersoft [[5L 2s|diatonic]], basic [[5L 3s|oneirotonic]], and semihard [[4L 3s|smitonic]].
+Choosing to not use RTT to interpret an edo's intervals doesn't mean you have to ignore the effect of JI entirely. For example, 5edo's perfect fifth is about 18 cents sharp of 3/2, and its proximity to this JI interval is very important to its sound. 3\5 sounds the way it does both because it's close to 3/2 and because it's quite sharp of 3/2. An RTT approach can view 3\5 as an imperfect representation of 3/2, but a more sound-based approach can view it as an interval with a unique character, whose sound is informed by its proximity to JI. You can also look at JI proximity while rejecting the linearity of RTT. You may consider 3\5 to be a tweaked version of 3/2, but not consider 1\5 to be related to 9/8. RTT would say that the interval formed by stacking two 3/2 approximations must "represent" 9/8.
 One common feature discussions of EDOs outside the ET paradigm is the concept of "mood", first used in the writings of Ivor Darreg. An EDO's mood is typically described as a subjective and unquantifiable property that can only be ascertained through listening to and composing in the EDO. The moods of EDOs are often described rather prosaically, using terms such as "bright", "narcotic", "spicy", "mysterious", "strident", or "strong". In the absence of the objective quantitative dimension of the ET paradigm, such subjective qualitative approaches are really the only possible way to discuss EDOs. One of the benefits of this is that it avoids creating a hierarchy of EDOs, and thus avoids creating a cognitive bias against the use of certain EDOs.