User:Eboone/EDO Impressions: Difference between revisions

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The first draft of everything up to 20edo is now complete, with the 20edo and 22edo demos also currently available. Tier placements of 20-29edo are also accurate, and all tier placements after that are outdated.

~~There is currently only one~~ overarching ~~point~~, ~~and my next task is to add more.~~ I will ~~then~~ go back and edit what I've already written thus far, making sure to maximize the efficiency of the reading experience.

I may add more overarching points next, then I will go back and edit what I've already written thus far, making sure to maximize the efficiency of the reading experience. The 17edo vs 19edo smackdown may also be completely switched up in favor of 19edo, in hindsight.

''Start date: May 13, 2024. Most recent update: June 8, 2024. '''26 days''' have been spent.''

''Start date: May 13, 2024. Most recent update: June 10, 2024. '''28 days''' have been spent.''

—————————[[File:EDO Tier List.png|alt=A tier list of EDOs. S Tier - 12, 34, 31, 53, 72, 50, 22, 17, 15, 26, 19. A Tier - 27, 10, 46, 21, 20, 7, 16, 41, 55. B Tier - 18, 24, 36, 48, 65, 58, 25, 63, 13, 11, 56, 29, 23, 9, 5, 8. C Tier - 40, 43, 28, 14, 70, 68, 57, 37, 30, 60, 54, 49, 47, 44, 38, 67, 59, 35, 6. F Tier - 52, 62, 66, 71, 64, 39, 51, 61, 32, 39, 42, 45, 33.|thumb|['''''OUTDATED''', will be updated when I'm finished with this page''] Ebooone's tier list of edos 5-72]]

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== Overarching Points ==

After working on this project for about a month, I realized there were many points I found myself repeating. This isn't inherently an issue, but I thought it would make more sense if I sectioned off a few points that apply to multiple edos. I will be referencing these points directly in the descriptions of many edos, so it may be wise to '''read them before continuing'''.

=== Approximating [[Just intonation|Just Intonation]] ===

My introduction to microtonality was the '''[[harmonic series]]'''. One of the properties of the harmonic series is that intervals found closer to the fundamental tend to be more '''resonant''', i.e. they are '''consonant''' by definition. Consonance is a very desirable property for any given temperament to have, as it is the basis for making music sound "pleasant." Of course, there are plenty of mathematically consonant intervals that can be treated as dissonances — such as [[7/5]] and [[16/15]] — but even these intervals have that unmistakable resonant quality that makes them "easier" to listen to.

Keep in mind, I'm using quotation marks a lot because all of this is largely subjective. If you want to use extreme intervals with no shred of consonance, feel free to do that. It's just generally desirable to have access to consonant intervals because of their beauty and versatility.

Intervals of certain [[Prime limit|'''prime limits''']] have their own flavors of resonance, like how [[5/4]] and [[6/5]] "lock in" in a similar fashion that is recognizable as 5-limit resonance. This, among other reasons, is why it is useful to analyze edos in terms of their abilities to approximate certain prime limits.

I say "approximate" because it is impossible for any [[regular temperament]] to perfectly represent more than one prime limit exactly (due to the fact that all prime numbers are coprime with each other). In the case of edos, the only prime limit that is uncompromised is the [[2-limit]], which grants the substantial ability to make use of perfect octaves. Maintaining the perfect octave is seen as a first priority in most cases due to the concept of [[octave equivalence]], so any other prime limit can only ever be approximated when using edos.

Lower prime limits are generally seen as more important, especially the 3-limit. The perfect fifth is absolutely paramount to the structure of Western music because 1) it is very well approximated in 12edo and 2) it is the [[generator]] for the [[diatonic scale]]. So, edos with poor approximations of the 3rd harmonic are generally at a disadvantage in terms of versatility and resonance. 5-limit systems are also very desirable, as are 7-limit, 11-limit, and 13-limit systems. I don't usually care for an edo's ability to approximate anything past that.

In short, edos with more consonant intervals are at a distinct advantage compared to other edos. That certainly doesn't mean there won't be any exceptions, but it is true in many cases.

=== Multiples of EDOs ===

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==== Fewer Notes, Easier ====

This isn't always the case, but it generally is. One of the primary appeals of edos in general is their convenience and versatility, especially in comparison to just intonation. The extreme end of this is very small edos, and as you venture into larger edos, their compactness becomes less important as they strive to be closer to [[just intonation]]. Essentially, a small edo that works very well in a certain JI subset (such as 12edo) gains less from being multiplied, as the amount of new notes isn't justified by a newfound ability to approximate JI.

This isn't always the case, but it generally is. One of the primary appeals of edos in general is their convenience and versatility, especially in comparison to just intonation. The extreme end of this is very small edos, and as you venture into larger edos, their compactness becomes less important as they strive to be closer to just intonation. Essentially, a small edo that works very well in a certain JI subset (such as 12edo) gains less from being multiplied, as the amount of new notes isn't justified by a newfound ability to approximate JI.

For example, 72edo does a much better job in the 2.3.5.7.11 subgroup than 12edo, but now you have to deal with 72 notes. At that point, why not just use just intonation?* Smaller examples include 9edo and and 10edo being placed above 18edo and 20edo respectively.

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==== The Novelty of Fewer ====

Yeah, I'm literally just saying smaller edos are cooler (unless they're too small; see F tier). I mean, think about it. Equally dividing the octave seems to clash with the logarithmic nature of the [[harmonic series]], so the fact that this even works at all is already cool enough. Yet, all you need are 12 equal partitions to completely dominate the music industry. Sure, your ears may not care about the novelty factor of using a small number of notes within the uncountably infinite realm of pitch, but conceptually I find it incredibly neat.

Yeah, I'm literally just saying smaller edos are cooler (unless they're too small; see F tier). I mean, think about it. Equally dividing the octave seems to clash with the logarithmic nature of the harmonic series, so the fact that this even works at all is already cool enough. Yet, all you need are 12 equal partitions to completely dominate the music industry. Sure, your ears may not care about the novelty factor of using a small number of notes within the uncountably infinite realm of pitch, but conceptually I find it incredibly neat.

5edo is a great example. Like, really? Only 5? But it sounds so cute! Then there's 7edo, a surprisingly versatile temperament that often gets me thinking "wait... there's only 7 notes here?" It's kind of like my brain is drawing parallels from the idea of simple ratios being consonant. In a way, small edos are "consonant" and large edos are "dissonant" in terms of their conception.

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''Tier: '''F+'''''

[[File:3edo groove.mp3|none|thumb|3edo foggy ambience on D augmented]]

This edo is just an '''augmented triad''', but specifically one that is derived by stacking three 400¢ major thirds. This means it closes the octave and, as such, is its own inversion. So, once again, the melodic and harmonic capabilities of this triad in isolation are pretty slim compared to, say, one derived by stacking [[5/4]]. Yet, the symmetrical nature of this chord allows for quite a "sturdy" sound that helps to keep the temperament together.

This edo is just an '''augmented triad''', but specifically one that is derived by stacking three 400¢ major thirds. This means it closes the octave and, as such, is its own inversion. So, once again, the melodic and harmonic capabilities of this triad in isolation are pretty slim compared to, say, one derived by stacking 5/4. Yet, the symmetrical nature of this chord allows for quite a "sturdy" sound that helps to keep the temperament together.

The major third itself is 13.79¢ sharp of 5/4, enough for noticeable inharmonicity to occasionally take place in timbres with a prominent 5th harmonic (such as a piano). This contributes to the "foggy" sound of the temperament, as the demo illustrates.

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''Tier: '''F+'''''

[[File:4edo groove.mp3|none|thumb|4edo fanfare on B diminished]]

While I wouldn't necessarily say 2edo is a better version of 1edo, I will say 4edo is a better version of 2edo. It's a '''diminished tetrad''', but specifically one that is derived by stacking four 300¢ minor thirds. This means it closes the octave and, as such, is its own inversion. So, as always, the melodic and harmonic capabilities of this tetrad in isolation are pretty slim compared to, say, one derived by stacking [[6/5]]. Yet, the symmetrical nature of this chord allows for quite a, dare I say, "sturdy" sound that helps to keep the temperament together... how familiar.

While I wouldn't necessarily say 2edo is a better version of 1edo, I will say 4edo is a better version of 2edo. It's a '''diminished tetrad''', but specifically one that is derived by stacking four 300¢ minor thirds. This means it closes the octave and, as such, is its own inversion. So, as always, the melodic and harmonic capabilities of this tetrad in isolation are pretty slim compared to, say, one derived by stacking 6/5. Yet, the symmetrical nature of this chord allows for quite a, dare I say, "sturdy" sound that helps to keep the temperament together... how familiar.

The minor third is sat between 6/5 and [[32/27]], giving it a mildly dark sound that is quite apt for the diminished tetrad. It's a powerful minor third, but without the 5-limit resonance of 6/5. Overall, a great minor third.

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Ah yes, the homeland of [[Mosh|'''mosh''']]. 10edo is actually my favorite of the edos less than 12, although 9edo does come pretty close. This edo is the largest one with a singular third, that being the 360¢ submajor third which is extremely close to [[16/13]]. That means this temperament also features a stunning approximation of the [[13/8|'''13th harmonic''']], coming in just 0.53¢ sharp. It also features all of the interseptimal intervals from 5edo, meaning it includes the same approximation of the 7th harmonic, as well as a relatively stable fifth (especially compared to 9edo).

10edo is a very xen tuning — more so than even 7edo in my opinion — because it doesn't even pretend to represent the [[diatonic scale]] (5L 2s). Instead, as mentioned previously, the heptatonic [[MOS scale|'''mos''']] of 10edo is 3L 4s, or mosh. This is an incredibly intriguing mos that can be difficult to wrap your head around, but the sounds it produces are quite beautiful. [https://www.youtube.com/watch?v=uOsrbFEC6bQ ''Apartment in the Sky''] by [[HEHEHE I AM A SUPAHSTAR SAGA]] is an example of the brightest mode of mosh, dril.

10edo is a very xen tuning — more so than even 7edo in my opinion — because it doesn't even pretend to represent the diatonic scale (5L 2s). Instead, as mentioned previously, the heptatonic [[MOS scale|'''mos''']] of 10edo is 3L 4s, or mosh. This is an incredibly intriguing mos that can be difficult to wrap your head around, but the sounds it produces are quite beautiful. [https://www.youtube.com/watch?v=uOsrbFEC6bQ ''Apartment in the Sky''] by [[HEHEHE I AM A SUPAHSTAR SAGA]] is an example of the brightest mode of mosh, dril.

The demo I wrote for this edo is in the middle mode, bish. It oscilates between "tonic" and "dominant," and it sounds quite reminiscent of such movement in the diatonic mos, until you realize it's totally different. This mode of mosh could be mapped to the [[wikipedia:Double_harmonic_scale|double harmonic major scale]] in a diatonic system, yet doing that results in a very different sound. I was incredibly surprised when I retuned the demo to be in A♭ double harmonic major, only to be subjected to something the original could only dream of being as weird as. Of course, I probably only say that because I wasn't used to it.

@@ Line 3: / Line 3: @@
 The first draft of everything up to 20edo is now complete, with the 20edo and 22edo demos also currently available. Tier placements of 20-29edo are also accurate, and all tier placements after that are outdated.
-There is currently only one overarching point, and my next task is to add more. I will then go back and edit what I've already written thus far, making sure to maximize the efficiency of the reading experience.
+I may add more overarching points next, then I will go back and edit what I've already written thus far, making sure to maximize the efficiency of the reading experience. The 17edo vs 19edo smackdown may also be completely switched up in favor of 19edo, in hindsight.
-''Start date: May 13, 2024. Most recent update: June 8, 2024. '''26 days''' have been spent.''
+''Start date: May 13, 2024. Most recent update: June 10, 2024. '''28 days''' have been spent.''
 —————————[[File:EDO Tier List.png|alt=A tier list of EDOs. S Tier - 12, 34, 31, 53, 72, 50, 22, 17, 15, 26, 19. A Tier - 27, 10, 46, 21, 20, 7, 16, 41, 55. B Tier - 18, 24, 36, 48, 65, 58, 25, 63, 13, 11, 56, 29, 23, 9, 5, 8. C Tier - 40, 43, 28, 14, 70, 68, 57, 37, 30, 60, 54, 49, 47, 44, 38, 67, 59, 35, 6. F Tier - 52, 62, 66, 71, 64, 39, 51, 61, 32, 39, 42, 45, 33.|thumb|['''''OUTDATED''', will be updated when I'm finished with this page''] Ebooone's tier list of edos 5-72]]
@@ Line 39: / Line 39: @@
 == Overarching Points ==
 After working on this project for about a month, I realized there were many points I found myself repeating. This isn't inherently an issue, but I thought it would make more sense if I sectioned off a few points that apply to multiple edos. I will be referencing these points directly in the descriptions of many edos, so it may be wise to '''read them before continuing'''.
+=== Approximating [[Just intonation|Just Intonation]] ===
+My introduction to microtonality was the '''[[harmonic series]]'''. One of the properties of the harmonic series is that intervals found closer to the fundamental tend to be more '''resonant''', i.e. they are '''consonant''' by definition. Consonance is a very desirable property for any given temperament to have, as it is the basis for making music sound "pleasant." Of course, there are plenty of mathematically consonant intervals that can be treated as dissonances — such as [[7/5]] and [[16/15]] — but even these intervals have that unmistakable resonant quality that makes them "easier" to listen to.
+Keep in mind, I'm using quotation marks a lot because all of this is largely subjective. If you want to use extreme intervals with no shred of consonance, feel free to do that. It's just generally desirable to have access to consonant intervals because of their beauty and versatility.
+Intervals of certain [[Prime limit|'''prime limits''']] have their own flavors of resonance, like how [[5/4]] and [[6/5]] "lock in" in a similar fashion that is recognizable as 5-limit resonance. This, among other reasons, is why it is useful to analyze edos in terms of their abilities to approximate certain prime limits.
+I say "approximate" because it is impossible for any [[regular temperament]] to perfectly represent more than one prime limit exactly (due to the fact that all prime numbers are coprime with each other). In the case of edos, the only prime limit that is uncompromised is the [[2-limit]], which grants the substantial ability to make use of perfect octaves. Maintaining the perfect octave is seen as a first priority in most cases due to the concept of [[octave equivalence]], so any other prime limit can only ever be approximated when using edos.
+Lower prime limits are generally seen as more important, especially the 3-limit. The perfect fifth is absolutely paramount to the structure of Western music because 1) it is very well approximated in 12edo and 2) it is the [[generator]] for the [[diatonic scale]]. So, edos with poor approximations of the 3rd harmonic are generally at a disadvantage in terms of versatility and resonance. 5-limit systems are also very desirable, as are 7-limit, 11-limit, and 13-limit systems. I don't usually care for an edo's ability to approximate anything past that.
+In short, edos with more consonant intervals are at a distinct advantage compared to other edos. That certainly doesn't mean there won't be any exceptions, but it is true in many cases.
 === Multiples of EDOs ===
@@ Line 44: / Line 57: @@
 ==== Fewer Notes, Easier ====
-This isn't always the case, but it generally is. One of the primary appeals of edos in general is their convenience and versatility, especially in comparison to just intonation. The extreme end of this is very small edos, and as you venture into larger edos, their compactness becomes less important as they strive to be closer to [[just intonation]]. Essentially, a small edo that works very well in a certain JI subset (such as 12edo) gains less from being multiplied, as the amount of new notes isn't justified by a newfound ability to approximate JI.
+This isn't always the case, but it generally is. One of the primary appeals of edos in general is their convenience and versatility, especially in comparison to just intonation. The extreme end of this is very small edos, and as you venture into larger edos, their compactness becomes less important as they strive to be closer to just intonation. Essentially, a small edo that works very well in a certain JI subset (such as 12edo) gains less from being multiplied, as the amount of new notes isn't justified by a newfound ability to approximate JI.
 For example, 72edo does a much better job in the 2.3.5.7.11 subgroup than 12edo, but now you have to deal with 72 notes. At that point, why not just use just intonation?* Smaller examples include 9edo and and 10edo being placed above 18edo and 20edo respectively.
@@ Line 51: / Line 64: @@
 ==== The Novelty of Fewer ====
-Yeah, I'm literally just saying smaller edos are cooler (unless they're too small; see F tier). I mean, think about it. Equally dividing the octave seems to clash with the logarithmic nature of the [[harmonic series]], so the fact that this even works at all is already cool enough. Yet, all you need are 12 equal partitions to completely dominate the music industry. Sure, your ears may not care about the novelty factor of using a small number of notes within the uncountably infinite realm of pitch, but conceptually I find it incredibly neat.
+Yeah, I'm literally just saying smaller edos are cooler (unless they're too small; see F tier). I mean, think about it. Equally dividing the octave seems to clash with the logarithmic nature of the harmonic series, so the fact that this even works at all is already cool enough. Yet, all you need are 12 equal partitions to completely dominate the music industry. Sure, your ears may not care about the novelty factor of using a small number of notes within the uncountably infinite realm of pitch, but conceptually I find it incredibly neat.
 edo is a great example. Like, really? Only 5? But it sounds so cute! Then there's 7edo, a surprisingly versatile temperament that often gets me thinking "wait... there's only 7 notes here?" It's kind of like my brain is drawing parallels from the idea of simple ratios being consonant. In a way, small edos are "consonant" and large edos are "dissonant" in terms of their conception.
@@ Line 90: / Line 103: @@
 ''Tier: '''F+'''''
 [[File:3edo groove.mp3|none|thumb|3edo foggy ambience on D augmented]]
-This edo is just an '''augmented triad''', but specifically one that is derived by stacking three 400¢ major thirds. This means it closes the octave and, as such, is its own inversion. So, once again, the melodic and harmonic capabilities of this triad in isolation are pretty slim compared to, say, one derived by stacking [[5/4]]. Yet, the symmetrical nature of this chord allows for quite a "sturdy" sound that helps to keep the temperament together.
+This edo is just an '''augmented triad''', but specifically one that is derived by stacking three 400¢ major thirds. This means it closes the octave and, as such, is its own inversion. So, once again, the melodic and harmonic capabilities of this triad in isolation are pretty slim compared to, say, one derived by stacking 5/4. Yet, the symmetrical nature of this chord allows for quite a "sturdy" sound that helps to keep the temperament together.
 The major third itself is 13.79¢ sharp of 5/4, enough for noticeable inharmonicity to occasionally take place in timbres with a prominent 5th harmonic (such as a piano). This contributes to the "foggy" sound of the temperament, as the demo illustrates.
@@ Line 103: / Line 116: @@
 ''Tier: '''F+'''''
 [[File:4edo groove.mp3|none|thumb|4edo fanfare on B diminished]]
-While I wouldn't necessarily say 2edo is a better version of 1edo, I will say 4edo is a better version of 2edo. It's a '''diminished tetrad''', but specifically one that is derived by stacking four 300¢ minor thirds. This means it closes the octave and, as such, is its own inversion. So, as always, the melodic and harmonic capabilities of this tetrad in isolation are pretty slim compared to, say, one derived by stacking [[6/5]]. Yet, the symmetrical nature of this chord allows for quite a, dare I say, "sturdy" sound that helps to keep the temperament together... how familiar.
+While I wouldn't necessarily say 2edo is a better version of 1edo, I will say 4edo is a better version of 2edo. It's a '''diminished tetrad''', but specifically one that is derived by stacking four 300¢ minor thirds. This means it closes the octave and, as such, is its own inversion. So, as always, the melodic and harmonic capabilities of this tetrad in isolation are pretty slim compared to, say, one derived by stacking 6/5. Yet, the symmetrical nature of this chord allows for quite a, dare I say, "sturdy" sound that helps to keep the temperament together... how familiar.
 The minor third is sat between 6/5 and [[32/27]], giving it a mildly dark sound that is quite apt for the diminished tetrad. It's a powerful minor third, but without the 5-limit resonance of 6/5. Overall, a great minor third.
@@ Line 197: / Line 210: @@
 Ah yes, the homeland of [[Mosh|'''mosh''']]. 10edo is actually my favorite of the edos less than 12, although 9edo does come pretty close. This edo is the largest one with a singular third, that being the 360¢ submajor third which is extremely close to [[16/13]]. That means this temperament also features a stunning approximation of the [[13/8|'''13th harmonic''']], coming in just 0.53¢ sharp. It also features all of the interseptimal intervals from 5edo, meaning it includes the same approximation of the 7th harmonic, as well as a relatively stable fifth (especially compared to 9edo).
-edo is a very xen tuning — more so than even 7edo in my opinion — because it doesn't even pretend to represent the [[diatonic scale]] (5L 2s). Instead, as mentioned previously, the heptatonic [[MOS scale|'''mos''']] of 10edo is 3L 4s, or mosh. This is an incredibly intriguing mos that can be difficult to wrap your head around, but the sounds it produces are quite beautiful. [https://www.youtube.com/watch?v=uOsrbFEC6bQ ''Apartment in the Sky''] by [[HEHEHE I AM A SUPAHSTAR SAGA]] is an example of the brightest mode of mosh, dril.
+edo is a very xen tuning — more so than even 7edo in my opinion — because it doesn't even pretend to represent the diatonic scale (5L 2s). Instead, as mentioned previously, the heptatonic [[MOS scale|'''mos''']] of 10edo is 3L 4s, or mosh. This is an incredibly intriguing mos that can be difficult to wrap your head around, but the sounds it produces are quite beautiful. [https://www.youtube.com/watch?v=uOsrbFEC6bQ ''Apartment in the Sky''] by [[HEHEHE I AM A SUPAHSTAR SAGA]] is an example of the brightest mode of mosh, dril.
 The demo I wrote for this edo is in the middle mode, bish. It oscilates between "tonic" and "dominant," and it sounds quite reminiscent of such movement in the diatonic mos, until you realize it's totally different. This mode of mosh could be mapped to the [[wikipedia:Double_harmonic_scale|double harmonic major scale]] in a diatonic system, yet doing that results in a very different sound. I was incredibly surprised when I retuned the demo to be in A♭ double harmonic major, only to be subjected to something the original could only dream of being as weird as. Of course, I probably only say that because I wasn't used to it.