User:Eufalesio/Harmonic tiers: Difference between revisions

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Harmonics are an extremely important basis to judge the efficacy of a tuning system, and though there are infinite harmonics, some are more important than others. How would one determine such rating? And which harmonics merit a rating?

One way is to determine some sort of harmonic uniqueness. Since harmonics are ~~essentially~~ integer frequency multiples, primes fit this category perfectly, as any composite harmonic can be uniquely built by a combination of prime harmonics. ~~This is an extension of~~ the '''Fundamental Theorem of Arithmetic''', applied to music. I ~~will~~

One way is to determine some sort of harmonic uniqueness. Since harmonics are integer frequency multiples, primes fit this category perfectly, as any composite harmonic can be uniquely built by a combination of prime harmonics. Basically, the '''Fundamental Theorem of Arithmetic''' applied to music. Here is how '''I''' would rate primes. You might not agree, but you're also not me.

== ~~Prime ratings~~ ==

== P-tier rating ==

This rating will deal predominantly with primes as a whole, not just the odds that appear.

This rating will deal predominantly with primes as a whole, not just the odds that appear - sorted by tiers. The tier is dependent on the denominator the prime octave-reduces to, starting with the octave at Tier 0... but that's a trivial category so I put it in Tier I.

=== P-Tier I ===

There are two members here: Primes 2 and 3, ~~octave reducing to...~~ the [[2/1|octave]]~~...~~ and the [[3/2|fifth]]~~. Technically speaking, octaves would belong to Tier 0, but that's a trivial category~~.

There are two members here: Primes 2 and 3, the [[2/1|octave]] and the octave-reduced [[3/2|just fifth]].

This tier is built from primes that are extremely important. A tuning system will very likely have these primes and tune them extremely precise, if not exact; for example, all edos by definition tune the octave perfectly, and edos that tune the fifth accurately are also very important~~, or use stack them to build all other intervals, as is the case with many rank-2 temperaments and historical temperaments~~. These are ~~the only~~ primes ~~can stack easily with everything else~~.

This tier is built from primes that are extremely important. A tuning system will very likely have these primes and tune them extremely precise, if not exact; for example, all edos by definition tune the octave perfectly, and edos that tune the fifth accurately are also very important. These are very stackable primes.

* Prime 2: So important that you only care about it when you work without it. Some people, like Aura, treat it differently as the rest of primes, in that it adds ''luminosity'' to an interval class, instead of being treated as its own interval class. I treat it as its own interval class, distinct from the unison (the fundamental). Melodically, it adds a lot of punch to melodies, but it should only be used once.

* Prime 2: So important that you only care about it when you work without it. Some people, like [[Aura]], treat it differently as the rest of primes, in that it adds ''luminosity'' to an interval class, instead of being treated as its own interval class. I treat it as its own interval class, distinct from the unison (the fundamental). Melodically, it adds a lot of punch to melodies, but it should only be used once.

* Prime 3: Extremely important. It's the first prime that adds sonic variety of any kind, and with octaves, it builds an infinite universe of ~~low complexity~~ called the 3-limit. Melodically, it makes for an incredible leap to drive momentum, and when stacked and octave reduced, it makes for incredible "low complexity" steps, such as 256/243 and 9/8.

* Prime 3: Extremely important. It's the first prime that adds sonic variety of any kind, and with octaves, it builds an infinite universe of called the 3-limit. Melodically, it makes for an incredible leap to drive momentum, and when stacked and octave reduced, it makes for incredible "low complexity" steps, such as 256/243 and 9/8.

=== P-Tier II ===

There are two members here: Primes 5 and 7, octave reducing to the [[5/4|just major third]] and the [[7/4|septimal harmonic seventh]].

This tier is built from primes that are quite important. A tuning system will probably make an effort to tune ~~this~~ accurately, but it's not a necessity. However, these primes add ''flavor'' to music. At least, that's what I think. Despite being ~~very~~ bad for building scales, they combine extremely well with each other.

This tier is built from primes that are quite important. A tuning system will probably make an effort to tune them accurately, but it's not a necessity. However, these primes add ''flavor'' to music. At least, that's what I think. Despite being kinda bad for building scales (horrible in case of 5), they combine extremely well with each other.

* Prime 5: Important mostly due to tradition, as it sits right beneath the unison and the fifth and makes for some sweet sweet 4:5:6 chords. It leaves a good taste on ~~the~~ mouth when using this interval to finish a cadence. Melodically, it is perfect for small leaps forward and it combines perfectly with 3. It allows many new flavors of tones and semitones, like 16/15, 25/24, 27/25, 135/128, 10/9... etc. Technically these already exist in the 3-limit, if you fudge the schisma.

* Prime 5: Important mostly due to tradition, as it sits right beneath the unison and the fifth and makes for some sweet sweet 4:5:6 chords. It leaves a good taste on my mouth when using this interval to finish a cadence. Melodically, it is perfect for small leaps forward and it combines perfectly with 3. It allows many new flavors of tones and semitones, like 16/15, 25/24, 27/25, 135/128, 10/9... etc. Technically these already exist in the 3-limit, if you fudge the schisma.

* Prime 7: Often overlooked in tradition, it's an interesting prime. A bit ~~alien~~, due to its notable ~~absence~~ in 12edo. Because its octave complement 8/7 is close to the unison, some people consider it a weird mix between consonance and dissonance. I personally consider it a consonance. Less than 5, but consonant nonetheless, specially in the chord 6:7:9. Melodically, it is a bit jarring, as it tends to create tiny intervals in scales ~~with plenty of~~ 7, namely 28/27 or 49/48~~, but it~~ can be well employed~~, specially~~ with less jarring limmoid steps like 21/20.

* Prime 7: Often overlooked in tradition, it's an interesting prime. A bit xenharmonic, due to its notable innacuracy in 12edo (which for some purposes... it may be just enough). Because its octave complement 8/7 is close to the unison, some people consider it a weird mix between consonance and dissonance. I personally consider it a consonance, but it makes a killer augmented sixth chord. Less than 5, but consonant nonetheless, specially in the chord 6:7:9. Melodically, it is a bit jarring, as it tends to create tiny intervals in scales based on 7, namely 28/27 or 49/48. It can still be well employed with less jarring limmoid steps like 21/20.

=== P-Tier III ===

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This tier is built from truly xenharmonic primes. Western music theory considers these harmonics truly alien, but I've grown so accustomed to these primes that they now feel natural. These primes are less consonant, but still can be treated as consonances.

Melodically, they are both the same. There are pairs of 11- and 13- limit intervals that sound alike, like 39/32~11/9, 27/22~16/13. The comma that equates both pairs is 352/351, so it can be safely fudged or tempered. They work extremely well melodically, giving a nice xenharmonic punch to any music rooted in the 5- or 7- limit.

Melodically, they are both basically the same. There are pairs of 11- and 13- limit intervals that sound alike, like 39/32~11/9, 27/22~16/13. The comma that equates both pairs is 352/351, so it can be safely fudged or tempered. They work extremely well melodically, giving a nice xenharmonic punch to any music rooted in the 5- or 7- limit.

* Prime 11: It's the first xenharmonic prime. Good flavor, but in my opinion, it's harder to use than 13. Since it can be considered a type of "fourth", "augmented fourth" or "half-augmented fourth" (depending on how precise or pedantic you want to be), I see it as a deviation from a very consonant interval: the fourth 4/3.

* Prime 11: It's the first truly xenharmonic prime. Good flavor, but in my opinion, it's harder to use than 13. Since it can be considered a type of "fourth", "augmented fourth" or "half-augmented fourth" (depending on how precise or pedantic you want to be), I see it as a deviation from a very consonant interval: the fourth 4/3, which is part of the reason why I don't like it as much. However, it adds a great otonal flavor from time to time.

* Prime 13: The second xenharmonic prime. Technically less consonant than 11, but it tends to be easier to work with. Part of this is the fact that it is extremely well approximated by 10n edos, and by Pythagorean tuning, placing it as the triple augmented fourth. It is also extremely easy to reach through the 7-limit too, connecting a chain of 5*7 to 13 through the schismina. The fact that it is a kind of sixth means that its interval class is not perfect, so it lends itself to readily accept xenharmonic (neutral) values. Might be just my bias, but I just really like it. And it also appears in the 12:13:14:15:16, which I really like too.

* Prime 13: The second xenharmonic prime. Technically less consonant than 11, but it tends to be easier to work with. Part of this is the fact that it is extremely well approximated by 10n edos, and by Pythagorean tuning placing it as the triple augmented fourth. It is also extremely easy to reach through the 7-limit too, connecting a chain of 5*7 to 13 through the schismina. The fact that it is a kind of sixth means that its interval class is not perfect, so it lends itself to readily accept xenharmonic (neutral) values. Might be just my bias, but I just really like it. And it also appears in the 12:13:14:15:16, which I really like too.

=== P-Tier IV ===

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It includes primes 17 and 19. They don't sound xenharmonic at all, as they are very well aproximated by 12edo.

* Prime 17: It's dissonant. It's good for making ~~locked-in harmonic~~ chords, but that's it. I don't like it harmonically. Melodically, works as ~~your typical diatonic~~ semitone, but I often just turn to 16/15 because it~~'s less complex~~.

* Prime 17: It's dissonant. It's good for making jazzy otonal chords, but that's it. I don't like it harmonically that much. Melodically, works as a 12edo-ish semitone, but I often just turn to 16/15 because I like it more.

* Prime 19: Despite being a very weak consonance, it is extremely close to a Pythagorean minor third, which automatically makes it very easy to use. It's great to build otonal minor chords and get somber feelings whilst being locked on the fundamental, and specially for ending cadences, for which 6/5 might make for a worse minor third. It combines great with 3 in the 16:19:24 chord. Melodically, it works the same as 5... so, great!

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It includes primes 23, 29, and 31, which sound quite xenharmonic.

* Prime 23: It's kind of a weird prime in my opinion. Since it sounds like a very sharp augmented fourth, but shares a similar consonance level to 19, so it should be in the same tier. ~~Its complexity makes~~ it ~~hard to~~ use. Melodically, I'd treat it like ~~an augmented fourth~~. ~~There are better options available, in my opinion~~.

* Prime 23: It's kind of a weird prime in my opinion. Since it sounds like a very sharp augmented fourth, but shares a similar consonance level to 19, it should be in the same tier. But I don't like it. The only time I'd use it is in a super wide otonal chord. Melodically, I'd treat it like a sharper 10/7, that is... a slightly sharper tritone.

* Prime 29: Despite being one of the less consonant primes, it makes for a nice supraminor flavor. It's very well approximated by 7n edos, so it also has that going for it. Melodically, I'd say it goes best when used in the tetrachord 24:26:29:32, serving as a more pungent, xenharmonic version of 15:16:18:20~~. The~~ interval 29/24 ~~is great for momentary supraminor flavors hidden in otonal passages~~.

* Prime 29: Despite being one of the less consonant primes, it makes for a nice supraminor flavor. Apart from that, since it is a kind of seventh, it accepts xenharmonic values more easily. It's very well approximated by 7n edos, so it also has that going for it. Melodically, I'd say it goes best when used in the tetrachord 24:26:29:32, serving as a more pungent, xenharmonic version of 15:16:18:20 (as the interval 29/24).

* Prime 31: It's arguably the most dissonant of all the primes seen so far. Its octave complement, 32/31 is also incredibly close to 33/32, so by fudging or tempering 1024/1023, one can get a good approximation of 31 by reducing it to the 11-limit, which at least makes it somewhat approachable, but its complexity makes it a very difficult prime to digest and work with. Melodically, it's the same as 64/33, so it is incredibly jarring. Its octave complement is no better either, being a quartertone.

* Prime 31: It's arguably the most dissonant of all the primes seen so far. Its octave complement, 32/31 is also incredibly close to 33/32, so by fudging or tempering 1024/1023, one can get a good approximation of 31 by reducing it to the 11-limit, which at least makes it somewhat approachable, but its integer complexity makes it a very difficult prime to digest and work with. Melodically, it's the same as 64/33, so it is incredibly jarring. Its octave complement is no better either, being a quartertone.

=== P-Tier V and beyond ===

There would theoretically be infinite tiers, each having exponentially more primes than the one before, but I really don't care for primes beyond 29. They become far too complex for my ears and I'm sure that their sonorities can be fudged into other less complex intervals. The only way they can be reliably and practically explored for most cases is by using the serendipitous 311edo, I'd say.

@@ Line 4: / Line 4: @@
 Harmonics are an extremely important basis to judge the efficacy of a tuning system, and though there are infinite harmonics, some are more important than others. How would one determine such rating? And which harmonics merit a rating?
-One way is to determine some sort of harmonic uniqueness. Since harmonics are essentially integer frequency multiples, primes fit this category perfectly, as any composite harmonic can be uniquely built by a combination of prime harmonics. This is an extension of the '''Fundamental Theorem of Arithmetic''', applied to music. I will
+One way is to determine some sort of harmonic uniqueness. Since harmonics are integer frequency multiples, primes fit this category perfectly, as any composite harmonic can be uniquely built by a combination of prime harmonics. Basically, the '''Fundamental Theorem of Arithmetic''' applied to music. Here is how '''I''' would rate primes. You might not agree, but you're also not me.
-== Prime ratings ==
+== P-tier rating ==
-This rating will deal predominantly with primes as a whole, not just the odds that appear.
+This rating will deal predominantly with primes as a whole, not just the odds that appear - sorted by tiers. The tier is dependent on the denominator the prime octave-reduces to, starting with the octave at Tier 0... but that's a trivial category so I put it in Tier I.
 === P-Tier I ===
-There are two members here: Primes 2 and 3, octave reducing to... the [[2/1|octave]]... and the [[3/2|fifth]]. Technically speaking, octaves would belong to Tier 0, but that's a trivial category.
+There are two members here: Primes 2 and 3, the [[2/1|octave]] and the octave-reduced [[3/2|just fifth]].
-This tier is built from primes that are extremely important. A tuning system will very likely have these primes and tune them extremely precise, if not exact; for example, all edos by definition tune the octave perfectly, and edos that tune the fifth accurately are also very important, or use stack them to build all other intervals, as is the case with many rank-2 temperaments and historical temperaments. These are the only primes can stack easily with everything else.
+This tier is built from primes that are extremely important. A tuning system will very likely have these primes and tune them extremely precise, if not exact; for example, all edos by definition tune the octave perfectly, and edos that tune the fifth accurately are also very important. These are very stackable primes.
-* <u>Prime 2:</u> So important that you only care about it when you work without it. Some people, like Aura, treat it differently as the rest of primes, in that it adds ''luminosity'' to an interval class, instead of being treated as its own interval class. I treat it as its own interval class, distinct from the unison (the fundamental).  Melodically, it adds a lot of punch to melodies, but it should only be used once.
+* <u>Prime 2:</u> So important that you only care about it when you work without it. Some people, like [[Aura]], treat it differently as the rest of primes, in that it adds ''luminosity'' to an interval class, instead of being treated as its own interval class. I treat it as its own interval class, distinct from the unison (the fundamental).  Melodically, it adds a lot of punch to melodies, but it should only be used once.
-* <u>Prime 3:</u> Extremely important. It's the first prime that adds sonic variety of any kind, and with octaves, it builds an infinite universe of low complexity called the 3-limit.  Melodically, it makes for an incredible leap to drive momentum, and when stacked and octave reduced, it makes for incredible "low complexity" steps, such as 256/243 and 9/8.
+* <u>Prime 3:</u> Extremely important. It's the first prime that adds sonic variety of any kind, and with octaves, it builds an infinite universe of called the 3-limit. Melodically, it makes for an incredible leap to drive momentum, and when stacked and octave reduced, it makes for incredible "low complexity" steps, such as 256/243 and 9/8.
 === P-Tier II ===
 There are two members here: Primes 5 and 7, octave reducing to the [[5/4|just major third]] and the [[7/4|septimal harmonic seventh]].
-This tier is built from primes that are quite important. A tuning system will probably make an effort to tune this accurately, but it's not a necessity. However, these primes add ''flavor'' to music. At least, that's what I think. Despite being very bad for building scales, they combine extremely well with each other.
+This tier is built from primes that are quite important. A tuning system will probably make an effort to tune them accurately, but it's not a necessity. However, these primes add ''flavor'' to music. At least, that's what I think. Despite being kinda bad for building scales (horrible in case of 5), they combine extremely well with each other.
-* <u>Prime 5:</u> Important mostly due to tradition, as it sits right beneath the unison and the fifth and makes for some sweet sweet 4:5:6 chords. It leaves a good taste on the mouth when using this interval to finish a cadence.  Melodically, it is perfect for small leaps forward and it combines perfectly with 3. It allows many new flavors of tones and semitones, like 16/15, 25/24, 27/25, 135/128, 10/9... etc. Technically these already exist in the 3-limit, if you fudge the schisma.
+* <u>Prime 5:</u> Important mostly due to tradition, as it sits right beneath the unison and the fifth and makes for some sweet sweet 4:5:6 chords. It leaves a good taste on my mouth when using this interval to finish a cadence. Melodically, it is perfect for small leaps forward and it combines perfectly with 3. It allows many new flavors of tones and semitones, like 16/15, 25/24, 27/25, 135/128, 10/9... etc. Technically these already exist in the 3-limit, if you fudge the schisma.
-* <u>Prime 7:</u> Often overlooked in tradition, it's an interesting prime. A bit alien, due to its notable absence in 12edo. Because its octave complement 8/7 is close to the unison, some people consider it a weird mix between consonance and dissonance. I personally consider it a consonance. Less than 5, but consonant nonetheless, specially in the chord 6:7:9. Melodically, it is a bit jarring, as it tends to create tiny intervals in scales with plenty of 7, namely 28/27 or 49/48, but it can be well employed, specially with less jarring limmoid steps like 21/20.
+* <u>Prime 7:</u> Often overlooked in tradition, it's an interesting prime. A bit xenharmonic, due to its notable innacuracy in 12edo (which for some purposes... it may be just enough). Because its octave complement 8/7 is close to the unison, some people consider it a weird mix between consonance and dissonance. I personally consider it a consonance, but it makes a killer augmented sixth chord. Less than 5, but consonant nonetheless, specially in the chord 6:7:9. Melodically, it is a bit jarring, as it tends to create tiny intervals in scales based on 7, namely 28/27 or 49/48. It can still be well employed with less jarring limmoid steps like 21/20.
 === P-Tier III ===
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 This tier is built from truly xenharmonic primes. Western music theory considers these harmonics truly alien, but I've grown so accustomed to these primes that they now feel natural. These primes are less consonant, but still can be treated as consonances.
-Melodically, they are both the same. There are pairs of 11- and 13- limit intervals that sound alike, like 39/32~11/9, 27/22~16/13. The comma that equates both pairs is 352/351, so it can be safely fudged or tempered. They work extremely well melodically, giving a nice xenharmonic punch to any music rooted in the 5- or 7- limit.
+Melodically, they are both basically the same. There are pairs of 11- and 13- limit intervals that sound alike, like 39/32~11/9, 27/22~16/13. The comma that equates both pairs is 352/351, so it can be safely fudged or tempered. They work extremely well melodically, giving a nice xenharmonic punch to any music rooted in the 5- or 7- limit.
-* <u>Prime 11</u>: It's the first xenharmonic prime. Good flavor, but in my opinion, it's harder to use than 13. Since it can be considered a type of "fourth", "augmented fourth" or "half-augmented fourth" (depending on how precise or pedantic you want to be), I see it as a deviation from a very consonant interval: the fourth 4/3.
+* <u>Prime 11</u>: It's the first truly xenharmonic prime. Good flavor, but in my opinion, it's harder to use than 13. Since it can be considered a type of "fourth", "augmented fourth" or "half-augmented fourth" (depending on how precise or pedantic you want to be), I see it as a deviation from a very consonant interval: the fourth 4/3, which is part of the reason why I don't like it as much. However, it adds a great otonal flavor from time to time.
-* <u>Prime 13</u>: The second xenharmonic prime. Technically less consonant than 11, but it tends to be easier to work with. Part of this is the fact that it is extremely well approximated by 10n edos, and by Pythagorean tuning, placing it as the triple augmented fourth. It is also extremely easy to reach through the 7-limit too, connecting a chain of 5*7 to 13 through the schismina. The fact that it is a kind of sixth means that its interval class is not perfect, so it lends itself to readily accept xenharmonic (neutral) values. Might be just my bias, but I just really like it. And it also appears in the 12:13:14:15:16, which I really like too.
+* <u>Prime 13</u>: The second xenharmonic prime. Technically less consonant than 11, but it tends to be easier to work with. Part of this is the fact that it is extremely well approximated by 10n edos, and by Pythagorean tuning placing it as the triple augmented fourth. It is also extremely easy to reach through the 7-limit too, connecting a chain of 5*7 to 13 through the schismina. The fact that it is a kind of sixth means that its interval class is not perfect, so it lends itself to readily accept xenharmonic (neutral) values. Might be just my bias, but I just really like it. And it also appears in the 12:13:14:15:16, which I really like too.
 === P-Tier IV ===
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 It includes primes 17 and 19. They don't sound xenharmonic at all, as they are very well aproximated by 12edo.
-* <u>Prime 17</u>: It's dissonant. It's good for making locked-in harmonic chords, but that's it. I don't like it harmonically.  Melodically, works as your typical diatonic semitone, but I often just turn to 16/15 because it's less complex.
+* <u>Prime 17</u>: It's dissonant. It's good for making jazzy otonal chords, but that's it. I don't like it harmonically that much. Melodically, works as a 12edo-ish semitone, but I often just turn to 16/15 because I like it more.
 * <u>Prime 19</u>: Despite being a very weak consonance, it is extremely close to a Pythagorean minor third, which automatically makes it very easy to use. It's great to build otonal minor chords and get somber feelings whilst being locked on the fundamental, and specially for ending cadences, for which 6/5 might make for a worse minor third. It combines great with 3 in the 16:19:24 chord. Melodically, it works the same as 5... so, great!
@@ Line 49: / Line 49: @@
 It includes primes 23, 29, and 31, which sound quite xenharmonic.
-* <u>Prime 23</u>: It's kind of a weird prime in my opinion. Since it sounds like a very sharp augmented fourth, but shares a similar consonance level to 19, so it should be in the same tier. Its complexity makes it hard to use.  Melodically, I'd treat it like an augmented fourth. There are better options available, in my opinion.
+* <u>Prime 23</u>: It's kind of a weird prime in my opinion. Since it sounds like a very sharp augmented fourth, but shares a similar consonance level to 19, it should be in the same tier. But I don't like it. The only time I'd use it is in a super wide otonal chord. Melodically, I'd treat it like a sharper 10/7, that is... a slightly sharper tritone.
-* <u>Prime 29</u>: Despite being one of the less consonant primes, it makes for a nice supraminor flavor. It's very well approximated by 7n edos, so it also has that going for it.  Melodically, I'd say it goes best when used in the tetrachord 24:26:29:32, serving as a more pungent, xenharmonic version of 15:16:18:20. The interval 29/24 is great for momentary supraminor flavors hidden in otonal passages.
+* <u>Prime 29</u>: Despite being one of the less consonant primes, it makes for a nice supraminor flavor. Apart from that, since it is a kind of seventh, it accepts xenharmonic values more easily. It's very well approximated by 7n edos, so it also has that going for it. Melodically, I'd say it goes best when used in the tetrachord 24:26:29:32, serving as a more pungent, xenharmonic version of 15:16:18:20 (as the interval 29/24).
-* <u>Prime 31</u>: It's arguably the most dissonant of all the primes seen so far. Its octave complement, 32/31 is also incredibly close to 33/32, so by fudging or tempering 1024/1023, one can get a good approximation of 31 by reducing it to the 11-limit, which at least makes it somewhat approachable, but its complexity makes it a very difficult prime to digest and work with.  Melodically, it's the same as 64/33, so it is incredibly jarring. Its octave complement is no better either, being a quartertone.
+* <u>Prime 31</u>: It's arguably the most dissonant of all the primes seen so far. Its octave complement, 32/31 is also incredibly close to 33/32, so by fudging or tempering 1024/1023, one can get a good approximation of 31 by reducing it to the 11-limit, which at least makes it somewhat approachable, but its integer complexity makes it a very difficult prime to digest and work with. Melodically, it's the same as 64/33, so it is incredibly jarring. Its octave complement is no better either, being a quartertone.
 === P-Tier V and beyond ===
 There would theoretically be infinite tiers, each having exponentially more primes than the one before, but I really don't care for primes beyond 29. They become far too complex for my ears and I'm sure that their sonorities can be fudged into other less complex intervals. The only way they can be reliably and practically explored for most cases is by using the serendipitous 311edo, I'd say.