User:Godtone: Difference between revisions

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=== Colourful EDOs ===

My above progression of "types"/"colours" can be used as a perhaps-interesting alternative to finding "good" EDOs for music by judging them not based on approximation of rationals of interest but instead based on their "colour palette"; not that these two methods are contradictory, and in fact I believe a combination of both is desirable. However, as a demonstration and a starting point, we will look at EDOs providing progressively more complex colour palettes, starting from a broad equalised 7-note approximation of the 5L2s diatonic scale (AKA 7 EDO) and considering only the 'seconds' and 'thirds' (and thus by octave complement, their inversions), with fourths and fifths not considered except to the extent that they should ideally not be too "out of tune", with "out-of-tune-ness" judged relative to approximating either 4/3 or 11/8 (but not both), as these are the simplest J.I fourths. Furthermore, we want the colour palettes to be generally symmetric about the "neutral" type (whether a system has a neutral type or not), so this excludes a large number of EDOs; this is intentional, as otherwise we would end up listing every EDO, and as it is a symmetry which I think is important or at least an interesting restriction.

# 7 EDO is the simplest/"trivial" EDO as it provides only the (at times very approximate) "neutral" colour. Note that its fourths are very out-of-tune; this EDO is mainly included as a trivial case. This corresponds to "1 type".

# [[7edo|7 EDO]] is the simplest/"trivial" EDO as it provides only the (at times very approximate) "neutral" colour. Note that its fourths are very out-of-tune; this EDO is mainly included as a trivial case. This corresponds to "1 type".

# 12 EDO is the next simplest as it provides (nova)major and (nova)minor seconds and thirds. Also a tone-efficient pure Pyth approximation so very good fourths. This corresponds to "2 types".

# [[12edo|12 EDO]] is the next simplest as it provides (nova)major and (nova)minor seconds and thirds. Also a tone-efficient pure Pyth approximation so very good fourths. This corresponds to "2 types".

# 15 EDO, in terms of colours, is similar to 12 EDO except that its minor third is a little sharper and that it now has 3 types of second which are roughly subminor, neutral and supermajor. The "subminor" and "supermajor" designations are used due to symmetry; in actuality the subminor second is closer to a neominor second and the supermajor second is closer to an ultramajor second. The prefixes may be omitted, or more exact colour terms may be used, making it have a "superneutral second". Note this EDO is more of an honourable mention due both to somewhat significant asymmetry and due to very off fifths, and also to explain why I won't include it in the "final" list..

# [[15edo|15 EDO]], in terms of colours, is similar to 12 EDO except that its minor third is a little sharper and that it now has 3 types of second which are roughly subminor, neutral and supermajor. The "subminor" and "supermajor" designations are used due to symmetry; in actuality the subminor second is closer to a neominor second and the supermajor second is closer to an ultramajor second. The prefixes may be omitted, or more exact colour terms may be used, making it have a "superneutral second". Note this EDO is more of an honourable mention due both to somewhat significant asymmetry and due to very off fifths, and also to explain why I won't include it in the "final" list..

# 17 EDO is the first EDO to truly have minor, neutral and major for both seconds and thirds, and is thus quite significant as a potential next step up from 12 EDO. More exactly, these come in the flavours of neominor, neutral and neomajor.

# [[17edo|17 EDO]] is the first EDO to truly have minor, neutral and major for both seconds and thirds, and is thus quite significant as a potential next step up from 12 EDO. More exactly, these come in the flavours of neominor, neutral and neomajor.

# 19 EDO is the next step up, having seconds and thirds of the ultraminor, minor, major and ultramajor varieties. It does this by conflating an ultramajor second as an ultraminor third, creating quite a distinct interval that escapes 5L2s categorisation. Note that 18 EDO does this too, but is less symmetric. Thus 19 (or 18 - if you are so inclined - which represents a sharpening of seconds and thirds) is the next step up as corresponding to "4 types".

# [[19edo|19 EDO]] is the next step up, having seconds and thirds of the ultraminor, minor, major and ultramajor varieties. It does this by conflating an ultramajor second as an ultraminor third, creating quite a distinct interval that escapes 5L2s categorisation. Note that 18 EDO does this too, but is less symmetric. Thus 19 (or 18 - if you are so inclined - which represents a sharpening of seconds and thirds) is the next step up as corresponding to "4 types".

Note that 17 and 19 have an interesting symmetry with each-other: while 17's new type is per 5L2s category and represents something between the types of major and minor, 19's new type is between 5L2s categories. Thus 17 respects diatonic interval categories more, which actually makes 19 the more novel system to me, for example I quite like the very distinct sound of the 5L4s "semaphore" scale; dyads and triads alike.

<br/>Furthermore, note that after this point I focus on EDOs with 'good enough' fourths.

# 22 EDO is next, having subminor, supraminor, submajor and supermajor seconds and thirds, although only approximately, and in this spirit "supraminor and submajor" can be shortened to "minor and major" for brevity. 22 is distinct from 19 in that it does not have any of its types overlap between categories.

# [[22edo|22 EDO]] is next, having subminor, supraminor, submajor and supermajor seconds and thirds, although only approximately, and in this spirit "supraminor and submajor" can be shortened to "minor and major" for brevity. 22 is distinct from 19 in that it does not have any of its types overlap between categories.

# 24 EDO is next, being easy to categorise as a colour extension of 12, and thus it has seconds and thirds of the ultraminor, novaminor, neutral, novamajor and ultramajor varieties, with the ultramajor second equal to the ultraminor third.

# [[24edo|24 EDO]] is next, being easy to categorise as a colour extension of 12, and thus it has seconds and thirds of the ultraminor, novaminor, neutral, novamajor and ultramajor varieties, with the ultramajor second equal to the ultraminor third.

# 26 EDO is next and mirrors 22 in its thirds but has one more type of second, and so the seconds are ''approximately'': subminor, minor, neutral, major, supermajor.

# [[26edo|26 EDO]] is next and mirrors 22 in its thirds but has one more type of second, and so the seconds are ''approximately'': subminor, minor, neutral, major, supermajor.

# 29 EDO next, (approximately) with seconds of types ("fifth-tone" =) ~ultraminor, neominor, supraminor, submajor, neomajor, ("semifourth" =) ~ultramajor, and with thirds of types ("semifourth" =) ultraminor, neominor, supraminor, submajor, neomajor, ("semisixth" =) ulramajor.

# [[29edo|29 EDO]] next, (approximately) with seconds of types ("fifth-tone" =) ~ultraminor, neominor, supraminor, submajor, neomajor, ("semifourth" =) ~ultramajor, and with thirds of types ("semifourth" =) ultraminor, neominor, supraminor, submajor, neomajor, ("semisixth" =) ulramajor.

# 31 EDO is then quite ideal, as it gives us seconds and thirds of subminor, minor, neutral, major and supermajor types!

# [[31edo|31 EDO]] is then quite ideal, as it gives us seconds and thirds of subminor, minor, neutral, major and supermajor types!

Beyond this point, it really becomes about narrowing down EDOs based on approximative or other considerations than just 'types'/'colours'.

# [[34edo|34 EDO]] is ideal if your main gripe with 31 EDO is the lack of semifourths/semisixths (which is my main gripe with it), although note that instead of a subminor third it has a flat neominor third, but its shrubmajor third ('''4.2''') is roughly the same as 31 EDO's; this corresponds to their opposite tendencies as systems.

Beyond this point, it really becomes about narrowing down EDOs based on approximative or other considerations than just 'types'/'colours'. Having said this, there are still yet a few EDOs to be considered, but we will require hereon a fifth strictly between that of [[31edo]] and that of [[34edo]] (AKA that of 17edo).

# [[41edo|41 EDO]] is ideal if your main gripe with 31 EDO is impurity of the fifth and lack of novamajor/novaminor as well as the unideal flatness of the supermajor third, although personally that is not my main gripe with 31 EDO.

# [[43edo|43 EDO]] is ideal if your main gripe with 31 EDO is lack of semifourths/semisixths and submajor/supraminor if you don't mind having a gothminor third ('''2.8''') instead of a subminor one ('''2.7''') and a neomajor third ('''4.1''') instead of a shrubmajor one ('''4.2''' or often we consider it less accurately a supermajor third ('''4.3''') in 31 EDO for symmetry with its subminor third ('''2.7''')).

# [[53edo|53 EDO]] is ideal if you want something that fixes the gripes stated in 34 EDO, 41 EDO and 43 EDO; that is: lack of semifourths/semisixths, lack of novamajor/novaminor & impurity of the fifth & lack of a proper supermajor third, and lack of (semifourths/semisixths &) submajor/supraminor / superneutral/subneutral, respectively, making it unusually complete & comprehensive as a system, corresponding to garibaldi+nestoria+catakleismic.

Beyond 53 EDO, what EDOs are notable as colour palettes becomes too subjective and dependent on various hyper-specific criteria that may have only very niche applicability to anyone, so beyond 53 EDO you are better off looking at the systems from a tuning perspective first and a colour palette perspective second rather than a colour palette perspective first and a tuning perspective second; 53 EDO is sort of unique as the middleground between these two ways of thinking, representing both a fairly complete palette and an excellent tuning system in the no-17's [[19-limit]] (with optional higher primes added depending on context/what you wanna achieve). It should perhaps be noted, however, that [[58edo]] is a colour palette with similar elegance to 53 but with less tuning accuracy.

<br/>

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 === Colourful EDOs ===
 My above progression of "types"/"colours" can be used as a perhaps-interesting alternative to finding "good" EDOs for music by judging them not based on approximation of rationals of interest but instead based on their "colour palette"; not that these two methods are contradictory, and in fact I believe a combination of both is desirable. However, as a demonstration and a starting point, we will look at EDOs providing progressively more complex colour palettes, starting from a broad equalised 7-note approximation of the 5L2s diatonic scale (AKA 7 EDO) and considering only the 'seconds' and 'thirds' (and thus by octave complement, their inversions), with fourths and fifths not considered except to the extent that they should ideally not be too "out of tune", with "out-of-tune-ness" judged relative to approximating either 4/3 or 11/8 (but not both), as these are the simplest J.I fourths. Furthermore, we want the colour palettes to be generally symmetric about the "neutral" type (whether a system has a neutral type or not), so this excludes a large number of EDOs; this is intentional, as otherwise we would end up listing every EDO, and as it is a symmetry which I think is important or at least an interesting restriction.
-# 7 EDO is the simplest/"trivial" EDO as it provides only the (at times very approximate) "neutral" colour. Note that its fourths are very out-of-tune; this EDO is mainly included as a trivial case. This corresponds to "1 type".
+# [[7edo|7 EDO]] is the simplest/"trivial" EDO as it provides only the (at times very approximate) "neutral" colour. Note that its fourths are very out-of-tune; this EDO is mainly included as a trivial case. This corresponds to "1 type".
-# 12 EDO is the next simplest as it provides (nova)major and (nova)minor seconds and thirds. Also a tone-efficient pure Pyth approximation so very good fourths. This corresponds to "2 types".
+# [[12edo|12 EDO]] is the next simplest as it provides (nova)major and (nova)minor seconds and thirds. Also a tone-efficient pure Pyth approximation so very good fourths. This corresponds to "2 types".
-# 15 EDO, in terms of colours, is similar to 12 EDO except that its minor third is a little sharper and that it now has 3 types of second which are roughly subminor, neutral and supermajor. The "subminor" and "supermajor" designations are used due to symmetry; in actuality the subminor second is closer to a neominor second and the supermajor second is closer to an ultramajor second. The prefixes may be omitted, or more exact colour terms may be used, making it have a "superneutral second". Note this EDO is more of an honourable mention due both to somewhat significant asymmetry and due to very off fifths, and also to explain why I won't include it in the "final" list..
+# [[15edo|15 EDO]], in terms of colours, is similar to 12 EDO except that its minor third is a little sharper and that it now has 3 types of second which are roughly subminor, neutral and supermajor. The "subminor" and "supermajor" designations are used due to symmetry; in actuality the subminor second is closer to a neominor second and the supermajor second is closer to an ultramajor second. The prefixes may be omitted, or more exact colour terms may be used, making it have a "superneutral second". Note this EDO is more of an honourable mention due both to somewhat significant asymmetry and due to very off fifths, and also to explain why I won't include it in the "final" list..
-# 17 EDO is the first EDO to truly have minor, neutral and major for both seconds and thirds, and is thus quite significant as a potential next step up from 12 EDO. More exactly, these come in the flavours of neominor, neutral and neomajor.
+# [[17edo|17 EDO]] is the first EDO to truly have minor, neutral and major for both seconds and thirds, and is thus quite significant as a potential next step up from 12 EDO. More exactly, these come in the flavours of neominor, neutral and neomajor.
-# 19 EDO is the next step up, having seconds and thirds of the ultraminor, minor, major and ultramajor varieties. It does this by conflating an ultramajor second as an ultraminor third, creating quite a distinct interval that escapes 5L2s categorisation. Note that 18 EDO does this too, but is less symmetric. Thus 19 (or 18 - if you are so inclined - which represents a sharpening of seconds and thirds) is the next step up as corresponding to "4 types".
+# [[19edo|19 EDO]] is the next step up, having seconds and thirds of the ultraminor, minor, major and ultramajor varieties. It does this by conflating an ultramajor second as an ultraminor third, creating quite a distinct interval that escapes 5L2s categorisation. Note that 18 EDO does this too, but is less symmetric. Thus 19 (or 18 - if you are so inclined - which represents a sharpening of seconds and thirds) is the next step up as corresponding to "4 types".
 Note that 17 and 19 have an interesting symmetry with each-other: while 17's new type is per 5L2s category and represents something between the types of major and minor, 19's new type is between 5L2s categories. Thus 17 respects diatonic interval categories more, which actually makes 19 the more novel system to me, for example I quite like the very distinct sound of the 5L4s "semaphore" scale; dyads and triads alike.
 <br/>Furthermore, note that after this point I focus on EDOs with 'good enough' fourths.
-# 22 EDO is next, having subminor, supraminor, submajor and supermajor seconds and thirds, although only approximately, and in this spirit "supraminor and submajor" can be shortened to "minor and major" for brevity. 22 is distinct from 19 in that it does not have any of its types overlap between categories.
+# [[22edo|22 EDO]] is next, having subminor, supraminor, submajor and supermajor seconds and thirds, although only approximately, and in this spirit "supraminor and submajor" can be shortened to "minor and major" for brevity. 22 is distinct from 19 in that it does not have any of its types overlap between categories.
-# 24 EDO is next, being easy to categorise as a colour extension of 12, and thus it has seconds and thirds of the ultraminor, novaminor, neutral, novamajor and ultramajor varieties, with the ultramajor second equal to the ultraminor third.
+# [[24edo|24 EDO]] is next, being easy to categorise as a colour extension of 12, and thus it has seconds and thirds of the ultraminor, novaminor, neutral, novamajor and ultramajor varieties, with the ultramajor second equal to the ultraminor third.
-# 26 EDO is next and mirrors 22 in its thirds but has one more type of second, and so the seconds are ''approximately'': subminor, minor, neutral, major, supermajor.
+# [[26edo|26 EDO]] is next and mirrors 22 in its thirds but has one more type of second, and so the seconds are ''approximately'': subminor, minor, neutral, major, supermajor.
-# 29 EDO next, (approximately) with seconds of types ("fifth-tone" =) ~ultraminor, neominor, supraminor, submajor, neomajor, ("semifourth" =) ~ultramajor, and with thirds of types ("semifourth" =) ultraminor, neominor, supraminor, submajor, neomajor, ("semisixth" =) ulramajor.
+# [[29edo|29 EDO]] next, (approximately) with seconds of types ("fifth-tone" =) ~ultraminor, neominor, supraminor, submajor, neomajor, ("semifourth" =) ~ultramajor, and with thirds of types ("semifourth" =) ultraminor, neominor, supraminor, submajor, neomajor, ("semisixth" =) ulramajor.
-# 31 EDO is then quite ideal, as it gives us seconds and thirds of subminor, minor, neutral, major and supermajor types!
+# [[31edo|31 EDO]] is then quite ideal, as it gives us seconds and thirds of subminor, minor, neutral, major and supermajor types!
-Beyond this point, it really becomes about narrowing down EDOs based on approximative or other considerations than just 'types'/'colours'.
+# [[34edo|34 EDO]] is ideal if your main gripe with 31 EDO is the lack of semifourths/semisixths (which is my main gripe with it), although note that instead of a subminor third it has a flat neominor third, but its shrubmajor third ('''4.2''') is roughly the same as 31 EDO's; this corresponds to their opposite tendencies as systems.
+Beyond this point, it really becomes about narrowing down EDOs based on approximative or other considerations than just 'types'/'colours'. Having said this, there are still yet a few EDOs to be considered, but we will require hereon a fifth strictly between that of [[31edo]] and that of [[34edo]] (AKA that of 17edo).
+# [[41edo|41 EDO]] is ideal if your main gripe with 31 EDO is impurity of the fifth and lack of novamajor/novaminor as well as the unideal flatness of the supermajor third, although personally that is not my main gripe with 31 EDO.
+# [[43edo|43 EDO]] is ideal if your main gripe with 31 EDO is lack of semifourths/semisixths and submajor/supraminor if you don't mind having a gothminor third ('''2.8''') instead of a subminor one ('''2.7''') and a neomajor third ('''4.1''') instead of a shrubmajor one ('''4.2''' or often we consider it less accurately a supermajor third ('''4.3''') in 31 EDO for symmetry with its subminor third ('''2.7''')).
+# [[53edo|53 EDO]] is ideal if you want something that fixes the gripes stated in 34 EDO, 41 EDO and 43 EDO; that is: lack of semifourths/semisixths, lack of novamajor/novaminor & impurity of the fifth & lack of a proper supermajor third, and lack of (semifourths/semisixths &) submajor/supraminor / superneutral/subneutral, respectively, making it unusually complete & comprehensive as a system, corresponding to garibaldi+nestoria+catakleismic.
+Beyond 53 EDO, what EDOs are notable as colour palettes becomes too subjective and dependent on various hyper-specific criteria that may have only very niche applicability to anyone, so beyond 53 EDO you are better off looking at the systems from a tuning perspective first and a colour palette perspective second rather than a colour palette perspective first and a tuning perspective second; 53 EDO is sort of unique as the middleground between these two ways of thinking, representing both a fairly complete palette and an excellent tuning system in the no-17's [[19-limit]] (with optional higher primes added depending on context/what you wanna achieve). It should perhaps be noted, however, that [[58edo]] is a colour palette with similar elegance to 53 but with less tuning accuracy.
 <br/>

Godtone

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