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 ~~for~~ the ~~thirds~~, 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.

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".

# 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".

# 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.

# 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".

<br/>~~Note~~ that in the EDOs

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. 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.

# 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.

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

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

<br/>

=== Philosophy ===

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== Intervals ==

[~~To be worked on.~~]

For now, I will leave tables for the interval/colour/type namings of the seconds and thirds of my two favourite highly-chromatic EDOs, both of which related to [[Tolermic family|17-limit Tolermic]] due to this family tempering many commas I am interested in tempering, and due to the resulting intervals providing a good framework for thinking about interval colours:<br/>

~~# "comma"~~, ~~"chroma"~~: <~~35c~~

==== 80 EDO interval colours/types (seconds and thirds) ====

~~# "fifth~~-tone~~", "quarter~~-tone~~", "~~subminor ~~semitone", "ultraminor second", "tiny tone": 35c to 60c~~ (~~includes 21~~/20 and ~~19/18~~)

<pre>

~~# "third~~-tone~~", "minor semitone", "~~subminor ~~second", "mini~~-tone~~": 60c to 75c~~

45c ultraminor (AKA ~quarter-tone)

~~# "small semitone", "~~supraminor ~~semitone", "~~minor ~~second": 75c to 85c~~

60c subminor (AKA ~third-tone)

~~# "medium semitone" (or "~~minor ~~semitone"):~~

75c neominor

90c novaminor

105c minor

120c supraminor

135c minor neutral

150c major neutral

165c submajor

180c major

195c novamajor

210c neomajor

225c supermajor

240c ultramajor

255c ultraminor (AKA ~semifourth)

270c subminor

285c neominor

300c novaminor

315c minor

330c supraminor

345c subneutral

360c superneutral

375c submajor

390c major

405c novamajor

420c neomajor

435c supermajor

450c ultramajor (AKA ~semisixth)

</pre>

==== 87 EDO interval colours/types (seconds and thirds) ====

<pre>

41.4c fifth-tone

55.2c ultraminor (aka quarter-tone)

69.0c subminor (aka third-tone)

82.8c neominor

96.6c novaminor

110.3c minor

124.1c supraminor

137.9c minor neutral

151.7c major neutral

165.5c submajor

179.3c major

193.1c novamajor

206.9c neomajor

220.7c supermajor

234.5c ultramajor

248.3c semifourth

262.1c ultraminor

275.9c subminor

289.7c neominor

303.4c novaminor

317.2c minor

331.0c supraminor

344.8c subneutral

358.6c superneutral

372.4c submajor

386.2c major

400.0c novamajor

413.8c neomajor

427.6c supermajor

441.4c ultramajor

455.2c semisixth

</pre>

@@ Line 51: / Line 51: @@
 === 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 for the thirds, 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.
+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".
 # 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".
+# 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.
 # 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".
-<br/>Note that in the EDOs
+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. 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.
+# 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.
+# 29 EDO next, (approximately) with thirds of types ("fifth-tone" =) ~ultraminor, neominor, supraminor, submajor, neomajor, ("semifourth" =) ~ultramajor, and with seconds 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!
+Beyond this point, it really becomes about narrowing down EDOs based on approximative or other considerations than just 'types'/'colours'.
+<br/>
 === Philosophy ===
@@ Line 65: / Line 73: @@
 == Intervals ==
-[To be worked on.]
+For now, I will leave tables for the interval/colour/type namings of the seconds and thirds of my two favourite highly-chromatic EDOs, both of which related to [[Tolermic family|17-limit Tolermic]] due to this family tempering many commas I am interested in tempering, and due to the resulting intervals providing a good framework for thinking about interval colours:<br/>
-# "comma", "chroma": <35c
+==== 80 EDO interval colours/types (seconds and thirds) ====
-# "fifth-tone", "quarter-tone", "subminor semitone", "ultraminor second", "tiny tone": 35c to 60c (includes 21/20 and 19/18)
+<pre>
-# "third-tone", "minor semitone", "subminor second", "mini-tone": 60c to 75c
+c ultraminor (AKA ~quarter-tone)
-# "small semitone", "supraminor semitone", "minor second": 75c to 85c
+c subminor (AKA ~third-tone)
-# "medium semitone" (or "minor semitone"):
+c neominor
+c novaminor
+c minor
+c supraminor
+c minor neutral
+c major neutral
+c submajor
+c major
+c novamajor
+c neomajor
+c supermajor
+c ultramajor
+c ultraminor (AKA ~semifourth)
+c subminor
+c neominor
+c novaminor
+c minor
+c supraminor
+c subneutral
+c superneutral
+c submajor
+c major
+c novamajor
+c neomajor
+c supermajor
+c ultramajor (AKA ~semisixth)
+</pre>
+==== 87 EDO interval colours/types (seconds and thirds) ====
+<pre>
+.4c fifth-tone
+.2c ultraminor (aka quarter-tone)
+.0c subminor (aka third-tone)
+.8c neominor
+.6c novaminor
+.3c minor
+.1c supraminor
+.9c minor neutral
+.7c major neutral
+.5c submajor
+.3c major
+.1c novamajor
+.9c neomajor
+.7c supermajor
+.5c ultramajor
+.3c semifourth
+.1c ultraminor
+.9c subminor
+.7c neominor
+.4c novaminor
+.2c minor
+.0c supraminor
+.8c subneutral
+.6c superneutral
+.4c submajor
+.2c major
+.0c novamajor
+.8c neomajor
+.6c supermajor
+.4c ultramajor
+.2c semisixth
+</pre>