Keemic chords: Difference between revisions

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A '''~~magical seventh~~''' is an [[11-limit]] [[essentially tempered ~~dyadic~~ chord]] ~~consisting of three sharp~~ [[~~6/5|minor thirds~~]] ~~and an~~ [[8/7]]~~, in other words a 6/5-6/5-6/5-8/7 chord, which closes at the~~ [[~~octave~~]] ~~since both~~ [[~~100/99~~]] ~~and~~ [[385/384]] ~~(and therefore~~ [[~~875/864~~]]) are ~~[[Tempering out|tempered out]]~~. ~~This means that in the 11-limit 4&7&15 [[Planar temperaments|planar temperament]] tempering~~ these ~~out~~, ~~the chord is the tempering of 1-6/5-16/11-7/4~~.

A '''keemic chord''' is an [[11-odd-limit]] [[essentially tempered chord]] in the [[keemic]] temperament. Since [[100/99]] is [[tempering out|tempered out]], [[ptolemismic chords]] are also keemic chords; since [[385/384]] is tempered out, [[keenanismic chords]] are also keemic chords. Aside from these, there are also essentially keemic tempered chords.

In an optimized tuning for the ~~4&7&15~~ temperament, the [[marvel]] comma [[225/224]] shrinks in size and may reverse direction, and adding it to the list of commas does little tuning [[damage]]; this results in 11-limit [[magic]] temperament, which has the same [[optimal patent val]] ([[104edo]]). Hence [[magic]] (19&22) temperament is practically the most accurate temperament that include this chord. Magic, however, does give it a [[Graham complexity]] of 12, so it ~~doesn't~~ appear that often.

The most basic of these is the '''magical seventh chord''', consisting of three sharp [[6/5|classical minor thirds]] and a [[8/7|septimal whole tone]], which closes at the [[octave]] since both [[100/99]] and [[385/384]] (and therefore [[875/864]]) are [[tempering out|tempered out]]. This means the chord is the tempering of

* 1–6/5–16/11–7/4 with steps 6/5, 6/5, 6/5, 8/7.

In an optimized tuning for the keemic temperament, the [[marvel]] comma [[225/224]] shrinks in size and may reverse direction, and adding it to the list of commas does little tuning [[damage]]; this results in 11-limit [[magic]] temperament, which has the same [[optimal patent val]] ([[104edo]]). Hence [[magic]] (19&22) temperament is practically the most accurate temperament that include this chord. Magic, however, does give it a [[Graham complexity]] of 12, so it does not appear that often.

Other temperaments that feature this chord prominently include 11-limit [[keemun]], [[superkleismic]], [[porcupine]] and [[doublewide]].

[[Category:11-odd-limit]]

For other tetrads, there are

* 1–5/4–16/11–9/5 with steps 5/4, 7/6, 5/4, 10/9;

* 1–12/11–5/4–9/5 with steps 12/11, 8/7, 16/11, 10/9, and its inverse

* 1–12/11–6/5–7/4 with steps 12/11, 10/9, 16/11, 8/7.

For pentads, there are

* 1–12/11–5/4–3/2–9/5 with steps 12/11, 8/7, 6/5, 6/5, 10/9, and its inverse

* 1–6/5–11/8–3/2–5/3 with steps 6/5, 8/7, 12/11, 10/9, 6/5;

* 1–5/4–11/8–3/2–12/7 with steps 5/4, 11/10, 12/11, 8/7, 7/6, and its inverse

* 1–12/11–6/5–3/2–7/4 with steps 12/11, 11/10, 5/4, 7/6, 8/7.

The count of keemic chords is therefore tetrads: 4, and pentads: 4, for a total of 8.

[[Category:11-odd-limit chords]]

[[Category:Essentially tempered chords]]

[[Category:Tetrads]]

[[Category:Pentads]]

[[Category:~~Supermagic~~]]

[[Category:Keemic]]

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-A '''magical seventh''' is an [[11-limit]] [[essentially tempered dyadic chord]] consisting of three sharp [[6/5|minor thirds]] and an [[8/7]], in other words a 6/5-6/5-6/5-8/7 chord, which closes at the [[octave]] since both [[100/99]] and [[385/384]] (and therefore [[875/864]]) are [[Tempering out|tempered out]]. This means that in the 11-limit 4&amp;7&amp;15 [[Planar temperaments|planar temperament]] tempering these out, the chord is the tempering of 1-6/5-16/11-7/4.
+A '''keemic chord''' is an [[11-odd-limit]] [[essentially tempered chord]] in the [[keemic]] temperament. Since [[100/99]] is [[tempering out|tempered out]], [[ptolemismic chords]] are also keemic chords; since [[385/384]] is tempered out, [[keenanismic chords]] are also keemic chords. Aside from these, there are also essentially keemic tempered chords.
-In an optimized tuning for the 4&amp;7&amp;15 temperament, the [[marvel]] comma [[225/224]] shrinks in size and may reverse direction, and adding it to the list of commas does little tuning [[damage]]; this results in 11-limit [[magic]] temperament, which has the same [[optimal patent val]] ([[104edo]]). Hence [[magic]] (19&amp;22) temperament is practically the most accurate temperament that include this chord. Magic, however, does give it a [[Graham complexity]] of 12, so it doesn't appear that often.
+The most basic of these is the '''magical seventh chord''', consisting of three sharp [[6/5|classical minor thirds]] and a [[8/7|septimal whole tone]], which closes at the [[octave]] since both [[100/99]] and [[385/384]] (and therefore [[875/864]]) are [[tempering out|tempered out]]. This means the chord is the tempering of
+* 1–6/5–16/11–7/4 with steps 6/5, 6/5, 6/5, 8/7.
+In an optimized tuning for the keemic temperament, the [[marvel]] comma [[225/224]] shrinks in size and may reverse direction, and adding it to the list of commas does little tuning [[damage]]; this results in 11-limit [[magic]] temperament, which has the same [[optimal patent val]] ([[104edo]]). Hence [[magic]] (19&amp;22) temperament is practically the most accurate temperament that include this chord. Magic, however, does give it a [[Graham complexity]] of 12, so it does not appear that often.
 Other temperaments that feature this chord prominently include 11-limit [[keemun]], [[superkleismic]], [[porcupine]] and [[doublewide]].
-[[Category:11-odd-limit]]
+For other tetrads, there are
+* 1–5/4–16/11–9/5 with steps 5/4, 7/6, 5/4, 10/9;
+* 1–12/11–5/4–9/5 with steps 12/11, 8/7, 16/11, 10/9, and its inverse
+* 1–12/11–6/5–7/4 with steps 12/11, 10/9, 16/11, 8/7.
+For pentads, there are
+* 1–12/11–5/4–3/2–9/5 with steps 12/11, 8/7, 6/5, 6/5, 10/9, and its inverse
+* 1–6/5–11/8–3/2–5/3 with steps 6/5, 8/7, 12/11, 10/9, 6/5;
+* 1–5/4–11/8–3/2–12/7 with steps 5/4, 11/10, 12/11, 8/7, 7/6, and its inverse
+* 1–12/11–6/5–3/2–7/4 with steps 12/11, 11/10, 5/4, 7/6, 8/7.
+The count of keemic chords is therefore tetrads: 4, and pentads: 4, for a total of 8.
+[[Category:11-odd-limit chords]]
 [[Category:Essentially tempered chords]]
+[[Category:Tetrads]]
 [[Category:Pentads]]
-[[Category:Supermagic]]
+[[Category:Keemic]]