5L 2s/Interval categories: Difference between revisions

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== Unisons and octaves ==

{{Infobox|Title=Perfect 0-mosstep|Header 1=MOS|Data 1=[[5L 2s]]|Header 2=Other names|Data 2=Perfect unison|Header 3=Generator span|Data 3=0 generators|Header 4=Basic tuning|Data 4=0c}}

Unisons and octaves are the distances between one note and a note with the same letter name. For example, you can take an octave from D to D an octave up, or a chromatic semitone from D to D#.{{Infobox|Title=Perfect 0-mosstep|Header 1=MOS|Data 1=[[5L 2s]]|Header 2=Other names|Data 2=Perfect unison|Header 3=Generator span|Data 3=0 generators|Header 4=Basic tuning|Data 4=0c}}

=== Perfect unison and perfect octave ===

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== Seconds and sevenths ==

Seconds are the steps of the diatonic scale, falling between one note and the next in the scale, for example D and E, or E and F. Larger intervals are considered skips, as they skip scale degrees.

=== Diatonic semitone and major seventh ===

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** The major second is generated the same way in diatonic systems, by stacking 2 fifths.

The minor seventh is the octave complement of the major second, and appears as the fourth note in dominant 7th chords in Western harmony. Additionally, the '''harmonic seventh''' is the interval 7/4, which appears in the harmonic series alongside the ratios of the major chord, and can be played alongside them to form a JI chord of 4:5:6:7.

The minor seventh is the octave complement of the major second, and appears as the fourth note in dominant 7th chords in Western harmony. Additionally, the '''harmonic seventh''' is the interval 7/4, the octave complement of the supermajor second 8/7, which appears in the harmonic series alongside the ratios of the major chord, and can be played alongside them to form a JI chord of 4:5:6:7.

The major second can be used to construct a sus2 chord, which has the property of being composed of consecutive generator steps.

The major second and minor seventh are conventionally considered dissonances, however they are somewhat on the line between dissonance and imperfect consonance, and can be used as a consonance in context.

== Thirds and sixths ==

Thirds are the backbone of diatonic harmony, providing scales, chords, and other harmonic elements a distinct "major" and "minor" flavor. Both varieties of third are considered imperfect consonances. Since they are both a good distance away from the unison fifth-wise, changing the tuning of the fifth can drastically alter the thirds, and thus, the flavor of the scale.

=== Minor third and major sixth ===

{{Infobox|Title=Minor 2-mosstep|Header 1=MOS|Data 1=[[5L 2s]]|Header 2=Other names|Data 2=Minor third|Header 3=Generator span|Data 3=-3 generators|Header 4=Basic tuning|Data 4=300c}}The minor third is the smaller variety of third, used to generate minor chords, scales, and keys. 4 minor thirds occur in the C major diatonic scale, on D, E, A, and B.

* In 12edo, it is represented by the interval of 300 cents.

* In 13-limit just intonation, the inframinor third of [[15/13]] is found at about 247 cents, and leads to the inframinor or arto chord 26:30:39.

* In 7-limit just intonation, the subminor third of [[7/6]] is found at about 267 cents, and leads to the subminor or '''zo''' chord 6:7:9.

* In 5-limit just intonation, the minor third of [[6/5]] is found at about 316 cents, and leads to the classical minor or '''gu''' chord 10:12:15.

* In 3-limit just intonation, the minor third of [[32/27]] is found at about 294 cents, and leads to the Pythagorean minor or '''wa''' chord 54:64:81.

** In generalized diatonic systems, the minor third is generated the same way, leading to an interval between 240 and 343 cents, by stacking -3 fifths.

The minor third is found in the diminished and minor chords in Western harmony.

The major sixth is its octave complement.

=== Major third and minor sixth ===

The major third is the larger variety of third, used to generate major chords, scales, and keys. 3 major thirds occur in the C major diatonic scale, on C, F, and G.{{Infobox|Title=Major 2-mosstep|Header 1=MOS|Data 1=[[5L 2s]]|Header 2=Other names|Data 2=Major third|Header 3=Generator span|Data 3=+4 generators|Header 4=Basic tuning|Data 4=400c}}

* In 12edo, it is represented by the interval of 400 cents.

* In 13-limit just intonation, the ultramajor third of [[13/10]] is found at about 454 cents, and leads to the ultramajor or tendo chord 10:13:15.

* In 7-limit just intonation, the supermajor third of [[9/7]] is found at about 436 cents, and leads to the supermajor or '''ru''' chord 14:18:21.

* In 5-limit just intonation, the major third of [[5/4]] is found at about 386 cents, and leads to the classical major or '''yo''' chord 4:5:6, which is also a harmonic series fragment that can be extended to 4:5:6:7 to produce the '''harmonic seventh''' chord.

* In 3-limit just intonation, the major third of [[81/64]] is found at about 408 cents, and leads to the Pythagorean major or '''lawa''' chord 64:81:96.

** In generalized diatonic systems, the major third is generated the same way, leading to an interval between 343 and 480 cents, by stacking 4 fifths.

The major third is found in the augmented and major chords in Western harmony.

The minor sixth is its octave complement.

== Fourths and fifths ==

Fourths and fifths are the generators of the diatonic scale, and the simplest possible intervals other than the unison and octave.

=== Perfect fourth and perfect fifth ===

{{Infobox|Title=Perfect 3-mosstep|Header 1=MOS|Data 1=[[5L 2s]]|Header 2=Other names|Data 2=Perfect fourth|Header 3=Generator span|Data 3=-1 generators|Header 4=Basic tuning|Data 4=500c}}The perfect fourth (and its octave complement, the perfect fifth) are the most basic diatonic intervals. A perfect fifth appears in most diatonic chords.

* In 12edo, the perfect fourth is represented by the interval of 500 cents.

* In just intonation, the perfect fourth is usually represented, regardless of limit, by the interval [[4/3]] of about 498 cents.

** In generalized diatonic systems, the perfect fourth is the dark generator, meaning it is generated by stacking -1 fifths, and can be used as a generator itself, as opposed to fifths.

The perfect fourth can be used to construct a sus4 chord, which has the same properties as the sus2 chord, being composed of consecutive generator steps. The perfect fourth and perfect fifth are considered perfect consonances.

=== Augmented fourth and diminished fifth ===

{{Infobox|Title=Augmented 3-mosstep|Header 1=MOS|Data 1=[[5L 2s]]|Header 2=Other names|Data 2=Augmented fourth|Header 3=Generator span|Data 3=6 generators|Header 4=Basic tuning|Data 4=600c}}The augmented fourth and diminished fifth, in contrast to the perfect counterparts, are harsh dissonances. They are collectively considered "tritones", as they are the size of approximately 3 whole tones. Only one of each occurs in the diatonic scale, an augmented fourth between F and B and a diminished fifth between B and F.

* In 12edo, they are enharmonic, represented by the interval of 600 cents.

* In just intonation, an augmented fourth can be represented by the perfect fourth stacked with any chromatic semitone.

** In the 7-limit, this becomes [[72/49]], which is closer to a fifth than a fourth, at 667 cents.

*** A 7-limit interval that is closer to the expected size for a tritone is 7/5, at 583 cents.

** In the 5-limit, this is [[25/18]], the classical augmented fourth of about 569 cents.

** In the 3-limit, this is [[729/512]], the Pythagorean augmented fourth of about 612 cents.

* In generalized diatonic systems, the augmented fourth is generated by stacking 6 generators.

The diminished fifth appears as the fifth of a diminished chord.

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 == Unisons and octaves ==
-{{Infobox|Title=Perfect 0-mosstep|Header 1=MOS|Data 1=[[5L 2s]]|Header 2=Other names|Data 2=Perfect unison|Header 3=Generator span|Data 3=0 generators|Header 4=Basic tuning|Data 4=0c}}
+Unisons and octaves are the distances between one note and a note with the same letter name. For example, you can take an octave from D to D an octave up, or a chromatic semitone from D to D#.{{Infobox|Title=Perfect 0-mosstep|Header 1=MOS|Data 1=[[5L 2s]]|Header 2=Other names|Data 2=Perfect unison|Header 3=Generator span|Data 3=0 generators|Header 4=Basic tuning|Data 4=0c}}
 === Perfect unison and perfect octave ===
@@ Line 31: / Line 31: @@
 == Seconds and sevenths ==
+Seconds are the steps of the diatonic scale, falling between one note and the next in the scale, for example D and E, or E and F. Larger intervals are considered skips, as they skip scale degrees.
 === Diatonic semitone and major seventh ===
@@ Line 56: / Line 57: @@
 ** The major second is generated the same way in diatonic systems, by stacking 2 fifths.
-The minor seventh is the octave complement of the major second, and appears as the fourth note in dominant 7th chords in Western harmony. Additionally, the '''harmonic seventh''' is the interval 7/4, which appears in the harmonic series alongside the ratios of the major chord, and can be played alongside them to form a JI chord of 4:5:6:7.
+The minor seventh is the octave complement of the major second, and appears as the fourth note in dominant 7th chords in Western harmony. Additionally, the '''harmonic seventh''' is the interval 7/4, the octave complement of the supermajor second 8/7, which appears in the harmonic series alongside the ratios of the major chord, and can be played alongside them to form a JI chord of 4:5:6:7.
+The major second can be used to construct a sus2 chord, which has the property of being composed of consecutive generator steps.
 The major second and minor seventh are conventionally considered dissonances, however they are somewhat on the line between dissonance and imperfect consonance, and can be used as a consonance in context.
 == Thirds and sixths ==
+Thirds are the backbone of diatonic harmony, providing scales, chords, and other harmonic elements a distinct "major" and "minor" flavor. Both varieties of third are considered imperfect consonances. Since they are both a good distance away from the unison fifth-wise, changing the tuning of the fifth can drastically alter the thirds, and thus, the flavor of the scale.
 === Minor third and major sixth ===
+{{Infobox|Title=Minor 2-mosstep|Header 1=MOS|Data 1=[[5L 2s]]|Header 2=Other names|Data 2=Minor third|Header 3=Generator span|Data 3=-3 generators|Header 4=Basic tuning|Data 4=300c}}The minor third is the smaller variety of third, used to generate minor chords, scales, and keys. 4 minor thirds occur in the C major diatonic scale, on D, E, A, and B.
+* In 12edo, it is represented by the interval of 300 cents.
+* In 13-limit just intonation, the inframinor third of [[15/13]] is found at about 247 cents, and leads to the inframinor or arto chord 26:30:39.
+* In 7-limit just intonation, the subminor third of [[7/6]] is found at about 267 cents, and leads to the subminor or '''zo''' chord 6:7:9.
+* In 5-limit just intonation, the minor third of [[6/5]] is found at about 316 cents, and leads to the classical minor or '''gu''' chord 10:12:15.
+* In 3-limit just intonation, the minor third of [[32/27]] is found at about 294 cents, and leads to the Pythagorean minor or '''wa''' chord 54:64:81.
+** In generalized diatonic systems, the minor third is generated the same way, leading to an interval between 240 and 343 cents, by stacking -3 fifths.
+The minor third is found in the diminished and minor chords in Western harmony.
+The major sixth is its octave complement.
 === Major third and minor sixth ===
+The major third is the larger variety of third, used to generate major chords, scales, and keys. 3 major thirds occur in the C major diatonic scale, on C, F, and G.{{Infobox|Title=Major 2-mosstep|Header 1=MOS|Data 1=[[5L 2s]]|Header 2=Other names|Data 2=Major third|Header 3=Generator span|Data 3=+4 generators|Header 4=Basic tuning|Data 4=400c}}
+* In 12edo, it is represented by the interval of 400 cents.
+* In 13-limit just intonation, the ultramajor third of [[13/10]] is found at about 454 cents, and leads to the ultramajor or tendo chord 10:13:15.
+* In 7-limit just intonation, the supermajor third of [[9/7]] is found at about 436 cents, and leads to the supermajor or '''ru''' chord 14:18:21.
+* In 5-limit just intonation, the major third of [[5/4]] is found at about 386 cents, and leads to the classical major or '''yo''' chord 4:5:6, which is also a harmonic series fragment that can be extended to 4:5:6:7 to produce the '''harmonic seventh''' chord.
+* In 3-limit just intonation, the major third of [[81/64]] is found at about 408 cents, and leads to the Pythagorean major or '''lawa''' chord 64:81:96.
+** In generalized diatonic systems, the major third is generated the same way, leading to an interval between 343 and 480 cents, by stacking 4 fifths.
+The major third is found in the augmented and major chords in Western harmony.
+The minor sixth is its octave complement.
 == Fourths and fifths ==
+Fourths and fifths are the generators of the diatonic scale, and the simplest possible intervals other than the unison and octave.
 === Perfect fourth and perfect fifth ===
+{{Infobox|Title=Perfect 3-mosstep|Header 1=MOS|Data 1=[[5L 2s]]|Header 2=Other names|Data 2=Perfect fourth|Header 3=Generator span|Data 3=-1 generators|Header 4=Basic tuning|Data 4=500c}}The perfect fourth (and its octave complement, the perfect fifth) are the most basic diatonic intervals. A perfect fifth appears in most diatonic chords.
+* In 12edo, the perfect fourth is represented by the interval of 500 cents.
+* In just intonation, the perfect fourth is usually represented, regardless of limit, by the interval [[4/3]] of about 498 cents.
+** In generalized diatonic systems, the perfect fourth is the dark generator, meaning it is generated by stacking -1 fifths, and can be used as a generator itself, as opposed to fifths.
+The perfect fourth can be used to construct a sus4 chord, which has the same properties as the sus2 chord, being composed of consecutive generator steps. The perfect fourth and perfect fifth are considered perfect consonances.
 === Augmented fourth and diminished fifth ===
+{{Infobox|Title=Augmented 3-mosstep|Header 1=MOS|Data 1=[[5L 2s]]|Header 2=Other names|Data 2=Augmented fourth|Header 3=Generator span|Data 3=6 generators|Header 4=Basic tuning|Data 4=600c}}The augmented fourth and diminished fifth, in contrast to the perfect counterparts, are harsh dissonances. They are collectively considered "tritones", as they are the size of approximately 3 whole tones. Only one of each occurs in the diatonic scale, an augmented fourth between F and B and a diminished fifth between B and F.
+* In 12edo, they are enharmonic, represented by the interval of 600 cents.
+* In just intonation, an augmented fourth can be represented by the perfect fourth stacked with any chromatic semitone.
+** In the 7-limit, this becomes [[72/49]], which is closer to a fifth than a fourth, at 667 cents.
+*** A 7-limit interval that is closer to the expected size for a tritone is 7/5, at 583 cents.
+** In the 5-limit, this is [[25/18]], the classical augmented fourth of about 569 cents.
+** In the 3-limit, this is [[729/512]], the Pythagorean augmented fourth of about 612 cents.
+* In generalized diatonic systems, the augmented fourth is generated by stacking 6 generators.
+The diminished fifth appears as the fifth of a diminished chord.