5L 3s: Difference between revisions

(5 intermediate revisions by 2 users not shown)

Line 174:

== Approaches ==

* [[5L 3s/Temperaments]]

== 5L 3s as diatonic with a fifth sharper than 5edo ==

You may have heard how [[2L 5s]] can be viewed as diatonic ([[5L 2s]]) generated by a fifth flatter than [[7edo]], in which case the large and small steps flip sizes. Similar to that, 5L 3s can be viewed as diatonic generated by a fifth sharper than [[5edo]]. To understand how this works and how you suddenly get an extra small step, let's first start with fifths slightly flat of 5edo and 5edo's fifth itself.

In diatonic scales generated by fifths slightly flat of [[5edo]]'s fifth, the small step (found by 8 octaves - 5 fifths) becomes increasingly narrow, for example in [[27edo]] (which has a fifth of 711.1 cents) the diatonic small step is 44.4 cents, and in [[32edo]] (which has a fifth of 712.5 cents) the diatonic small step is even smaller at 37.5 cents. The fifth-generated major scale in 32edo in [[cent]]s is 225-450-487.5-712.5-937.5-1162.5-1200 – notice how the "major third" (450) and "perfect fourth" (487.5) are quite close to each other, as are the "major seventh" (1162.5) and octave (1200). As the fifth gets closer and closer to 5edo's fifth, the small step becomes more and more narrow.

In [[5edo]] itself, if you attempt to generate a diatonic scale by stacking 5edo's 720-centfifth, the diatonic small step becomes 0 cents, which means two pairs of notes in the diatonic scale will be the same, collapsing it into a 5-note scale (this is why a 1:0 step ratio is known as "collapsed" in [[TAMNAMS]]). For example, attempting to create the major scale in 5edo results in (in cents 240-480-480-720-960-1200-1200 – now the "major third" and "perfect fourth" are ''identical'', as are the "major seventh" and "octave".

But what happens if you were to make the fifth even sharper than 5edo's? Well now the small step is going to become even smaller than 0 cents, that is, it will become negative in size. Let's try to create a major scale by stacking 28edo's 728.6-cent fifth, which is just slightly larger than 5edo's: 257.1-514.3-471.4-728.6-985.7-1242.9-1200. Now the so-called "major third" is ''larger'' than the "perfect fourth", and likewise for the "major seventh" being larger than the octave.

Obviously it is quite nonsensical to have an octave-repeating scale with the notes out of order, so let's put these in order and octave-reduce the 1242.9-cent "major seventh" to 42.9 cents: 42.9-257.1-471.4-514.3-728.6-985.7-1200. This scale is now in order but it is not a MOS because there are 3 step sizes, because there is now the 42.9-cent small step, the 214.3-cent large step and a single 257.1-cent "extra large" step (between 728.6 and 985.7). If you want to turn it back into a MOS scale, then the extra-large step can be simply split into one large step and one small step, which adds an extra note. This results in the "Celephaïsian" mode of 28edo's oneirotonic scale, 42.9-257.1-471.4-514.3-728.6-942.9-985.7-1200. This approach will work for any fifth between 720 and 750 cents (the generator range of oneirotonic). So the oneirotonic scale can be seen as the result of a diatonic scale with fifths even sharper than that of 5edo, just with an adjustment to make it into a normal properly-ordered MOS scale.

== Samples ==

@@ Line 174: / Line 174: @@
 == Approaches ==
 * [[5L&nbsp;3s/Temperaments]]
+== 5L 3s as diatonic with a fifth sharper than 5edo ==
+You may have heard how [[2L 5s]] can be viewed as diatonic ([[5L 2s]]) generated by a fifth flatter than [[7edo]], in which case the large and small steps flip sizes. Similar to that, 5L 3s can be viewed as diatonic generated by a fifth sharper than [[5edo]]. To understand how this works and how you suddenly get an extra small step, let's first start with fifths slightly flat of 5edo and 5edo's fifth itself.
+In diatonic scales generated by fifths slightly flat of [[5edo]]'s fifth, the small step (found by 8 octaves - 5 fifths) becomes increasingly narrow, for example in [[27edo]] (which has a fifth of 711.1 cents) the diatonic small step is 44.4 cents, and in [[32edo]] (which has a fifth of 712.5 cents) the diatonic small step is even smaller at 37.5 cents. The fifth-generated major scale in 32edo in [[cent]]s is 225-450-487.5-712.5-937.5-1162.5-1200 – notice how the "major third" (450) and "perfect fourth" (487.5) are quite close to each other, as are the "major seventh" (1162.5) and octave (1200). As the fifth gets closer and closer to 5edo's fifth, the small step becomes more and more narrow.
+In [[5edo]] itself, if you attempt to generate a diatonic scale by stacking 5edo's  720-centfifth, the diatonic small step becomes 0 cents, which means two pairs of notes in the diatonic scale will be the same, collapsing it into a 5-note scale (this is why a 1:0 step ratio is known as "collapsed" in [[TAMNAMS]]). For example, attempting to create the major scale in 5edo results in (in cents 240-480-480-720-960-1200-1200 – now the "major third" and "perfect fourth" are ''identical'', as are the "major seventh" and "octave".
+But what happens if you were to make the fifth even sharper than 5edo's? Well now the small step is going to become even smaller than 0 cents, that is,  it will become negative in size. Let's try to create a major scale by stacking 28edo's 728.6-cent fifth, which is just slightly larger than 5edo's: 257.1-514.3-471.4-728.6-985.7-1242.9-1200. Now the so-called "major third" is ''larger'' than the "perfect fourth", and likewise for the "major seventh" being larger than the octave.
+Obviously it is quite nonsensical to have an octave-repeating scale with the notes out of order, so let's put these in order and octave-reduce the 1242.9-cent "major seventh" to 42.9 cents: 42.9-257.1-471.4-514.3-728.6-985.7-1200. This scale is now in order but it is not a MOS because there are 3 step sizes, because there is now the 42.9-cent small step, the 214.3-cent large step and a single 257.1-cent "extra large" step (between 728.6 and 985.7). If you want to turn it back into a MOS scale,  then the extra-large step can be simply split into one large step and one small step, which adds an extra note. This results in the "Celephaïsian" mode of 28edo's oneirotonic scale, 42.9-257.1-471.4-514.3-728.6-942.9-985.7-1200. This approach will work for any fifth between 720 and 750 cents (the generator range of oneirotonic). So the oneirotonic scale can be seen as the result of a diatonic scale with fifths even sharper than that of 5edo, just with an adjustment to make it into a normal properly-ordered MOS scale.
 == Samples ==

5L 3s: Difference between revisions

Lucius Chiaraviglio (talk | contribs)