5L 3s: Difference between revisions

Line 175:

* [[5L 3s/Temperaments]]

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

== 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.

Line 184:

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. 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 28edo's oneirotonic scale, 42.9-257.1-300.0-514.3-728.6-771.4-985.7-1200. So the oneirotonic scale can be seen as essentially being the result of a diatonic scale with fifths even sharper than that of 5edo, just with some adjustments to make it back into a normal properly-ordered MOS scale.

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. 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 28edo's oneirotonic scale, 42.9-257.1-300.0-514.3-728.6-771.4-985.7-1200. 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 essentially being the result of a diatonic scale with fifths even sharper than that of 5edo, just with some adjustments to make it back into a normal properly-ordered MOS scale.

== Samples ==

@@ Line 175: / Line 175: @@
 * [[5L&nbsp;3s/Temperaments]]
-== 5L 3s as diatonic with a fifth sharper than 5edo ===
+== 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.
@@ Line 184: / Line 184: @@
 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. 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 28edo's oneirotonic scale, 42.9-257.1-300.0-514.3-728.6-771.4-985.7-1200. So the oneirotonic scale can be seen as essentially being the result of a diatonic scale with fifths even sharper than that of 5edo, just with some adjustments to make it back into a normal properly-ordered MOS scale.
+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. 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 28edo's oneirotonic scale, 42.9-257.1-300.0-514.3-728.6-771.4-985.7-1200. 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 essentially being the result of a diatonic scale with fifths even sharper than that of 5edo, just with some adjustments to make it back into a normal properly-ordered MOS scale.
 == Samples ==

5L 3s: Difference between revisions

CompactStar (talk | contribs)