Yer: Difference between revisions

Line 277:

]]

== ~~Blume~~ comma ==

== Yama comma ==

The reason why 13 and 11 * 19 are so close is that it turns out there is a *another* comma existing in this ~~world~~ of 11’s, 13’s, 17’s, and 19’s ~~— not nearly as exciting or colorful~~ of ~~one, but it’s there~~. If we move by an eleventh, a seventeenth, and another eleventh, we end up right back where you started, off by 7.591 cents~~. 13 and 11 * 19 are off from each~~ other ~~by an amount of~~ one Blumeyer comma ~~and one~~ Blume comma~~. These two commas are just both so small that it doesn’t much matter. 7.591 + 0.712 is still just 8.303 cents~~.

The reason why 13 and 11 * 19 are so close is that it turns out there is *another* comma existing as a step in this tuning of 11’s, 13’s, 17’s, and 19’s. If we move by an eleventh, then a nineteenth, and then down a thirteenth, we end up right back where we started, off by 8.303 cents. That's the yama comma.

{| class="wikitable"

|+

!name

!value

!cents

!monzo

|-

|yama comma

|209/208

|8.303296728

|<nowiki>| -4 0 0 0 1 -1 0 1 ></nowiki>

|}

Another consequence of the yama comma is that a couple “extra” lattice connections appear. These are drawn in dotted green on the lattices in the introduction.

If we move by an eleventh, a seventeenth, and another eleventh, we end up right back where you started, off by 7.591 cents, or in other words, one yama comma minus one Blumeyer comma. We call this the Blume comma.

{| class="wikitable"

|+

Line 294:

Line 308:

Blume comma visualized

]]

~~Another consequence of the Blume comma is that a couple “extra” lattice connections appear. These are drawn in dotted green on the lattices in the introduction.~~

The idea is that if you were to try to go from 11 * 17 by an eleventh to 11 * 11 * 17, well, you’re not really allowed to do that because the EFG does not duplicate factors (you can’t have two 11’s), but since 11 * 11 * 17 is essentially 1, we’ll permit it.

Line 303:

Line 315:

And as long as we're changing the angle we look on the cube to bring the right pairs of pitches together, we have also taken care to balance these dotted lines with the real 11 lines, so that it’s the zig to the zag of the real 11, more strongly suggesting the dimension of the 11th harmonic’s relationship to that of the 17th (i.e that two 11's make a 17).

Now we could have drawn dotted lines connecting 11 * 13 * 17 to 13, but declined,

Now we could have drawn dotted lines connecting 11 * 13 * 17 to 13, but declined, considering that the ability to move between these two pitches is already achievable by modulating from from 11 * 13 * 17 to 19, then moving by that 11 to 11 * 19, then modulating to 13. The same goes for the connection between 11 * 17 * 19 and 19. This would be pretty obvious on the cycle view, because we’d just be drawing a dotted line right alongside an existing solid one.

considering that the ability to move between these two pitches is already achievable

by modulating from from 11 * 13 * 17 to 19, then moving by that 11 to 11 * 19, then modulating to 13. The same goes for the connection between 11 * 17 * 19 and 19. This would be pretty obvious on the cycle view, because we’d just be drawing a dotted line right alongside an existing solid one.

Here’s something else interesting: we can move by four 11’s in a row. We can move from 17 * 19 to 11 * 17 * 19, which can be shifted to 13 * 17, then move to 11 * 13 * 17, which can be shifted to 19, then move to 11 * 19, which can be shifted to 13, then move to 11 * 13.

@@ Line 277: / Line 277: @@
 ]]
-== Blume comma ==
+== Yama comma ==
-The reason why 13 and 11 * 19 are so close is that it turns out there is a *another* comma existing in this world of 11’s, 13’s, 17’s, and 19’s — not nearly as exciting or colorful of one, but it’s there. If we move by an eleventh, a seventeenth, and another eleventh, we end up right back where you started, off by 7.591 cents. 13 and 11 * 19 are off from each other by an amount of one Blumeyer comma and one Blume comma. These two commas are just both so small that it doesn’t much matter. 7.591 + 0.712 is still just 8.303 cents.
+The reason why 13 and 11 * 19 are so close is that it turns out there is *another* comma existing as a step in this tuning of 11’s, 13’s, 17’s, and 19’s. If we move by an eleventh, then a nineteenth, and then down a thirteenth, we end up right back where we started, off by 8.303 cents. That's the yama comma.
+{| class="wikitable"
+|+
+!name
+!value
+!cents
+!monzo
+|-
+|yama comma
+|209/208
+|8.303296728
+|<nowiki>| -4 0 0 0 1 -1 0 1 ></nowiki>
+|}
+Another consequence of the yama comma is that a couple “extra” lattice connections appear. These are drawn in dotted green on the lattices in the introduction.
+If we move by an eleventh, a seventeenth, and another eleventh, we end up right back where you started, off by 7.591 cents, or in other words, one yama comma minus one Blumeyer comma. We call this the Blume comma.
 {| class="wikitable"
 |+
@@ Line 294: / Line 308: @@
 Blume comma visualized
 ]]
-Another consequence of the Blume comma is that a couple “extra” lattice connections appear. These are drawn in dotted green on the lattices in the introduction.
 The idea is that if you were to try to go from 11 * 17 by an eleventh to 11 * 11 * 17, well, you’re not really allowed to do that because the EFG does not duplicate factors (you can’t have two 11’s), but since 11 * 11 * 17 is essentially 1, we’ll permit it.
@@ Line 303: / Line 315: @@
 And as long as we're changing the angle we look on the cube to bring the right pairs of pitches together, we have also taken care to balance these dotted lines with the real 11 lines, so that it’s the zig to the zag of the real 11, more strongly suggesting the dimension of the 11th harmonic’s relationship to that of the 17th (i.e that two 11's make a 17).
-Now we could have drawn dotted lines connecting 11 * 13 * 17 to 13, but declined,
+Now we could have drawn dotted lines connecting 11 * 13 * 17 to 13, but declined, considering that the ability to move between these two pitches is already achievable by modulating from from 11 * 13 * 17 to 19, then moving by that 11 to 11 * 19, then modulating to 13. The same goes for the connection between 11 * 17 * 19 and 19. This would be pretty obvious on the cycle view, because we’d just be drawing a dotted line right alongside an existing solid one.
-considering that the ability to move between these two pitches is already achievable
-by modulating from from 11 * 13 * 17 to 19, then moving by that 11 to 11 * 19, then modulating to 13. The same goes for the connection between 11 * 17 * 19 and 19. This would be pretty obvious on the cycle view, because we’d just be drawing a dotted line right alongside an existing solid one.
 Here’s something else interesting: we can move by four 11’s in a row. We can move from 17 * 19 to 11 * 17 * 19, which can be shifted to 13 * 17, then move to 11 * 13 * 17, which can be shifted to 19, then move to 11 * 19, which can be shifted to 13, then move to 11 * 13.