User:TallKite/The delta method: Difference between revisions

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WORK IN PROGRESS

TO DO:

* clean up natural generators section

* split up into multiple pages: Delta method,

* make a new page and category, "mental math", that links to all these pages

* make a new page, how to find an edo's 5th, about the divide-by-5 rule from my Notation Guide for Edos 5-72

The delta method is a way to find the [[wikipedia:Stern–Brocot_tree|Stern-Brocot]] ancestors of a reasonably-sized ratio quickly and easily in one's head without using paper and pencil. It can be used to estimate the size in cents of a ratio. It also has applications in comparing edos.

The usual way of finding a Stern-Brocot ancestor is to use the [[wikipedia:Extended_Euclidean_algorithm|extended Euclidean algorithm]], which requires a computer. But there's times when computer usage isn't appropriate. You might be at a company retreat, and your boss's boss is telling a long anecdote. You lose interest, and start to wonder, which interval in 31edo is the farthest from 41edo? Or you're attending a performance of your friend's 17-minute-long minimalist 23-limit piece, and you start to wonder, roughly how many cents is 23/13 anyway? The delta method allows you to solve such problems in your head.

The usual way of finding a Stern-Brocot ancestor is to use the [[wikipedia:Extended_Euclidean_algorithm|extended Euclidean algorithm]], which requires a computer. But there's times when computer usage isn't appropriate. You might be at a company retreat, and your boss's boss is telling a long anecdote. You lose interest, and start to wonder, which interval in 31edo is the farthest from 41edo? Or you're attending a performance of your friend's 17-minute-long minimalist 23-limit piece, and you start to wonder, roughly how many cents is 23/13 anyway?

The delta method allows you to solve such problems in your head.

== Background and terminology ==

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All of the above generalizes to [[Edonoi|EDONOIs]].

==== Finding the natural generator ====

Every non-perfect edo has a "natural" heptatonic generator. For 13edo, it's a 2\13 2nd (and its octave inverse of course), because seven 2\13's falls only one edostep away from the octave. Thus the sharp means "sharpened by one edostep", major is one edostep wider than minor, and ups and downs aren't needed.

The natural heptatonic generator of N-edo is simply the nearest miss of that edo and 7edo. The natural generator is always one of these:

* the perfect 2nd (or the perfect 7th) - edos 8, 13, 15, 20, 22, 27, 29...

* the perfect 3rd (or the perfect 6th) - edos 10, 11, 17, 18, 24, 25...

* the perfect 5th (or the perfect 4th) - edos 9, 12, 16, 19, 23, 26... (sharp-1 and flat-1 edos)

* the perfect 2nd, 3rd, 4th, 5th, 6th or 7th - edos 7, 14, 21, 28... (sharp-0 edos)

The usual genchain of fifths runs ...d5 - m2 - m6 - m4 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 - A4... This can be generalized to any generator: The generator is always perfect, so the middle three intervals of the genchain are always perfect. One side of the genchain is always major or augmented, and the other side is always minor or diminished. For heptatonic notation, there are four major and four minor intervals. For pentatonic, there's two of each. In general, N-3 of each. The major side is usually chosen so that major is wider than minor. The only exception is for fifth-generated notation of superflat EDOs, when major may be on the left even when it should be on the right, in order to preserve familiar interval arithmetic.

For 13edo, the genchain runs in 2nds: ...5 - 6 - 7 - 1 - 2 - 3 - 4 - 5... The righthand 5th is the sum of four perfect 2nds, and equals 4 * (2\13) = 8\13. The lefthand 5th is the octave minus three perfect 2nds, and equals 13\13 - 3 * (2\13) = 7\13. The righthand one is larger and therefore major. Thus the 13edo genchain is ...d8 - d2 - m3 - m4 - m5 - m6 - P7 - P1 - P2 - M3 - M4 - M5 - M6 - A7 - A1...

For 17edo, the generator is the 5\17 3rd. The genchain runs in 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

There is a natural pentatonic generator for all edos...

TO DO: explore octotonic, etc.

=== Finding the fifthspan ===

@@ Line 1: / Line 1: @@
 WORK IN PROGRESS
+TO DO:
+* clean up natural generators section
+* split up into multiple pages: Delta method,
+* make a new page and category, "mental math", that links to all these pages
+* make a new page, how to find an edo's 5th, about the divide-by-5 rule from my Notation Guide for Edos 5-72
 The delta method is a way to find the [[wikipedia:Stern–Brocot_tree|Stern-Brocot]] ancestors of a reasonably-sized ratio quickly and easily in one's head without using paper and pencil. It can be used to estimate the size in cents of a ratio. It also has applications in comparing edos.
-The usual way of finding a Stern-Brocot ancestor is to use the [[wikipedia:Extended_Euclidean_algorithm|extended Euclidean algorithm]], which requires a computer. But there's times when computer usage isn't appropriate. You might be at a company retreat, and your boss's boss is telling a long anecdote. You lose interest, and start to wonder, which interval in 31edo is the farthest from 41edo? Or you're attending a performance of your friend's 17-minute-long minimalist 23-limit piece, and you start to wonder, roughly how many cents is 23/13 anyway? The delta method allows you to solve such problems in your head.
+The usual way of finding a Stern-Brocot ancestor is to use the [[wikipedia:Extended_Euclidean_algorithm|extended Euclidean algorithm]], which requires a computer. But there's times when computer usage isn't appropriate. You might be at a company retreat, and your boss's boss is telling a long anecdote. You lose interest, and start to wonder, which interval in 31edo is the farthest from 41edo? Or you're attending a performance of your friend's 17-minute-long minimalist 23-limit piece, and you start to wonder, roughly how many cents is 23/13 anyway?
+The delta method allows you to solve such problems in your head.
 == Background and terminology ==
@@ Line 81: / Line 90: @@
 All of the above generalizes to [[Edonoi|EDONOIs]].
+==== Finding the natural generator ====
+Every non-perfect edo has a "natural" heptatonic generator. For 13edo, it's a 2\13 2nd (and its octave inverse of course), because seven 2\13's falls only one edostep away from the octave. Thus the sharp means "sharpened by one edostep", major is one edostep wider than minor, and ups and downs aren't needed.
+The natural heptatonic generator of N-edo is simply the nearest miss of that edo and 7edo. The natural generator is always one of these:
+* the perfect 2nd (or the perfect 7th) - edos 8, 13, 15, 20, 22, 27, 29...
+* the perfect 3rd (or the perfect 6th) - edos 10, 11, 17, 18, 24, 25...
+* the perfect 5th (or the perfect 4th) - edos 9, 12, 16, 19, 23, 26... (sharp-1 and flat-1 edos)
+* the perfect 2nd, 3rd, 4th, 5th, 6th or 7th - edos 7, 14, 21, 28... (sharp-0 edos)
+The usual genchain of fifths runs ...d5 - m2 - m6 - m4 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 - A4... This can be generalized to any generator: The generator is always perfect, so the middle three intervals of the genchain are always perfect. One side of the genchain is always major or augmented, and the other side is always minor or diminished. For heptatonic notation, there are four major and four minor intervals. For pentatonic, there's two of each. In general, N-3 of each. The major side is usually chosen so that major is wider than minor. The only exception is for fifth-generated notation of superflat EDOs, when major may be on the left even when it should be on the right, in order to preserve familiar interval arithmetic.
+For 13edo, the genchain runs in 2nds: ...5 - 6 - 7 - 1 - 2 - 3 - 4 - 5... The righthand 5th is the sum of four perfect 2nds, and equals 4 * (2\13) = 8\13. The lefthand 5th is the octave minus three perfect 2nds, and equals 13\13 - 3 * (2\13) = 7\13. The righthand one is larger and therefore major. Thus the 13edo genchain is ...d8 - d2 - m3 - m4 - m5 - m6 - P7 - P1 - P2 - M3 - M4 - M5 - M6 - A7 - A1...
+For 17edo, the generator is the 5\17 3rd. The genchain runs in 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...
+There is a natural pentatonic generator for all edos...
+TO DO: explore octotonic, etc.
 === Finding the fifthspan ===