User:Oak Blood Three: Difference between revisions

(4 intermediate revisions by the same user not shown)

Line 1:

I am a professional electrical engineer and [https://soundcloud.com/oakbloodthree amateur musician].

I first started to see the cracks in equal temperament when tuning a guitar to pure fourths and a third. At the time, I didn't know the source of the error in the pitch of the high E, or that it was called the [[Syntonic comma]]. I thought maybe the intonation was off on my guitar.

I first started to see the cracks in equal temperament as a highschooler when tuning a guitar to pure fourths and a third. At the time, I didn't know the source of the error in the pitch of the high E, or that it was called the [[Syntonic comma]]. I thought maybe the intonation was off on my guitar.

I first learned in earnest about tuning theory as a university student. The pianos in the rehearsal rooms at my [[Wikipedia:University of Illinois at Urbana–Champaign|university]] were badly out of tune. I got tuning tools and learned how to tune by ear. (I also learned a bit about "well" temperaments.)

I'm mostly interested in just intonations, but not exclusively.

I'm mostly interested in just intonations (and things approximating them, and alterations of them to accommodate [[Wikipedia:inharmonicity|inharmonicity]]), but not exclusively.

In fact, here are some [[Well temperament|Well temperaments]] I devised and [https://docs.google.com/spreadsheets/d/1_jBmq1HOHUjMxPgUsWaLLh8iIVhWlZs2muKwljPmUMc/copy the spreadsheet used to calculate them.]

They are two related tunings from a family of 12-tone "almost-equal" temperaments, where fifths squeeze and stretch sinusoidally as you go around the circle.

The first variation is symmetrical about D, with the narrowest fifths (696.05¢) at G–D and D-A and the widest fifths (703.95¢) at C♯–G♯ and G♯–D♯. The amount of stretch is selected to make the major third from C–E just (5/4). This leaves the major third at F♯–A♯ very wide at 407.82¢ or 5.87¢ wider than Pythagorean.

! Cosine D Well C-E Just.scl

!

Kloba's Cosine Well D-Centered

12

!

89.21

193.16

297.11

386.31

501.06

586.31

697.11

793.16

889.21

1000.00

1085.26

2/1

The second variation is also symmetrical about D, with the amount of stretch selected to make the major third from F♯–A♯ Pythagorean (81/64). The narrowest fifths are 697.74¢ (at G–D and D-A) and the widest fifths are 702.26¢ (at C♯–G♯ and G♯–D♯). This leaves the major third from C–E a little wide at 392.18¢ or 5.87¢ wider than just.

! Cosine D Well F#-A# Pyth.scl

!

Kloba's Cosine Well D-Centered

12

!

93.83

196.09

298.35

392.18

500.60

592.18

698.35

796.09

893.83

1000.00

1091.58

2/1

[[/sandbox]]

@@ Line 1: / Line 1: @@
 I am a professional electrical engineer and [https://soundcloud.com/oakbloodthree amateur musician].
-I first started to see the cracks in equal temperament when tuning a guitar to pure fourths and a third.  At the time, I didn't know the source of the error in the pitch of the high E, or that it was called the [[Syntonic comma]].  I thought maybe the intonation was off on my guitar.
+I first started to see the cracks in equal temperament as a highschooler when tuning a guitar to pure fourths and a third.  At the time, I didn't know the source of the error in the pitch of the high E, or that it was called the [[Syntonic comma]].  I thought maybe the intonation was off on my guitar.
 I first learned in earnest about tuning theory as a university student.  The pianos in the rehearsal rooms at my [[Wikipedia:University of Illinois at Urbana–Champaign|university]] were badly out of tune.  I got tuning tools and learned how to tune by ear.  (I also learned a bit about "well" temperaments.)
-I'm mostly interested in just intonations, but not exclusively.
+I'm mostly interested in just intonations (and things approximating them, and alterations of them to accommodate [[Wikipedia:inharmonicity|inharmonicity]]), but not exclusively.
+In fact, here are some [[Well temperament|Well temperaments]] I devised and [https://docs.google.com/spreadsheets/d/1_jBmq1HOHUjMxPgUsWaLLh8iIVhWlZs2muKwljPmUMc/copy the spreadsheet used to calculate them.]
+They are two related tunings from a family of 12-tone "almost-equal" temperaments, where fifths squeeze and stretch sinusoidally as you go around the circle.
+The first variation is symmetrical about D, with the narrowest fifths (696.05¢) at G–D and D-A and the widest fifths (703.95¢) at C♯–G♯ and G♯–D♯.  The amount of stretch is selected to make the major third from C–E just (5/4). This leaves the major third at F♯–A♯ very wide at 407.82¢ or 5.87¢ wider than Pythagorean.
+ ! Cosine D Well C-E Just.scl
+   !
+   Kloba's Cosine Well D-Centered
+   !
+.21
+.16
+.11
+.31
+.06
+.31
+.11
+.16
+.21
+.00
+.26
+/1
+The second variation is also symmetrical about D, with the amount of stretch selected to make the major third from F♯–A♯ Pythagorean (81/64). The narrowest fifths are 697.74¢ (at G–D and D-A) and the widest fifths are 702.26¢ (at C♯–G♯ and G♯–D♯). This leaves the major third from C–E a little wide at 392.18¢ or 5.87¢ wider than just.
+ ! Cosine D Well F#-A# Pyth.scl
+   !
+   Kloba's Cosine Well D-Centered
+   !
+.83
+.09
+.35
+.18
+.60
+.18
+.35
+.09
+.83
+.00
+.58
+/1
+[[/sandbox]]