Thursday, August 29, 2019

A Cosine-derived Well Temperament

Ever since I read about Brad Lehman's "Bach" tuning (some time in 2006), I have been interested in experimenting with unequal "well" temperaments for keyboard instruments.  But, it was only recently that I got a hardware synthesizer with reasonable capabilities for custom tunings.

While there seems to be insufficient proof that Lehman's tuning truly recreates anything used by J. S. Bach, it does adhere to the general goal that's implied in other historical well temperaments I've since read about:  A gradual change in "coloration" of the tuning as you go around the circle of fifths, with the most "just" major third (i.e. flat of equal-tempered) occurring in the key of C and the most "Pyhagorean" major third (i.e. sharp of ET) at the other side of the circle at F♯.  The width of the fifths in these temperaments appear to have been derived by trial and error to close the circle while achieving the desired thirds.  Precision was of course limited by the mechanical precision of the piano and the capabilities of the human tuner.

I decided to develop a well temperament mathematically, using a cosine function to adjust the width of the fifths gradually.  I used a spreadsheet to derive the thirds, and below are links (to Google Sheets) and a copy of the output, with two variations. The amount of stretch was chosen so that the widest fifths would be perfect (i.e. 3:2, about 701.955¢). For each of version, the flattest third (C-E) ends up about 7¢ sharp of just, and the sharpest third is about 20¢ sharp of just (but not quite Pythagorean).

Centered on C, with perfect fifths at C♯-G♯ and G♯-D♯:

-->
NoteIndexCosineFifthDegreeM3∆ Just
C00.7071698.5688400.000000C-E:393.2276786.913964
G10.9659698.044999698.568840G-B:394.1349977.821284
D20.9659698.044999196.613839D-F#:396.61383910.300125
A30.7071698.568840894.658838A-C#:400.00000013.686286
E40.2588699.476159393.227678E-G#:403.38616117.072447
B5-0.2588700.5238411092.703837B-D#:405.86500319.551289
F#6-0.7071701.431160593.227678F#-A#:406.77232220.458608
C#7-0.9659701.95500194.658838C#-F:405.86500319.551289
G#8-0.9659701.955001796.613839G#-C:403.38616117.072447
D#9-0.7071701.431160298.568840D#-G:400.00000013.686286
A#10-0.2588700.5238411000.000000A#-D:396.61383910.300125
F110.2588699.476159500.523841F-A:394.1349977.821284

Centered between C and G
, with a perfect fifth at G♯-D♯:

-->
NoteIndexCosineFifthDegreeM3∆ Just
C00.5000699.0225000.000000C-E:393.6813387.367624
G10.8660698.306920699.022500G-B:393.6813387.367624
D21.0000698.044999197.329419D-F#:395.3744189.060704
A30.8660698.306920895.374418A-C#:398.30692011.993206
E40.5000699.022500393.681338E-G#:401.69308015.379367
B50.0000700.0000001092.703837B-D#:404.62558218.311868
F#6-0.5000700.977500592.703837F#-A#:406.31866220.004948
C#7-0.8660701.69308093.681338C#-F:406.31866220.004948
G#8-1.0000701.955001795.374418G#-C:404.62558218.311868
D#9-0.8660701.693080297.329419D#-G:401.69308015.379367
A#10-0.5000700.977500999.022500A#-D:398.30692011.993206
F110.0000700.000000500.000000F-A:395.3744189.060704

Here are the tunings in Scala format if you would like to use them:

! Cosine C Well.scl
!
C-Centered Cosine Well
 12
!
 94.65884
 196.61384
 298.56884
 393.22768
 500.52384
 593.22768
 698.56884
 796.61384
 894.65884
 1000.00000
 1092.70384
 2/1

! Cosine CG Well.scl
!
C/G-Centered Cosine Well
 12
!
 93.68134
 197.32942
 297.32942
 393.68134
 500.00000
 592.70384
 699.02250
 795.37442
 895.37442
 999.02250
 1092.70384
 2/1

Incidentally, these are rather nice-looking when viewed through Scala's temperament radar:

Monday, September 3, 2018

Stopping the Squeaks on a Burghardt Scissor-Jack Piano Bench

With a good piano, decent microphones, a [Really] nice preamp, and a not-bad A/D converter, you'd hope that playing ability would be the only thing to prevent a good piano recording in a home studio…  Then you play back to find that the squeaking piano bench is louder than the all but the loudest passages. Judging from the numerous discussion threads on piano-related online forums, I presume it's a common problem.  I discovered a relatively simple solution that might be helpful to others.

My bench is an adjustable Burghardt with a "scissor jack" style lifting mechanism, but my squeak solution should be applicable to some other types of benches.



I searched online for some solutions:

  • "Frequently tightening bolted joints." – Considering the bench came with its own wrench, the manufacturer must have been expecting this approach, but eventually it stops working.  It also only works for squeaks between the legs and apron board.
  • "Ensuring the bench is upright, on a level surface before tightening leg bolts." – I loosened the leg bolts and tried, but there was no difference.
  • "Putting Vaseline (or other lubricants) in wood-to-wood joints between apron and legs." – It seemed like this would be bad for the wood finish.
  • "WD40 at various locations." – Also bad for the wood finish, and not a good long term solution.

As I was reading these aloud, my wife suggested inserting a piece of fabric into the joints, and offered a scrap of nylon. That's when I remembered I had a partial roll of Rockler "Nylo-Tape" from a furniture project.  Nylo-tape is adhesive-backed nylon tape. I had 3/4" wide × 10 mil thick on hand, which worked just right.

First, I removed the legs (keeping track of which goes where!) and applied short pieces of Nylo-tape to the ends of each apron board, where they bear against the legs.

Then I applied long strips of Nylo-tape between the the lifting mechanism and the two wooden frames.  I just cut them a little longer than needed, marked the backing with the position of each hole, and then used a standard handheld office paper punch to make holes for the screws in each strip of tape.  (I actually removed each frame from the mechanism in turn, rather than all at once, to make it easier to put back in the right orientation.)



I did not apply any Nylo-tape between the corner brackets and the apron, partly because the squeaking seemed to stop after the first round of fixes, and partly because I ran out of Nylo-tape.  If new squeaks develop there, I may have to apply some later:

Tuesday, January 26, 2016

The Tube Technology behind Ghost in the Museum

A brief update because Ghost in the Museum by Diane Marie Kloba was released today, and I used some of my homebrew gear in recording my parts for the album:

My BoxHead amplifier was used for the baritone guitar part that's featured throughout the track "Rescued".  It was close-mic'd with an Electro-Voice N/D-468 dynamic, one of my favorite all purpose dynamics. The guitar itself is the Danelectro 63 that I upgraded a while ago:



My re-worked LK5H DI Box was used for the baritone guitar (including EBow passage) on "Only Summer Brings":



The LK5H was also used for the parallel EBow guitar parts on "A Thousand Pretty Strings". I really liked using it on the EBow because it simplified recording, and because the guitars have such a "hot" signal level when driven by the EBow that the pentode input on the LK5H gets some nice clipping character:



Bonus track: This was released on Soundcloud last year and is not part of any album. I played the lead guitar part on "Aching" through my Buffalo Box B175 and my Lama Kazu 8SE Amplifier.  I recorded with a Cascade Vin-Jet ribbon mic.