Musical Tunings

Musical Tunings
	Part 1	Part 2

The TUNING structure is useful to manage the intervals and the frequencies of a musical scale. For example:

(in-package :scratch)

(defvar *tun*
  (make-tuning :notes '(16/15 9/8 6/5 5/4 4/3 7/5 3/2 8/5 5/3 9/5 15/8 2/1)
               :description "Basic JI with 7-limit tritone. Robert Rich: Geometry"))

The pitch values are rational numbers and/or floating point values (cents). The first note 1/1 (or 0.0) is implicit. It is compatible with a popular scale file format.

SCRATCH> (tuning-ratios *tun*)
#(1 16/15 9/8 6/5 5/4 4/3 7/5 3/2 8/5 5/3 9/5 15/8 2)

SCRATCH> (tuning-cents *tun*)
#(0.0 111.731285 203.90999 315.6413 386.3137 498.045 582.5122 701.95496
  813.6863 884.3587 1017.59625 1088.2687 1200.0)

SCRATCH> (tuning-description *tun*)
"Basic JI with 7-limit tritone. Robert Rich: Geometry"

We can also set the tuning of an external synthesizer by sending a MIDI bulk tuning dump message.

The list of the PortMidi devices is

SCRATCH> (pm:print-devices-info :output)

         ;; Create a PM:OUTPUT-STREAM
SCRATCH> (defvar *midiout* (pm:open YOUR-DEVICE-ID :direction :output))

In this example the bulk tuning dump message is recorded in a MIDI file for simplicity. From a shell (or any MIDI sequencer):

arecordmidi -p YOUR-ALSA-PORT ji_12.mid

and from Incudine:

SCRATCH> (midi-tuning-sysex *tun* *midiout*)

The default for device-id and program is zero.

Alternatively, if SINGLE-NOTE-TUNING-P is non-NIL, it sends 128 single note tuning change messages (Exclusive Real Time). It is also possible to set a different function to compute the checksum of the SysEx.

Ok, we'll use this MIDI file later on.

The reference frequency of a TUNING, the related keynum and the degree index are:

SCRATCH> (list (tuning-freq-base *tun*)
               (tuning-keynum-base *tun*)
               (tuning-degree-index *tun*))
(440.0d0 69 9)

and the frequencies from keynum 60 to keynum 72 are

SCRATCH> (loop for k from 60 to 72 collect (tuning-cps *tun* k))
(264.0d0 281.6d0 297.0d0 316.8d0 330.0d0 352.0d0 369.59999999999997d0
 396.0d0 422.40000000000003d0 440.0d0 475.2d0 495.0d0 528.0d0)

With a different reference frequency:

SCRATCH> (set-tuning-reference *tun* 60 261.625 0)

SCRATCH> (loop for k from 60 to 72 collect (tuning-cps *tun* k))
(261.625d0 279.06666666666666d0 294.328125d0 313.95d0 327.03125d0
 348.8333333333333d0 366.275d0 392.4375d0 418.6d0 436.0416666666667d0
 470.925d0 490.546875d0 523.25d0)

We can use SET-TUNING to change the TUNING notes:

SCRATCH> (set-tuning *tun* '(63.8 127.6 191.4 255.2 319.0 382.8 446.6
                             510.4 574.2 638.0 701.8 765.6 829.4
                             893.2 957.0 1020.8 1084.6 1148.4 1212.2
                             1276.0 1339.8 1403.6)
           "Wendy Carlos' Beta scale with perfect fifth divided by eleven"
           69 440 9)
#<TUNING "Wendy Carlos' Beta scale with perfect fifth divided by eleven">

SCRATCH> (loop for k from 60 to 82 collect (tuning-cps *tun* k))
(315.7983193277311d0 327.65332578481366d0 339.95337000168405d0
 352.71512648605324d0 365.95599379894475d0 379.6938919825375d0
 393.94752650215275d0 408.73624910271997d0 424.0801065278395d0
 440.0d0 456.51748012974946d0 473.6550443880858d0 491.4359701061084d0
 509.8844048742105d0 529.0253944812085d0 548.8848610028032d0
 569.4898901817311d0 590.8683747069908d0 613.0494578503176d0
 636.0632050985532d0 659.9409416603221d0 684.7149924512975d0
 710.4189488799282d0)

TUNING-SAVE is useful to save the tuning to a text file in scale file format:

SCRATCH> (tuning-save *tun* "/tmp/carlos_beta.scl")
#P"/tmp/carlos_beta.scl"

The file's contents:

! carlos_beta.scl
!
Wendy Carlos' Beta scale with perfect fifth divided by eleven
22
!
63.800
127.600
191.400
255.200
319.0
382.800
446.600
510.400
574.200
638.0
701.800
765.600
829.400
893.200
957.0
1020.800
1084.600
1148.400
1212.200
1276.0
1339.800
1403.600

LOAD-SCLFILE is useful to get the intervals from a scl file:

SCRATCH> (load-sclfile "/tmp/carlos_beta.scl")
(63.8 127.6 191.4 255.2 319.0 382.8 446.6 510.4 574.2 638.0 701.8 765.6
 829.4 893.2 957.0 1020.8 1084.6 1148.4 1212.2 1276.0 1339.8 1403.6)
22
"Wendy Carlos' Beta scale with perfect fifth divided by eleven"

We can also change the frequencies of a TUNING with the data received from a MIDI bulk tuning dump message.

Note: the frequency-table of a VOICER:MIDI-EVENT is automatically updated if the VOICER:MIDI-EVENT responder receives a MIDI bulk tuning dump message. I omit the example for the VOICER because it is trivial.

Now is the time to use the previously recorded MIDI file:

;; Create a PM:INPUT-STREAM
(defvar *midiin* (pm:open (pm:get-default-input-device-id)))

(recv-start *midiin*)

(make-responder *midiin*
                (lambda (st d1 d2)
                  (declare (ignore d1 d2))
                  (when (pm:sysex-message-p st)
                    (set-tuning-from-midi *tun* *midiin*))))

From a shell (or any MIDI sequencer):

aplaymidi -p YOUR-ALSA-PORT ji_12.mid

Done.

SCRATCH> (recv-stop *midiin*)
SCRATCH> (remove-all-responders *midiin*)
SCRATCH> (pm:terminate)

Now the frequencies are again:

SCRATCH> (loop for k from 60 to 72 collect (tuning-cps *tun* k))
(264.0003435899381d0 281.60031310507037d0 296.9997429143919d0
 316.79966571076d0 329.99968619907844d0 351.99959854058784d0
 369.59944167764706d0 396.0000863018157d0 422.40001196906144d0
 440.00005835720447d0 475.2006589983342d0 495.0007380822791d0
 528.0006871798762d0)

but the intervals…

SCRATCH> (tuning-ratios *tun*)
#(1 6191/5967 3814/3543 4309/3858 7318/6315 6501/5407 4799/3847 4288/3313
  1500/1117 2618/1879 3202/2215 5581/3721 6469/4157 3121/1933 16948/10117
  4851/2791 7464/4139 3830/2047 9883/5091 5269/2616 2468/1181 7760/3579
  8373/3722)

… are not updated, because it's impossible to know the number of the notes of a musical scale from a MIDI bulk tuning dump message. TUNING-NOTES-FROM-DATA is useful for this work:

SCRATCH> (tuning-notes-from-data *tun* 60 72 "Basic JI with 7-limit tritone")
#<TUNING "Basic JI with 7-limit tritone">

SCRATCH> (tuning-ratios *tun*)
#(1 16/15 9/8 6/5 5/4 4/3 7/5 3/2 8/5 5/3 9/5 15/8 2)

SCRATCH> (tuning-cents *tun*)
#(0.0 111.731285 203.90999 315.6413 386.3137 498.045 582.5122 701.95496
  813.6863 884.3587 1017.59625 1088.2687 1200.0)

Sparkling!

Note: TUNING-NOTES-FROM-DATA uses a particular algorithm that minimizes a rational number by introducing an error in the significand of the floating point number. This error is 0.0005% by default. If you want use the original RATIONALIZE from Common Lisp, set the significand-error to zero.

SET-TUNING also works with a scl file:

SCRATCH> (set-tuning *tun* "/tmp/carlos_beta.scl")
#<TUNING "Wendy Carlos' Beta scale with perfect fifth divided by eleven">

SCRATCH> (tuning-cents *tun*)
#(0.0 63.8 127.6 191.4 255.2 319.0 382.8 446.6 510.4 574.2
  638.0 701.8 765.6 829.4 893.2 957.0 1020.8 1084.6 1148.4
  1212.2 1276.0 1339.8 1403.6)

Both the structures BUFFER and TUNING extend BUFFER-BASE, therefore it is possible to change the frequencies of a TUNING by using the following utilities:

MAP-BUFFER
MAP-INTO-BUFFER
SCALE-BUFFER
NORMALIZE-BUFFER
RESCALE-BUFFER
SORT-BUFFER
CIRCULAR-SHIFT
QUANTIZE

also BUFFER->LIST works with a TUNING:

SCRATCH> (buffer->list *tun*)
(34.6038332657699d0 35.902855233562235d0 37.2506395151566d0
 38.649022046482d0 40.09989580486158d0 41.60523693859668d0
 ...
 3465.1280017082695d0 3595.2076701213614d0 3730.1710550898856d0)

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