The CALCULATE command (which can be abbreviated like all commands) evaluates a pitch expression and shows the result. One can use this result subsequently as input to other commands. For example:
calc 3/2^12-2/1^7
which means take 3/2 to the power 12 and divide by 2/1 to the power 7. In other words 12 pure fifths minus 7 pure octaves. It displays the following:
531441/524288 = 2^-19.3^12 = 1.01364326477 size is 23.46001038465 cents, 5.884553 savarts/eptamerides Pythagorean comma
Parentheses can be added if necessary. Ratios and values in cents can be used together in an expression. There is also a longer output format which gives the value in units of a (modifiable) list of intervals. See CALCULATE for the list of operations.
The SHOW SCALE command (which can be abbreviated to SHOW or even SH) displays the working scale in the following form:
Ptolemy's Intense Diatonic, also Zarlino's scale 0: 1/1 0.000000 unison, perfect prime 1: 9/8 203.9100 major whole tone 2: 5/4 386.3139 major third 3: 4/3 498.0452 perfect fourth 4: 3/2 701.9553 perfect fifth 5: 5/3 884.3591 major sixth 6: 15/8 1088.269 classic major seventh 7: 2/1 1200.000 octave
The first column is the scale degree number. The first note is always number 0 with value 1/1 or 0 cents. The second column contains the frequency ratio with respect to degree 0. This need not be a ratio. If it is not, then the value is given in cents. The third column gives the frequency ratio in cents in the preceding example. However the contents of this column can be chosen from a wide range of options such as note names, prime factorisations, linear factors, harmonic complexity measures, interval sizes, logarithmic sizes in any unit, and so on. See also SET ATTRIBUTE.
The SHOW DATA command displays a list of values and properties. For example, given the following scale:
1/4-comma mean-tone scale. Pietro Aaron's temperament (1523). 0: 1/1 0.000000 unison, perfect prime 1: 76.049 cents 76.04904 2: 193.157 cents 193.1569 3: 310.265 cents 310.2648 4: 5/4 386.3139 major third 5: 503.422 cents 503.4218 6: 579.471 cents 579.4708 7: 696.578 cents 696.5787 8: 25/16 772.6278 classic augmented fifth 9: 889.735 cents 889.7357 10: 1006.843 cents 1006.843 11: 1082.892 cents 1082.892 12: 2/1 1200.000 octave
the SHOW DATA command displays:
Number of notes : 12 -- Interval properties -- Smallest interval : 76.04900 cents, class 1 Average step (divided formal octave): 100.0000 cents Largest one step interval : 117.10786 cents Average / Smallest step : 1.314942 Largest / Average step : 1.171079 Largest / Smallest step : 1.539900 Median interval of one step : 117.10786 cents, amount: 7 Most common interval of one step : 117.10786 cents, amount: 7 Least squares average step : 99.04196 cents, oct.: 1188.50352 cents Scale is strictly proper Scale has Myhill's property chromas: 41.05886 cents and 34.99014 cents generators: 5 of 503.42157 cents and 7 of 696.57843 cents ET pair of linear temperament : 12&19 Scale is maximally even for L / S <= 2 Scale is distributional even Scale is a mode of a 789-tone equal temperament with octave 2/1 degrees: 50 127 204 254 331 381 458 508 585 662 712 789 generators: 331 and 458 Number of contiguous 1-step segments: 0 Interval pattern alph. order: ABBABABABBAB Interval pattern size order : SLLSLSLSLLSL Scale is a Constant Structure, by a margin of 34.99015 cents Ratio Whole / Half tone: 1.649393 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.623627 Rothenberg efficiency : 0.722222 redundancy: 0.277778 Efficiency x scale size : 8.666667 Number of different interval sizes : 22 = 2.00000 / class Number of one step interval sizes : 2 Highest interval variety : 2 Mean interval variety : 2.00000 = 2 Lowest interval variety : 2 Smallest interval difference : 34.99015 cents Most common intervals : 503.42157 cents & inv., amount: 11 Scale is a chain of 6 triads 0.0 117.108 696.578 cents Most common triad is 0.0 503.422 696.578 cents, amount: 10 Number of recognisable fifths : 11, average 696.5784 cents Scale contains two identical disjunct hexachords Best fifths form a closed circle Formal octave complements present : 7 = 58.3333% Limited inverse transpositions : 11 Inversional symmetry on intervals : 5-6 11-12 -- Rational properties -- Prime limit : 5 (not all pitches rational)
There are many more commands for obtaining structural and other properties of scales such as COMPARE, LATTICE, SHOW/FREQUENCY, SHOW INTERVALS, SHOW BEATS, SHOW ET_DIFF, SHOW TEMPERINGS, SHOW TRANSPOSE, etc. There's also a built-in procedure for accurately calculating lengths for fretted string instruments: SHOW STRINGLEN.
If the pitches in a scale are close or equal to those of an equal temperament, a notation system for this equal temperament can be selected with the command SET NOTATION (follow this link to see which systems are provided). Take 26-tone equal temperament for example. It's shown as
0: 1/1 C unison, perfect prime 1: 46.154 cents C# Dbbb 2: 92.308 cents Cx Dbb 3: 138.462 cents Cx# Db 4: 184.615 cents D 5: 230.769 cents D# Ebbb 6: 276.923 cents Dx Ebb 7: 323.077 cents Dx# Eb 8: 369.231 cents E 9: 415.385 cents E# Fbb 10: 461.538 cents Ex Fb 11: 507.692 cents F 12: 553.846 cents F# Gbbb 13: 600.000 cents Fx Gbb 14: 646.154 cents Fx# Gb 15: 692.308 cents G 16: 738.462 cents G# Abbb 17: 784.615 cents Gx Abb 18: 830.769 cents Gx# Ab 19: 876.923 cents A 20: 923.077 cents A# Bbbb 21: 969.231 cents Ax Bbb 22: 1015.385 cents Ax# Bb 23: 1061.538 cents B 24: 1107.692 cents B# Cbb 25: 1153.846 cents Bx Cb 26: 2/1 C octave
The command EQUALTEMP/DATA provides a set of properties of a given equal temperament, not necessarily an octave division. This for example is the output for 31-tone equal temperament:
1200.0000 cents divided by 31, step = 38.7097 cents
Nearest to 5/4 : 10, 387.0968 cents, diff. 0.020229 steps, 0.7831 cents
Nearest to 3/2 : 18, 696.7742 cents, diff. -0.133837 steps, -5.1808 cents
Nearest to 7/4 : 25, 967.7419 cents, diff. -0.028002 steps, -1.0840 cents
Nearest to 11/8 : 14, 541.9355 cents, diff. -0.242380 steps, -9.3825 cents
Misfit numbers M1-M5 : 26.84076 27.45395 28.62894 116.65947 239.54205
Relative errors R1-R5 : 53.5350 30.8133 24.2759 42.4449 56.8654 % of average
Highest errors H1-H5 : 26.7675 30.8133 30.8133 57.5808 94.2502 % of half step size
Pepper ambiguities 1-5: 0.15452 0.18213 0.18213 0.40431 0.89126
TOP errors T1-T5 : 3.2687 3.2687 3.2687 3.2687 3.2687 cents
TOP-RMS errors 1-5 : 2.3113 1.8972 1.6543 1.9133 2.1322 cents
Combined error factor : 1.4792 (3 - 7)
Combined error factor : 11.5745 (3 - 11)
Weighted triad dissonance : 3.4217 cents, 0.088394 steps
Major triad min/maj third beat ratio : 11.40708
Highest harmonic represented consistently : 12
highest error 9/5 : 26, diff. -0.287904 steps, -11.1447 cents, level 1
Highest harmonic represented uniquely : 9
highest error 9/5 : 26, diff. -0.287904 steps, -11.1447 cents, level 1
Highest harm. represented uniquely inv. equiv.: 8
highest error 5/3 : 23, diff. 0.154066 steps, 5.9639 cents, level 3
Consistency levels : 3: 3 5: 3 7: 3 9: 1 11: 1
Diameters : 3: 15 5: 4 7: 2 9: 2 11: 2
Consistency 12 region : 30.85557 .. 31.07329 tones/octave or
1197.1696 cents .. 1205.6168 cents
Embedded divisions: none
Number of possible generators: 30
Coprime sums: all
Pyth. maj. third : 10, 387.0968 cents, diff. 0.020229 steps, 0.7831 cents
Pyth. dim. fourth : 11, 425.8065 cents, diff. 1.020229 steps, 39.4927 cents
Basic fifth : 18, 696.7742 cents, diff. -0.133837 steps, -5.1808 cents,
-0.24090 syntonic commas, -0.22084 Pythagorean commas ( 2/9 )
Number of recognisable fifths : 1
Number of recognisable thirds : 2
Best Pyth. comma :-1, Basic Pyth. comma :-1
Best diesis : 1, Basic diesis : 1
Syntonic comma : 0, 2nd-best syntonic comma : 1
Diaschisma : 1, Schisma :-1
Diatonic semitone : 3, Chromatic semitone : 2
Minor tone : 5, Minor chroma : 2
Pythagorean limma : 3, Apotome : 2, Pyth. whole tone : 5
R = W / H : 1.666667 = 5/3
Hyperoche : 1, Eschatum : 0
Kleisma : 1, Würschmidt comma : 0
Septimal kleisma : 0, Septimal comma : 1
Septimal diesis : 1, Harrison comma : 0
Undecimal comma : 1, Tridecimal comma : 1
Some enharmonic equivalences:
Bx = Dbb
Cx = Ebbb Cx# = Ebb
Shortest one-step 3-limit interval : 1162261467/1073741824 Pythagorean-19 comma
Shortest one-step 5-limit interval : 128/125 minor diesis, diesis
Shortest one-step 7-limit interval : 50/49 Erlich's decatonic comma, tritonic diesis
Shortest one-step 11-limit interval : 50/49 Erlich's decatonic comma, tritonic diesis
Shortest one-step 13-limit interval : 78/77
Shortest vanishing 5-limit interval : 81/80 syntonic comma, Didymus comma
Shortest vanishing 7-limit interval : 126/125 small septimal comma, Starling comma
Shortest vanishing 11-limit interval : 99/98 small undecimal comma
Shortest vanishing 13-limit interval : 66/65
Shortest reversed 3-limit interval : 531441/524288 Pythagorean comma, ditonic comma
Shortest reversed 5-limit interval : 78732/78125 medium semicomma, Sensi comma
Shortest reversed 7-limit interval : 118098/117649 stearnsma
Shortest reversed 11-limit interval : 43923/43750
Shortest reversed 13-limit interval : 847/845 Cuthbert comma
Largest superparticular 1-step interval : 27/26 tridecimal comma
Largest superparticular 0-step interval : 54/53
Largest superparticular -1-step interval : 222/221
Standard 13-limit val : <31 49 72 87 107 115|
TOP error: 3.268725 cents, TOP-RMS error: 2.335681 cents
3-limit TOP octave : 1201.636592 cents, error: 1.636592 cents
5-limit TOP octave : 1201.467532 cents, error: 1.805191 cents
7-limit TOP octave : 1201.467532 cents, error: 1.805191 cents
11-limit TOP octave : 1201.467532 cents, error: 1.805191 cents
13-limit TOP octave : 1200.136551 cents, error: 3.132547 cents
3-limit TOP-RMS octave : 1201.634357 cents, error: 1.636590 cents
5-limit TOP-RMS octave : 1200.975747 cents, error: 1.627544 cents
7-limit TOP-RMS octave : 1200.828262 cents, error: 1.432402 cents
11-limit TOP-RMS octave : 1201.205316 cents, error: 1.486858 cents
13-limit TOP-RMS octave : 1200.502314 cents, error: 2.072213 cents
3-limit MSR octave : 1203.277654 cents, error: 3.277654 cents
5-limit MSR octave : 1203.277654 cents, error: 3.277654 cents
7-limit MSR octave : 1201.344124 cents, error: 4.400348 cents
11-limit MSR octave : 1209.003745 cents, error: 9.003745 cents
13-limit MSR octave : 1185.592928 cents, error: 14.407072 cents
Minimal steps generators to primes 3-13 : 13 13 8 7 2
Number of generator steps : 1 4 10 11 14
Best 13-limit pentadic subgroup : 2 5 11/9 7 7/5, error: 1.8670 cents
Best 13-limit pentadic integer subgroup : 2 5 7 15 3, error: 5.1808 cents
Number of cycles of basic fifths : 1 of 31 tones
Number of basic fifths in basic third : 4 or -27
Number of basic fifths in best seventh : 10 or -21
Number of basic fifths in best sixth : 3 or -28
Number of basic thirds in best seventh : 18 or -13
Number of cycles of basic thirds : 1 of 31 tones
Number of basic thirds in basic fifth : 8 or -23
Number of cycles of best sevenths : 1 of 31 tones
Number of best sevenths in basic fifth : 28 or -3
Number of best sevenths in basic third : 19 or -12
Nearest diff. tone is class 30: -2, diff. -0.022872 steps, -0.8854 cents
Nearest summ. tone is class 2: 32, diff. 0.011178 steps, 0.4327 cents
Number of different Myhill subsets: 157
Number of different triads: 145, with inversional equivalence: 80
Number of different tetrads: 560
Possible Pythagorean chromatic distribution 7X 5Y:
0 2 5 8 10 13 15 18 20 23 26 28 31 : strictly proper
C C# D Eb E F F# G G# A Bb B C
0 2 3 5 7 8 10 12 13 15 16 18 20 21 23 25 26 28 30 31
C C# Db D D# Eb E E# F F# Gb G G# Ab A A# Bb B B# C
Possible classical chromatic distribution :
0 3 5 8 10 13 15 18 21 23 26 28 31 : strictly proper
C C# D Eb E F F# G G# A Bb B C
Possible diatonic distributions of form 5L 2s :
0 5 10 13 18 23 28 31 : strictly proper
C d e - f G a b - C
Possible distributions of form 4L 3Z :
0 7 14 15 22 23 30 31 : not proper
C d e f g a b C
Possible distributions of form 6X Y :
0 5 10 15 20 25 30 31 : strictly proper
C d e f g a b C
Possible distributions of form 3L 2M 2s :
0 7 10 12 19 22 29 31 : not proper
C d e f g a b C
It shows how good the basic intervals major third, perfect fifth and harmonic
seventh are approximated in an equal temperament.
Misfit numbers are an invention of Fokker and are sums of squared errors
in cents. M1 only counts 3/2, M2 is the sum of squared errors of 3/2 and 5/4,
M3 of 3/2, 5/4 and 7/4, M4 adds 11/8 and M5 adds 13/8.
Relative errors are a percentage of the average deviation in terms of
step size. The average error is a quarter step, because the error varies
between zero and half a step. R1 only counts 3/2, R2 is the sum of relative
errors of 3/2 and 5/4 divided by two, R3 of 3/2, 5/4 and 7/4 divided by three,
etc. like with the misfit numbers.
Combined error factor, from Joel Mandelbaum, is the sum of the squared errors
of the first basic three divided by the size of half a step. This makes it a
kind of normalised version of the misfit number M3.
The second combined error factor is given if the highest consistently
represented harmonic is 9 or higher. It then includes the errors of the higher
odd consistent harmonics reduced by the octave like 9/8, 11/8, etc.
The number of possible Pythagorean generators is the value of Euler's totient
function for the given division. It is the amount of numbers from 1 to the
division with a greatest common divisor of 1 with the division. If the greatest
common divisor of a scale degree with the number of notes is larger than 1 then
it cannot be a Pythagorean generator for the complete scale.
Two Pythagorean thirds are shown. The first is the value of 4 times the best
fifth minus two octaves and the second is five octaves minus eight best fifths.
A list of basic intervals and commas is given with their sizes in number of
scale steps. For example the syntonic comma is four times the nearest degree to
3/2 minus two times the number of steps in the octave minus the nearest degree to 5/4.
Defining one's own methods for constructing scales can be done with files containing Scala commands. The following example makes a Fokker double-tie circular mirroring. If the filename is dt.cmd, it can be invoked by entering @dt at the prompt.
echo Create a double-tie circular mirroring echo enter the 3 triad factors input/factor 2 ? ? ? %3 := -%1 %4 := -%2 %5 := %2-%1 %6 := -%5 %7 := 2/1 normalize 2/1 show scale
In this example, the user enters three factors which define some triad chord and then the rest of the notes of the resulting 7-note scale are calculated automatically. The question marks cause waiting for user input. The percent sign refers to the scale degree with the given number. In these scripts, pitch variables can be used and the command syntax is the same as with interactive use.
Mapping of scale degrees to keyboard keys is achieved by using keyboard mapping files. Often they won't be needed, such as with scales with 12 tones per octave. Keyboard mappings can be created with any text editor, and look like the following. Lines beginning with an exclamation mark are comments.
! Example keyboard mapping for a 10-tone scale. Two tones are duplicated. ! Other ones may be more convenient depending on the scale. ! ! Size of map. The pattern repeats every so many keys: 12 ! First MIDI note number to retune: 0 ! Last MIDI note number to retune: 127 ! Middle note where the first entry of the mapping is mapped to: 60 ! Reference note for which frequency is given: 60 ! Frequency to tune the above note to (floating point e.g. 440.0): 261.6256 ! Scale degree to consider as formal octave (determines difference in pitch ! between adjacent mapping patterns): 10 ! Mapping. ! The numbers represent scale degrees mapped to keys. The first entry is for ! the given middle note, the next for subsequent higher keys. 0 1 2 3 3 4 5 6 7 8 8 9
Once a keyboard mapping is loaded into Scala, the correspondences of keys with notes and notes with keys can be viewed with the SHOW MAPPING command:
Range : 0.C .. 127.G Middle : 60.C Reference : 261.6256 Hertz at note 60.C Octave degree : 10 Mapping : 60.C : 0 0: 60.C 61.C#: 1 1: 61.C# 62.D : 2 2: 62.D 63.Eb: 3 3: 63.Eb 64.E 64.E : 3 4: 65.F 65.F : 4 5: 66.F# 66.F#: 5 6: 67.G 67.G : 6 7: 68.G# 68.G#: 7 8: 69.A 70.Bb 69.A : 8 9: 71.B 70.Bb: 8 10: 72.C 71.B : 9 11: 73.C# 72.C : 10 12: 74.D
A set of keyboard mappings for scales of various sizes is supplied with Scala. They can be used as examples or be adapted to your own needs.
How a given scale is approximated by equal temperaments is indicated by the command FIT/MODE. If we take Ptolemy's Intense Diatonic scale from above, then FIT/MODE will give:
7: 1 1 1 1 1 1 1 SP B ME I SD: 33.3285 c. M: 59.6973 c. Lebeng 10: 2 1 1 2 1 2 1 P M ME S SD: 25.8629 c. M: 44.3587 c. Sethares Neutral, Beatles-7, Dichotic-7 12: 2 2 1 2 2 2 1 P M ME S SD: 9.2003 c. M:-15.6413 c. G.Lydian, M.Ionian, M.Hypolydian, Major, Bilaval That, Mela Shankarabharanam, Ghana Heptatonic, Peruvian Major, 4th plagal Byzantine, Ararai: Ethiopia, Makam Cargah, Ajam Ashiran, 19: 3 3 2 3 3 3 2 SP M ME S SD: 9.0995 c. M: 14.5845 c. Nineteen-tone Major 22: 4 3 2 4 3 4 2 P T3 SD: 8.1766 c. M:-14.2718 c. Twenty-two tone "Just" Major 29: 5 4 3 5 4 5 3 SP T3 SD: 9.2386 c. M: 15.3932 c. Twenty-nine tone "Just" Major 31: 5 5 3 5 5 5 3 SP M DE S SD: 5.5622 c. M: 10.3616 c. Thirty-one tone Major, Intense Diatonic Lydian, M.Ionian 34: 6 5 3 6 5 6 3 P T3 SD: 4.3829 c. M:-7.8547 c. Thirty-four tone "Just" Major 41: 7 6 4 7 6 7 4 SP T3 SD: 3.8489 c. M: 6.3099 c. Forty-one tone "Just" Major 46: 8 7 4 8 7 8 4 P T3 SD: 4.1494 c. M:-7.3835 c. Forty-six tone "Just" Major 53: 9 8 5 9 8 9 5 SP T3 SD: 0.9247 c. M: 1.4763 c. Fifty-three tone "Just" Major 118: 20 18 11 20 18 20 11 SP T3 SD: 0.2902 c. M: 0.5202 c.The first column is the number of equal tempered steps per octave. Then the following numbers are the number of steps that approximate each successive interval in the scale, so a higher number represents a larger interval. Some properties are shown with the following codes: P = proper, SP = strictly proper, N = not proper, M = Myhill's property, B = block repeats, G = generated, ME = maximal even, DE = distributional even, D3 = 3-distributional even, T3 = trivalent, I = self-inverse, S = symmetrical. The number after SD: is the standard deviation of the scale from the approximation in units of cents. If the given equal temperament mode is in the list of modes, then its name is also given.
Computer sound cards and instruments which have no built-in tuning table(s) can also be used to play MIDI files generated by Scala with retuning via pitch bend commands. There are two methods to create these MIDI files. One is with an input file with text in Scala sequence format and the other by conversion of an existing MIDI file. MIDI files can also be converted to the sequence format so the tuning can be changed at any point in time. Keyboard mappings can also be applied. The command to be used is EXAMPLE.
Because pitch bend commands affect all notes on a MIDI channel, different channels have to be used and chosen dynamically based on the programs (voices) and pitch bend values. This poses some restriction on the amount of simultaneous notes with different voice and tuning. The algorithm that does the channel allocation minimises the amount of pitch bend messages.
The sequence file format is quite powerful. One can use different tunings without having to change the input file. The file may specify a default tuning. Notes can be specified by scale degree number, frequency ratio, or note name/octave number pairs. MIDI program change, tempo change and controller messages can be inserted. Tracks can be specified. Time can be specified either in absolute values or relative to the end of the previous note statement. There are statements for changing the scale, key, or base frequency at any time. MIDI channels can also be excluded from being used. This is a short example:
! Easley Blackwood: harmonization of a mode in 15-tET, ! example 29 of "Modes and chord progressions in equal tunings", p. 195 ! Perspectives of New Music vol. 29/2, 1991. ! 0 exclude 10 0 tempo 800_000 0 program 5 0 velocity 64 0 frequency 261.6255653 0 equal 15 0 notation E15 ! 0 track 1 0 note C.1 480 0 note E\.1 480 480 note B\ 480 480 note D.1 480 960 note Bb/ 480 960 note D\.1 480 1440 note A\ 480 1440 note C.1 480 1920 note G 480 1920 note B\ 480 2400 note F#\ 480 2400 note A 480 2880 note F 480 2880 note A\ 480 3360 note E\ 480 3360 note G 480 3840 note Eb 480 3840 note G\ 480 4320 note D\ 480 4320 note F 480 4800 note C 960 4800 note E\ 960 ! 0 track 2 0 note C 480 0 note G 960 480 note G.-1 480 960 note Bb.-1 480 960 note F 960 1440 note F.-1 480 1920 note G.-1 480 1920 note D 960 2400 note D.-1 480 2880 note F.-1 480 2880 note C 960 3360 note C.-1 480 3840 note Eb.-1 480 3840 note Bb.-1 960 4320 note Bb.-2 480 4800 note C.-1 960 4800 note G.-1 960
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