Scales and Tunings

Introduction

The "cents" scale was invented by Alexander J. Ellis (1885).  By definition, there exist 1200 cents to an octave.  It's simplist to describe the relationships between scales by describing intervals in terms of cents. 

Cents are additive, therefore they are related logarithmically to pitch ratios:

	Pitch Interval = K * log( f1 / f2 )
	
whre f1 and f2 are frequencies of tones making up the interval.  By definition, there are 1200 cents in an octave:

1200 = K * Log[10,2]

which becomes

K = 3986.3137138648348175

Then if you want to calculate a frequency and know the interval in cents, use the following formula:

	Frequency = StartingFrequency * 10^( Interval / K )

where 
	Frequency = desired frequency, in hertz
	StartingFrequency = known frequency, in hertz
	Interval = interval between the two in cents
	
The following Mathematica program, written by Matthew Markert, takes three arguments:

	scaleType = a list of intervals, in cents
	startingFreq = a starting frequency, in hertz
	totalDuration = the duration for the entire scale, in seconds

PlayScale[scaleType_,startingFreq_,totalDuration_] :=
  Module[{scaleList,noteDuration},

    scaleList = 
    	Map[(N[startingFreq * 10^(#/(1200/Log[10,2]))])&, 
    		scaleType];
    noteDuration = Length[scaleList]/totalDuration;

    Play[
      Sin[ 2 Pi t scaleList[[ 1+Floor[t * noteDuration] ]] ],
      	{t,0,totalDuration-.000001}]
  ];

The following scales are defined below: 

	Pythagorean major and chromatic
	Just Intonation major and minor
	Mean-Tone major, minor, and chromatic
	Equal-Tempered major, minor, and chromatic
	Quarter-Tone chromatic
	Sixth-Tone chromatic

Pythagorean Tuning

Pythagoras is reputed to have lived in the 3rd or 4th century, B.C. No authentic writings of his survive, only those of his followers.

Two quotations attributed to Pythagoras are:

	Intervals in music are rather to be judged intellectually through 		numbers than 	sensibly through the ear.
	
	The simpler the ratio of the two parts into which the vibrating string is 	divided, the more perfect is the consonance of the two sounds.
	
The scale of Pythagoras was based entirely on octaves and fifths.  Pythagoras worked out ratios on lengths of strings.  The discussion here will be in terms of cents.

An octave is defined to have 1200 cents.

A perfect fifth (the ration of 3:2) has 702 cents.

So, if we start on the pitch C:

	C to G adds 702 cents
	G to D adds 702 cents again
	
	so C to D adds 1404 cents
	
	and subtract one octave from the D (1404 - 1200 = 204)
	
	then the major tone is 204 cents (9:8 ratio)

Successive fifth addition and octave subtraction yields A, E, B and fifth subtraction plus octave addition yields F.

So, the Pythagorean scale in cents above a starting pitch is:

	0	204	408	498	702	906	1110	1200
	
This same process can be continued upwards in order to generate the complete chromatic  scale, but E# will be higher than F (E# = 522, F = 498), and B# will be higher than C 
(B# = 1224, C = 1200).

Note: In tempered tunings systems (e.g., pianos) E# is considered to be equivalent to F, and B# is considered to be equivalent to C.

If the process is continued downwards, the flat degrees of the scale can be derived, but these frequencies will not be the same as the sharp frequencies, and Fb is lower than E 
(Fb = 384, E = 408), and Cb is lower than B (Cb = 1086, B = 1110).

Thus the comple chromatic scale requires 21 notes: 7 "natural" notes, 7 "sharp" notes,
and 7 "flat" notes.

C	B#	Db	C#	D	Eb	D#	Fb	E	F	E#	Gb
0	24	90	114	204	294	318	384	408	498	522	588

F#	G	Ab	G#	A	Bb	A#	Cb	B	C
612	702	798	816	906	996	1020	1086	1110	1200

PythagoreanChromatic = {0,24,90,114,204,294,318,384,408,498,
	522,588,612,702,798,816,906,996,1020,1086,1110,1200}

PythagoreanMajor = {0,204,408,498,702,906,1110,1200}

PlayScale[PythagoreanChromatic,440,6]

PlayScale[PythagoreanMajor,440,3]

Just Intonation

The tuning that today is called "Just Intonation" is the modification made by the theorist Zarlino in the 16th century of a scale attributed to Aristaxenus, who lived in the 3-4th centuries B.C.

The following table compare the ratio of intervals in the Pythagorean Major scale with those in Zarlino's modification of Aristaxenus:

note		interval		Zarlino		Pythagoras
C		unison			1			1
D		second			9:8			9:8
E		major third		5:4			81:64
F		fourth			4:3			4:3
G		fifth			3:2			3:2
A		major sixth		5:3			27:16
B		major seventh		15:8			243:128
C		octave			2			2

Zarlino's modification allowed all intervals of the major third (C-E, F-A, G-B) to be in tune, something not possible in the Pythagorean scale.  For example, notice that in Zarlino's scale the distance from C to E has the ratio of 5:4, but in Pythagoras' scale that same distance has the ratio of 81:64.

For those of you who may not know, about 100 to 150 years before Zarlino, composers had begun to give the major 3rd a structural prominence in their works.  During this time, the major 3rd was considered to be a "dissonant" interval, to be used only in passing contexts.  Zarlino's assertions were a means of making consistent and otherwise somewhat unruly compositional practice.

The major and minor scales in Just Intonation are given below, their intervals in cents:

JustMajor = {0,204,386.4,498,702,884.4,1088.4,1200}

JustMinor = {0,204,315.6,498,702,813.7,996.1,1200}

PlayScale[JustMajor,220,3]

PlayScale[JustMinor,220,3]

Mean-Tone Temperment

Zarlino's and Pythagoras' scales generated difficulties when the possibilities of modulation were introduced by composers in the 17th and 18th centuries:

	1. Which frequencies should be used for the chromatic scale, since chromatic 		equivalence did not exist in these tunings? (i.e., Ab was not the same as G#).
	
	2. The same interval in different keys did not have the same tuning.	
	
In the 18th century, Gottfried Silbermann, a famous organ builder during the time of J.S. Bach, devised  a tuning in which each of the fifths in the scale was flattened to result in perfect major thirds through most of the chromatic scale.

	Mean tone fifth = 696.6 cents
	Just Intonation fifth = 702 cents
	
A fully chromatic scale is developed by fifths and octaves, as the method of Pythagoras, shown above.

The Mean-Tone chromatic scale is this:

MeanChromatic = {0,76,193.2,310.3,386.3,503.4,579.5,
                696.6,772.6,889.7,1006.8,1082.9,1200}

From which can be extracted the major and the minor scales, as we call them today.

MeanMajor = {0,193.2,386.3,503.4,696.6,889.7,1082.9,1200}

MeanMinor = {0,193.2,310.3,503.4,696.6,772.6,1006.8,1200}

This tuning was the tuning preferred by organ builders until the middle of the 19th century.

PlayScale[MeanMajor,220,3]

PlayScale[MeanMinor,220,3]

PlayScale[MeanChromatic,220,6]

Equal Temperment

In Equal Temperment, the octave is divided into 12 equal subdivisions, each of which is a tempered semi-tone (100 cents).   This system is equivalent to making 12 fifths equal to 7 octaves, so 

	12 / 7 * 1200 = 700 cents is equal-temperment fifth
	
	in contradistinction to 702 cents for a Just Intonation and Pythagorean fifth, and
				696.6 cents for a Mean-Tone fifth

This guarentees chromatic equivalence between pitches (A# and Bb are the same, for example), and allows intervals to be the same in all keys.  However, except for the octave, none of the intervals is in tune, in terms of mathematical ratios and the logic Pythagoras developed from proportional lengths of strings.

Here are the major, minor, and chromatic tempered scales, given in cents:

TemperedMajor = {0,200,400,500,700,900,1100,1200}

TemperedMinor = {0,200,300,500,700,800,1000,1200}

TemperedChromatic = {0,100,200,300,400,500,600,700,
			800,900,1000,1100,1200}

PlayScale[TemperedMajor,220,3]

PlayScale[TemperedMinor,220,3]

PlayScale[TemperedChromatic,220,6]

Quarter-Tone and Sixth-Tone Scales

These two scales are made by dividing up the tempered semi-tone into two and three parts:

QuarterTone = {0,50,100,150,200,250,300,350,400,450,500,550,
                600,650,700,750,800,850,900,950,1000,1050,
                1100,1150,1200}

SixthTone = {0,33,66,100,133,166,200,233,266,300,333,366,
            400,433,466,500,533,566,600,633,666,700,733,
            766,800,833,866,900,933,966,1000,1033,1066,1100,
            1133,1166,1200}

PlayScale[QuarterTone,220,6]

PlayScale[SixthTone,220,6]

Harry Partch's 43 Tone Scale

Partch = {0,21.5,53.2,84.5,111.7,150.6,165.0,182.4,203.9,
			231.2,266.9,294.1,315.6,347.4,386.3,417.5,435.1,
			470.8,498.0,519.5,551.3,582.5,617.5,648.7,680.5,
			702.0,729.2,764.9,782.5,813.7,852.6,884.4,905.9,
			933.1,968.8,996.1,1017.6,1035.0,1049.4,1088.3,
			1115.5,1146.8,1178.5,1200}

PlayScale[Partch,100,6]