com.dixshtix.audio.pitch
Class AbstractTuning

java.lang.Object
  |
  +--com.dixshtix.audio.pitch.AbstractTuning

Direct Known Subclasses:

CyclicTemperament, LinearTemperament

public abstract class AbstractTuning

extends java.lang.Object

The object of temperament (literally tuning), is to render possible the expression of an indefinite number of intervals by means of limited number of tone without distressing the ear too much. The general practice has been, from the earliest invention of the keyboard of the organ to the present day, to make 12 notes in the octave suffice. This number has been in a very few instances increased to 14, 16, 19, and even to 31 and 53, but such instruments have never come into general use.

The system which tuners at the present day intend to follow, though none of them absolutely succeed in so doing, is to produce 12 notes reckoned from any tone exclusive to its octave inclusive, such that the octave should be just and the interval between any two consecutive notes, that is the ratio of their pitch numbers, should be always the same. This is known as Equal Temperament. The interval between any two notes in an Equal Semitone and its ratio is 1:2^(1/12). If we divide the Equal Semitone into 100 equal intervals, those intervals would be called "Cents," having the common ratio 1:2^(1/1200). As the human ear is, except is very rare cases, insensible to the interval of a cent, we need not divide further for psychoacoustic reasons, but may do so for mathematical precision.

No recurrence of notes formed by taking intervals of Fifths, major Thirds and Octaves is possible because every possible combination of powers of the numbers 3/2, 5/4, 2 is a unique ratio. There are, however, approximations possible.

Version:

0.1

Author:

Richard C. Penner II

Field Summary

static java.lang.String	A
static java.lang.String	Af
static java.lang.String	Aff
static java.lang.String	As
static java.lang.String	Ass
static java.lang.String	B
static java.lang.String	Bf
static java.lang.String	Bff
static java.lang.String	Bs
static java.lang.String	Bss
static java.lang.String	C
static java.lang.String	Cf
static java.lang.String	Cff
static java.lang.String[]	circleOfFifths
static double	comma Synonym for the didymusComma.
static java.lang.String	Cs
static java.lang.String	Css
static java.lang.String	D
static java.lang.String	Df
static java.lang.String	Dff
static double	didymusComma Four fifths up and two octaves plus a major third down give the Comma of Didymus.
static java.lang.String	doubleFlat
static java.lang.String	doubleSharp
static java.lang.String	Ds
static java.lang.String	Dss
static java.lang.String	E
static java.lang.String	Ef
static java.lang.String	Eff
static double	equalTemperedFifth By diminishing fifths be exactly 1/12 of a pythagoreanComma, an equally tempered fifth is obtained.
static double	equalTemperedThird An equally tempered third is sharper than a Pythagorean major third.
static java.lang.String	Es
static java.lang.String	Ess
static java.lang.String	F
static java.lang.String	Ff
static java.lang.String	Fff
static java.lang.String	flat
static java.lang.String	Fs
static java.lang.String	Fss
static java.lang.String	G
static java.lang.String	Gf
static java.lang.String	Gff
static java.lang.String	Gs
static java.lang.String	Gss
static double	hemholtzFifth If we employed fifths diminished by the undetectable interval of 1/8 of a Skhisma, eight of these fifths added to a just major third would exactly equal five octaves.
static double	justFifth
static double	justOctave
static double	majorThird
static double	meantoneFifth If we Diminish a fifth the the small, but detectable interval of a quarter of a comma, four of these Fifths would be precisely two Octaves plus an exact major third.
static double	mercatorialComma A precision system consists of dividing an octave into 53 equal parts, and not twelve as in the common music notation.
static double	mercatorialFifth A Mercatorial Fifth is flat by 1/53 of a Mercatorial Comma.
static double	pythagoreanComma Twelve fifths up and seven octaves down give the sum of a Comma of Didymus and a Skhisma, known as the Pythagorean Comma.
static double	pythagoreanDiesis One octave minus three major thirds is also the difference between two Commas of Didymus and a a Skhisma, known as the GreatDiĕsis.
static java.lang.String	sharp
static double	skhisma Eight fifths and a major third up and five Octaves down give the Skhisma.
static double	skhismicMajorThird If we diminish the major third by a Skhisma, giving a Skhismic major third, then this major third added to eight fifths will give exactly five octaves.

Constructor Summary

AbstractTuning()

Method Summary

static double	centsToInterval(double cents) A cent is a 1200th of an octave.
static double	intervalToCents(double interval) A cent is a 1200th of an octave.

Methods inherited from class java.lang.Object

, clone, equals, finalize, getClass, hashCode, notify, notifyAll, registerNatives, toString, wait, wait, wait

Field Detail

sharp

public static final java.lang.String sharp

flat

public static final java.lang.String flat

doubleSharp

public static final java.lang.String doubleSharp

doubleFlat

public static final java.lang.String doubleFlat

A

public static final java.lang.String A

Af

public static final java.lang.String Af

Aff

public static final java.lang.String Aff

As

public static final java.lang.String As

Ass

public static final java.lang.String Ass

B

public static final java.lang.String B

Bf

public static final java.lang.String Bf

Bff

public static final java.lang.String Bff

Bs

public static final java.lang.String Bs

Bss

public static final java.lang.String Bss

C

public static final java.lang.String C

Cf

public static final java.lang.String Cf

Cff

public static final java.lang.String Cff

Cs

public static final java.lang.String Cs

Css

public static final java.lang.String Css

D

public static final java.lang.String D

Df

public static final java.lang.String Df

Dff

public static final java.lang.String Dff

Ds

public static final java.lang.String Ds

Dss

public static final java.lang.String Dss

E

public static final java.lang.String E

Ef

public static final java.lang.String Ef

Eff

public static final java.lang.String Eff

Es

public static final java.lang.String Es

Ess

public static final java.lang.String Ess

F

public static final java.lang.String F

Ff

public static final java.lang.String Ff

Fff

public static final java.lang.String Fff

Fs

public static final java.lang.String Fs

Fss

public static final java.lang.String Fss

G

public static final java.lang.String G

Gf

public static final java.lang.String Gf

Gff

public static final java.lang.String Gff

Gs

public static final java.lang.String Gs

Gss

public static final java.lang.String Gss

circleOfFifths

public static final java.lang.String[] circleOfFifths

justFifth

public static final double justFifth

majorThird

public static final double majorThird

justOctave

public static final double justOctave

didymusComma

public static final double didymusComma

Four fifths up and two octaves plus a major third down give the Comma of Didymus. (3/2)^4 (1/2)^2 (4/5) = 3^4 / 5 * 2 ^ 4 = 81 / 80.

comma

public static final double comma

Synonym for the didymusComma.

meantoneFifth

public static final double meantoneFifth

If we Diminish a fifth the the small, but detectable interval of a quarter of a comma, four of these Fifths would be precisely two Octaves plus an exact major third. These are called meantone fifths and we long in use.

3/2 * (80/81)^(1/4) = 3/2 * (2/3) * 5 ^(1/4) = 5^(1/4).

skhisma

public static final double skhisma

Eight fifths and a major third up and five Octaves down give the Skhisma. Which is just smaller than two cents. 3^8 * 5 / 2^15;

hemholtzFifth

public static final double hemholtzFifth

If we employed fifths diminished by the undetectable interval of 1/8 of a Skhisma, eight of these fifths added to a just major third would exactly equal five octaves. These shall be called Helmholtz's fifths. 3/2 * (2^15/ 3^8 * 5 ) ^ (1/8) = ( 2^7 / 5) ^ (1/8)

skhismicMajorThird

public static final double skhismicMajorThird

If we diminish the major third by a Skhisma, giving a Skhismic major third, then this major third added to eight fifths will give exactly five octaves. This is the relation which Professor Helmholtz point out as existing in medieval Arabic scales, and will be called skhismic. majorThird / Skhisma = 2^13/3^8

pythagoreanComma

public static final double pythagoreanComma

Twelve fifths up and seven octaves down give the sum of a Comma of Didymus and a Skhisma, known as the Pythagorean Comma. 3^12/2^19 = (3^4 / 5 * 2 ^ 4)( 3^8 * 5 / 2 ^ 15) which is just short of 12 Skhismas by the interval 3^84 * 5^12 / 2^161 ( 8.87 parts per million ).

equalTemperedFifth

public static final double equalTemperedFifth

By diminishing fifths be exactly 1/12 of a pythagoreanComma, an equally tempered fifth is obtained. (700 Cents).

equalTemperedThird

public static final double equalTemperedThird

An equally tempered third is sharper than a Pythagorean major third. It is 400 cents.

pythagoreanDiesis

public static final double pythagoreanDiesis

One octave minus three major thirds is also the difference between two Commas of Didymus and a a Skhisma, known as the GreatDiĕsis. 2 * (4/5)^3 = 128/125 = (81/80)^2 * 2^15/(3^8 * 5)

mercatorialComma

public static final double mercatorialComma

A precision system consists of dividing an octave into 53 equal parts, and not twelve as in the common music notation. It is of higher precision than the equally tempered 12-note scale. Fifty-three fifths up and thirty-one octaves down give what we may call a "Mercatorial Comma" after Mercator's 53 division of the octave. 3^53 / 2^84

mercatorialFifth

public static final double mercatorialFifth

A Mercatorial Fifth is flat by 1/53 of a Mercatorial Comma.

Constructor Detail

AbstractTuning

public AbstractTuning()

Method Detail

centsToInterval

public static double centsToInterval(double cents)

A cent is a 1200th of an octave.

intervalToCents

public static double intervalToCents(double interval)

A cent is a 1200th of an octave.