Inspect atonal music Pitch-Class Sets in prime form, transposed, inverted and complementary form using Allen Forte labeling

Pitch-Class Set Graphical Toolkit - Version 2.4

Use this tool to analyse and inspect the properties of pitch-class sets. [See the user guidelines >>]

Use this page in landscape mode on mobile devices.

1. Pitch-Class set display and transformation tool

Select Pitch-Class Set

PC Set 1	Root	State			Interval	PC Set 2	Root	State			Interval
3-1 3-2 3-3 3-4 3-5 3-6 3-7 3-8 3-9 3-10 3-11 3-12 4-1 4-2 4-3 4-4 4-5 4-6 4-7 4-8 4-9 4-10 4-11 4-12 4-13 4-14 4-Z15 4-16 4-17 4-18 4-19 4-20 4-21 4-22 4-23 4-24 4-25 4-26 4-27 4-28 4-Z29 5-1 5-2 5-3 5-4 5-5 5-6 5-7 5-8 5-9 5-10 5-11 5-12 5-13 5-14 5-15 5-16 5-Z17 5-Z18 5-19 5-20 5-21 5-22 5-23 5-24 5-25 5-26 5-27 5-28 5-29 5-30 5-31 5-32 5-33 5-34 5-35 5-Z36 5-Z37 5-Z38 6-1 6-2 6-Z3 6-Z4 6-5 6-Z6 6-7 6-8 6-9 6-Z10 6-Z11 6-Z12 6-Z13 6-14 6-15 6-16 6-Z17 6-18 6-Z19 6-20 6-21 6-22 6-Z23 6-Z24 6-Z25 6-Z26 6-27 6-Z28 6-Z29 6-30 6-31 6-32 6-33 6-34 6-35 6-Z36 6-Z37 6-Z38 6-Z39 6-Z40 6-Z41 6-Z42 6-Z43 6-Z44 6-Z45 6-Z46 6-Z47 6-Z48 6-Z49 6-Z50 7-1 7-2 7-3 7-4 7-5 7-6 7-7 7-8 7-9 7-10 7-11 7-12 7-13 7-14 7-15 7-16 7-Z17 7-Z18 7-19 7-20 7-21 7-22 7-23 7-24 7-25 7-26 7-27 7-28 7-29 7-30 7-31 7-32 7-33 7-34 7-35 7-Z36 7-Z37 7-Z38 8-1 8-2 8-3 8-4 8-5 8-6 8-7 8-8 8-9 8-10 8-11 8-12 8-13 8-14 8-Z15 8-16 8-17 8-18 8-19 8-20 8-21 8-22 8-23 8-24 8-25 8-26 8-27 8-28 8-Z29 9-1 9-2 9-3 9-4 9-5 9-6 9-7 9-8 9-9 9-10 9-11 9-12	0	O	I	C		3-1 3-2 3-3 3-4 3-5 3-6 3-7 3-8 3-9 3-10 3-11 3-12 4-1 4-2 4-3 4-4 4-5 4-6 4-7 4-8 4-9 4-10 4-11 4-12 4-13 4-14 4-Z15 4-16 4-17 4-18 4-19 4-20 4-21 4-22 4-23 4-24 4-25 4-26 4-27 4-28 4-Z29 5-1 5-2 5-3 5-4 5-5 5-6 5-7 5-8 5-9 5-10 5-11 5-Z12 5-13 5-14 5-15 5-16 5-Z17 5-Z18 5-19 5-20 5-21 5-22 5-23 5-24 5-25 5-26 5-27 5-28 5-29 5-30 5-31 5-32 5-33 5-34 5-35 5-Z36 5-Z37 5-Z38 6-1 6-2 6-Z3 6-Z4 6-5 6-Z6 6-7 6-8 6-9 6-Z10 6-Z11 6-Z12 6-Z13 6-14 6-15 6-16 6-Z17 6-18 6-Z19 6-20 6-21 6-22 6-Z23 6-Z24 6-Z25 6-Z26 6-27 6-Z28 6-Z29 6-30 6-31 6-32 6-33 6-34 6-35 6-Z36 6-Z37 6-Z38 6-Z39 6-Z40 6-Z41 6-Z42 6-Z43 6-Z44 6-Z45 6-Z46 6-Z47 6-Z48 6-Z49 6-Z50 7-1 7-2 7-3 7-4 7-5 7-6 7-7 7-8 7-9 7-10 7-11 7-12 7-13 7-14 7-15 7-16 7-Z17 7-Z18 7-19 7-20 7-21 7-22 7-23 7-24 7-25 7-26 7-27 7-28 7-29 7-30 7-31 7-32 7-33 7-34 7-35 7-Z36 7-Z37 7-Z38 8-1 8-2 8-3 8-4 8-5 8-6 8-7 8-8 8-9 8-10 8-11 8-12 8-13 8-14 8-Z15 8-16 8-17 8-18 8-19 8-20 8-21 8-22 8-23 8-24 8-25 8-26 8-27 8-28 8-Z29 9-1 9-2 9-3 9-4 9-5 9-6 9-7 9-8 9-9 9-10 9-11 9-12	0	O	I	C
Comment						Comment

PC Set 1 Tonedisc	Overlapping Pitches	PC Set 2 Tonedisc
C           B   C#       Bb       D   A     +     Eb   G#       E       G   F           F#	C           B   C#       Bb       D   A     +     Eb   G#       E       G   F           F#	C           B   C#       Bb       D   A     +     Eb   G#       E       G   F           F#

PC Set 1 Transformations		PC Set 2 Transformations

Subset Analysis for two PC sets of different cardinal number (i.e., different size)
Number of Semitones n:	0	1	2	3	4	5	6	7	8	9	10	11
Transposition (Tn):
Inversion and Transposition (TnI):

Similarity Relations for PC sets of equal cardinal number (i.e., same size)
Number of Pitch Similarities									Interval Similarity
1	2	3	4	5	6	7	8	9	n1	n2	n3	n4	n5	n6

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2. Pitch-Class set analysis tool

Tick the Pitch Boxes

C	C#/Db	D	D#/Eb	E	F	F#/Gb	G	G#/Ab	A	A#/Bb	B

Set name:	Tick the pitches	Root:	-	Set state:	O	I

Guidelines for the user

This page contains a GUI for display, transformation and analysis of Pitch-Class (PC) sets. PC sets may be used as the structuring elements in atonal music. There is a User Guide movie on YouTube: How to use the Pitch-Class Set Tool (9:18).

1. Pitch-Class set display and transformation tool

This part of the tool is designed for display and transformation of pitch-class (PC) sets. The GUI has the following features:

• The screen is divided into three sections. The left and right section allow you to work on two PC sets independently. The centre section shows the overlap between the two PC sets and is automatically updated.
• The pull-down menu at the top allows you to Select a Pitch-Class set from a pre-determined list. The numbers in the list correspond to the labeling used by Allen Forte in Appendix 1 from his textbook The Structure of Atonal Music (Yale University Press, New Haven and London, ISBN 0-300-02120-8, 1973). After selection of a set number the PC Set Information Table and the Tone disc (at the centre of the section) will display PC set data.
• The Information Table has a number of fields:
  1. The current PC set number, e.g., 4-19.
  2. The current Root, e.g., C#, a note name from the chromatic scale. This field is updated after Transposition of the PC set.
  3. The current State of the PC-set. A set may displayed in various forms: either as the Original set (the O will be highlighted), as an Inverted set, i.e., the set structure is mirrored about the current root (the I will be highlighted), or as the Complement set, i.e., those notes from the chromatic scale that are not in the current set (the C will be highlighted). The O and I are mutually exclusive, the C is toggled on/off.
  4. The Interval Vector of the current set, e.g., 101310. This row vector contains six entries n1,n2,n3,n4,n5,n6, characterising the interval content of the PC set: n1 is the number of minor seconds in the PC set, n2 the number of major seconds, n3 the number of minor thirds, n4 the number of major thirds, n5 the number of perfect fourths and n6 the number of augmented fourths or diminished fifths.
  5. The Comment field contains text, e.g., Octatonic Scale. Here additional information is shown such as the similarity to a tonal chordal structure or melodic scale.
• The Tone Disc will colour the current note names of the PC set, enabling easy visual inspection of the musical contents of the set structure. The disc incorporates the cyclic nature of the chromatic scale.
• The Set Transformation menu contains four buttons:
  1. Pressing the Transpose Down button yields a downward transposition of the PC set by a minor second. This corresponds to a counter-clockwise rotation in the tone disc diagram.
  2. Pressing the Transpose Up button yields an upward transposition of the PC set by a minor second. This corresponds to a clockwise rotation in the tone disc diagram.
  3. Pressing the Complement button will highlight the notes from the full chromatic set (12 tones) that are currently not in the structure. Pressing this button once again will highlight the original structure content.
  4. Pressing the Invert button will mirror the PC structure about the current root.
  The Set Transformation menu will become active after a specific PC set has been selected. The four buttons may be pressed in any order.
• The Subset Analysis table shows for which n semitones transposition T_n (or transposition after inversion T_nI) the smaller PC set is a subset of the larger PC set (i.e, all pitches from the smaller set will also be contained in the larger PC set). First two PC sets must be selected from the top pull-down menu (and they obviously must be of non-equal size). Check the overview in this table by using the Set Transformation buttons above the table. Changing the selected PC set will automatically update the Subset Analysis table.
• The Similarity Relations table shows the pitch and interval similarity between two PC sets of equal cardinal number N. The maximum pitch similarity is N-1. This occurs frequently, but usually is combined with the interval similarity property. Here, the strongest possible similarity implies that four interval class counts are equal (e.g., n1,n3,n5,n6) and that the remaining two interval class counts (i.e., n2,n4) are swapped between the sets.
• The navigation menu at the top will lead you back to the Homepage of this Website, the Document Archive or to these Help Instructions. This Website contains also a paper form, equivalent to this GUI (see the Document Archive).

2. Pitch-Class set analysis tool

This part of the tool is designed for analysis (determination) of pitch-class sets. The GUI has the following features:

• The upper part of the screen has a set of twelve checkboxes, corresponding with the note names from the chromatic scale. Put tickmarks for the appropriate pitches contained in the pitch-class set before analysis. There must be between 3 and 9 tickmarks before analysis is enabled.
• Pushing the Analyse button will display the results of the Pitch-Class set analysis. You will see the following features in the information textfields:
  1. The current PC set number, e.g., 5-3.
  2. The current Root, e.g., Eb, a note name from the chromatic scale.
  3. The current State of the PC-set: after analysis the set may be found to be in either the original or inverted form (the O or I fields will be highlighted).
• Pushing the Clear all pitches button will remove the tickmarks in the checkboxes and clear the textfields in the information table.

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