Introduction to Combination Product Sets
Source:vignettes/introduction-to-combination-product-sets.Rmd
introduction-to-combination-product-sets.RmdAlternative Tunings Overview
Unless you’re into alternative tunings, your first question is probably “Alternative to what?” Of course, that depends on what culture you call home. But for most Americans and other native English speakers, most Europeans, and mass-market popular music creators and audiences nearly everywhere, music is tuned to 12-tone equal temperament, abbreviated as “12-TET”. You will also see this tuning referred to as “12-EDO”, which stands for “twelve equal divisions of the octave.”
So what are the alternatives? There are two main sources of alternative tunings:
Other cultures, and
Music theorists / composers / instrument builders / performers within the ancient Greek / Italian / French / German mainstream (with nationalist enhancements!) that now uses 12-EDO. For brevity we’ll call those “Western theoretical tunings”.
Other cultures
Notable examples of tunings from other cultures are Indonesian gamelan tuning, South Indian carnatic music, and various makams from Ottoman music. You can find detailed analyses of gamelan tunings, as well as Thai classical tunings and numerous theoretical and practical tunings in (Sethares 2013). The first edition of this book (Sethares 1998) is the book that sent me down the alternate tunings rabbit hole in 2001.
Western theoretical tunings
Numerous tunings and temperaments have derived by theorists, composers, and performers over time. This happened first as steps along the path from ancient Greek music to 12-EDO claiming the dominant market share it has today, and later as a counter to said market share.
In some cases, instrument builders constructed moderately complex instruments that used these tunings. It was the desire to mass-produce musical instruments, especially the piano, that led to the widespread adoption of 12-EDO as the standard.
There are two main types of such alternative tunings in use today:
Equal divisions of a period, usually the octave, and
Just intonation.
Brief guide to tuning and scales
Before moving onward, we need to define some concepts. Pitches are (sort of) the musical name for frequencies. Frequencies are the rate at which some string or air column or reed or membrane or loudspeaker or other physical object vibrates.
Frequencies are designated in cycles per second, called Hertz and abbreviated Hz. The typical range of human hearing is 20 - 20,000 Hertz, but not all of that range is commonly used in music.
A tuning describes a set of pitches a composer selects from in writing a piece, often dictated by the instruments that will be used to play it. Tunings usually consist of a set of repetitions of a scale. A scale is a set of pitches in ascending order of frequency.
The most common tuning of this type is equal divisions of the octave with more than twelve notes, generally called “N-TET” or “N-EDO”. For example, quarter-tone tunings are 24-EDO. Musicians create tunings of this kind for a variety of reasons, most often to provide more harmonic possibilities without giving up the ability to modulate to different keys.
Two such tunings that have achieved serious usage are 19-EDO and 31-EDO. If you’re
curious about such tunings, function et_scale_table in this
package can create scale tables for them, and function
keyboard_map can create a mapping for such a scale to a
synthesizer keyboard.
Here’s what the 19-EDO scale table and keyboard map look like:
data("edo19_names")
edo19_scale_table <- et_scale_table(edo19_names)
print(edo19_scale_table)
#> note_name ratio ratio_frac ratio_cents interval_cents degree
#> 1: C 1.000000 1 0.00000 NA 0
#> 2: C# 1.037155 977/942 63.15789 63.15789 1
#> 3: Db 1.075691 739/687 126.31579 63.15789 2
#> 4: D 1.115658 627/562 189.47368 63.15789 3
#> 5: D# 1.157110 1009/872 252.63158 63.15789 4
#> 6: Eb 1.200103 2333/1944 315.78947 63.15789 5
#> 7: E 1.244693 1114/895 378.94737 63.15789 6
#> 8: E#|Fb 1.290939 1553/1203 442.10526 63.15789 7
#> 9: F 1.338904 1442/1077 505.26316 63.15789 8
#> 10: F# 1.388651 1297/934 568.42105 63.15789 9
#> 11: Gb 1.440247 1639/1138 631.57895 63.15789 10
#> 12: G 1.493759 1077/721 694.73684 63.15789 11
#> 13: G# 1.549260 2406/1553 757.89474 63.15789 12
#> 14: Ab 1.606822 895/557 821.05263 63.15789 13
#> 15: A 1.666524 3893/2336 884.21053 63.15789 14
#> 16: A# 1.728444 1744/1009 947.36842 63.15789 15
#> 17: Bb 1.792664 1124/627 1010.52632 63.15789 16
#> 18: B 1.859271 1123/604 1073.68421 63.15789 17
#> 19: B#|Cb 1.928352 1884/977 1136.84211 63.15789 18
#> 20: C' 2.000000 2 1200.00000 63.15789 19
edo19_keyboard_map <- keyboard_map(edo19_scale_table)
print(edo19_keyboard_map)
#> note_number name_12edo octave_12edo note_name ratio_frac degree
#> 1: 0 C -1 Bb 1124/627 16
#> 2: 1 C# -1 B 1123/604 17
#> 3: 2 D -1 B#|Cb 1884/977 18
#> 4: 3 D# -1 C 1 0
#> 5: 4 E -1 C# 977/942 1
#> ---
#> 124: 123 D# 9 E 1114/895 6
#> 125: 124 E 9 E#|Fb 1553/1203 7
#> 126: 125 F 9 F 1442/1077 8
#> 127: 126 F# 9 F# 1297/934 9
#> 128: 127 G 9 Gb 1639/1138 10
#> period_number freq cents ref_keyname ref_octave ref_offset
#> 1: -4 29.31292 2210.526 A# 1 11
#> 2: -4 30.40205 2273.684 B 1 -26
#> 3: -4 31.53164 2336.842 B 1 37
#> 4: -3 32.70320 2400.000 C 2 0
#> 5: -3 33.91828 2463.158 C# 2 -37
#> ---
#> 124: 3 2605.14722 9978.947 E 8 -21
#> 125: 3 2701.94158 10042.105 E 8 42
#> 126: 3 2802.33234 10105.263 F 8 5
#> 127: 3 2906.45312 10168.421 F# 8 -32
#> 128: 3 3014.44252 10231.579 F# 8 32Just intonation
The other main class of derived tunings is called just intonation. Just intonation attempts to create perfect harmonies by dividing the octave into unequal intervals using ratios of small integers. For example, the just perfect fifth is 3/2, the just perfect fourth is 4/3, and the just major third is 5/4.
If you’ve heard a barbershop quartet, you’ve heard just intonation. Combination product sets are a form of just intonation.
Combination Product Sets
Combination product sets (Narushima 2019, chap. 6) are just one of theoretician Erv Wilson’s tuning constructs. The current release, v0.5.0, is focused on them, and the ability to play music with them on synthesizers that can be retuned, such as the Korg Minilogue XD, the Aodyo Anyma Phi, and the Ashun Sound Machines Hydrasynth.