Где послушать алгоритмическую компьютерную музыку

Соберу здесь хронологически все привлёкшие меня алгоритмические композиции, как они устроены и ссылки на записи (не включая докомпьютерную эпоху).

• обзор от IRCAM, с примерами аудио
• https://www.ircam.fr/article/detail/une-breve-chronologie-subjective-de-lusage-de-lintelligence-artificielle-en-composition-musicale/

• Lejaren Hiller, Leonard Isaacson - Illiac Suite (1957)
• https://www.youtube.com/watch?v=fojKZ1ymZlo
• В википедии есть краткое описание методов, 4 эксперимента:
1. the first is about the generation of cantus firmi
2. the second generates four-voice segments with various rules
3. the third deals with rhythm, dynamics and playing instructions
4. the fourth with various models and probabilities for generative grammars or Markov chains
• то есть - строгий контрапункт (в первых двух частях) (todo: есть прям схема, надо добавить)
• причём generated only pitch; registration, instrumentation, dynamics, and rhythm manually applied
• в третьем эксперименте - хроматическая музыка
• Generated pitch, rhythm, amplitude, and performance articulation
• и в четвёртом - марковские цепи
• Markov chains (zero and first order) for interval and harmony selection
• Models from music theory (Schenker)
• Only movement not produced from a combination of outputs
• Tempo, meter, dynamics added manually
• Рудольф Зарипов - "Уральские напевы" (1959)
• можно послушать трек mp3/ural.mp3
• из репозитория https://github.com/true-grue/algomusic
• или вот из лекции https://youtu.be/WNs8t5Yjm7o?t=2398 Петра Советова
• у него ручные правила, ручная настройка характеристик и рандом
• есть сканы нот
• а ведь вроде как он до самой смерти генерил композиции, интересно где они
• Не останавливаясь на вопросе подробнее, просто приведём перечень мелодий, сочинённых в разное время ЭВМ («Урал», «Урал-2», «БЭСМ-6») по программам P. X. Зарипова (рис. 9).
• 1. Соч. 1959 г. Самая первая мелодия из опубликованных «Уральских напевов». Напечатана в статье P. X. Зарипова «Об алгоритмическом описании процесса сочинения музыки» (Доклады АН СССР, 1960 г., т. 132, №6).
• 2. Соч. 1962 г. «Подражание венским классикам», вылившееся в детскую песенку. Опубликована в брошюре P. X. Зарипова «Кибернетика и музыка», 1963 г. О ней писал Лев Католин в кн. «Кибернетические путешествия» (М.: Знание, 1967, с. 88).
• 3. Соч. 1964г. Мелодия вальса. Исполнялась на вечере «Музыка и наука» в Доме композиторов в 1967 г. (председательствовал композитор Г. С. Фрид) виолончелисткой В. Яглинг – аспиранткой Московской консерватории, лауреатом международных конкурсов. См. об этом в кн.: И. Б. Гутчин. Кибернетические модели творчества (М.: Знание, 1969, с. 49).
• 4. Соч. 1964 г. Мелодия и аккомпанемент сочинены ЭВМ «Урал-2»; эту пьесу исполнял Т. Н. Хренников в фильме «Ищу законы творчества» (вып. 1966 г.), её передавали по радио в исполнении ансамбля электромузыкальных инструментов Всесоюзного Радио под управлением Вяч. Мещерина.
• 5. Соч. 1964 г. В экспериментах по сравнительной оценке мелодий, сочинённых машиной «Урал-2» и нашими профессиональными композиторами, которые проводились в 1968-1969 гг., эта мелодия заняла первое место. См. книгу P. X. Зарипова «Кибернетика и музыка» (1971, с. 201).
• 6. Соч. 1967 г. Мелодия на стихи «В лесу родилась ёлочка».
• 7. Соч. 1972 г. Машина БЭСМ-6 Вычислительного Центра АНСССР «методом Дунаевского» сочинила «Молодёжную» этого композитора; опубликована впервые в ДАН СССР (1973, т. 211, №3).
• 8 и 9. Соч. 1973 г. Сочинения ЭВМ БЭСМ-6-вальс и марш несколько раз передавались по Всесоюзному радио. Мелодии 7, 8, 9 опубликованы в кн. «Управление, информация, интеллект» (М.: «Мысль», 1976).
• Lambert Meertens - ...
• куда-то сюда надо видимо вставить David Cope'а
• но вообще ощущается какой-то громадный разрыв по датам, непонятно
• todo:
• Sony Flow Machines
• Hello World album
• https://www.helloworldalbum.net
• https://skyggewithai.bandcamp.com/album/hello-world
• и ещё один альбом вышел
• https://skyggewithai.bandcamp.com/album/american-folk-songs

todo:

- Iannis Xenakis - Achorripsis (1957)

    First work with a strong formal conception (done by hand).

    Sonic entities: equated with instrument categories.

    Sequential programming: density across the (musical) time.

- Enno Poppe ...

- Per Nørgård "Infinite Series" ...

- David Lang "The Passing Measures"

- Brian Eno ...

- Curtis Roads ...

- James Tenney ...

- Thomas Adès Violin Concerto (Concentric Paths "algorithmic")

- Pierre Boulez Dérive (1984)

    ■ work for piano + ensemble

      ■ piano ‘leads’ the texture, ensemble “prolongs, sustains, develops, emphasizes and echoes the phrases of (the piano)” (Goldman, p. 117)

    ■ Six harmonic fields (A–F) are developed from a 6-note row using ‘rotation’ technique

  использует серию SACHER

    которая крутится и получаются ещё 5 серий

  ■ Musical ramifications of rotation technique:

    ■ ‘PEDAL TONE’ (Ef) continues throughout

    ■ SINGLE SET-CLASS, but hexachord has enough intervallic variety to

    ensure developmental possibilities

    ■ ensures constant PITCH-CLASS VARIETY (harmonic flux)

  Problems

    ■ Repetition of ‘tonic’ note

    ■ If used melodically, intervals always move in same direction leads to predictability

      – cf. preference for inverted imitative counterpoint in, say, Bartók

    ■ Intervallic economy may become predictable

- ‘Rotation Technique’ in "Flourish with Fireworks" by Oliver Knussen

Человек + машина:

• Первый тут видимо Xenakis, про него и так везде и всюду можно почитать, пока что не буду расписывать.
• George Lewis - писал/пишет всякие системы для импровизации
• Rainbow Family (1984)
• https://www.youtube.com/watch?v=HhzC4Bm440k
• Voyager (1987)
• https://www.youtube.com/watch?v=hO47LiHsFtc
• "The KIM and I" (1979), for micro-computer and improvising musician
• "Chamber Music for Humans and Non-Humans" (1980), for micro-computer and improvising musician
• "Virtual Concerto" (2004), for improvising computer piano soloist and orchestra
• "Interactive Duo" (2007), for interactive computer-driven piano and human instrumentalist
• "Interactive Trio" (2007), for interactive computer-driven piano, human pianist, and additional instrumentalist
• Al Biles - GenJam (статья 1994 года)
• https://www.youtube.com/watch?v=rFBhwQUZGxg
• тоже система для импровизации
• Но вернемся к дням сегодняшним. Мы уже как-то писали об экспериментах швейцарца Eliyah Reichen, который совместно с Bösendorfer (одна из дочек Yamaha) решил поиграться с настройками и программированием MAX/MSP Ableton Live и Lemur, заложив определенные алгоритмы "отклика" инструмента на действия играющего.
• Позже эти наработки использовались уже непосредственно самой YAMAHA в сотрудничестве с Dan Tepfer. Они также совместно разработали ряд алгоритмов, основываясь на теории музыкальной импровизации и гармонии, которые стали частью проекта Yamaha Disklavier. Основа импровизации закладывается на наличие различных триггеров, разрешающих «первый ход» музыканта последующим «вторым ходом» машины. По признанию слушающих, далеко не всё удачно звучит, и порой машина «перебарщивает», но уже сейчас отмечают интерес к данному направлению и не отрицают положительные факторы развития направления.

Новые нейро-времена:

• todo: Google Magenta
• YACHT's new album is powered by ML + Artists
• https://magenta.tensorflow.org/chain-tripping
• Яндекс - Нейронная оборона - 404 (2016)
• https://music.yandex.ru/album/3558837
• IBM Watson Beat (2018?)
• какая-то демка есть - https://www.youtube.com/watch?v=Z5ymVzTUU6Y
• http://www.aiartonline.com/music/dj-beep-code/
• OpenAI MuseNet (2019)
• https://openai.com/blog/musenet/
• OpenAI Jukebox (2020)
• https://openai.com/blog/jukebox/

todo:

• Lejaren Hiller
• MUSICOMP
• Computer Cantata
• Доживём до понедельника (1968) - пример (?) музыки, сочинённой "электронным композитором" (звучит правда как Моцарт)
• https://youtu.be/TpTCCF9sEFA?t=3934
• 1:05:34 - 1:08:06
• William Shottstaedt - automatic species counterpoint program (75 rules)
• https://github.com/namin/metasolfeggio/blob/master/classics/fux.md
• Kemal Ebcioğlu - CHORAL (350 rules; в другом месте пишут про over 270 rules)
• http://global-supercomputing.com/people/kemal.ebcioglu/music/1989-01-28-Concert-Chorales.mp3
• fun fact:
• Following their initial success, Optikler played occasional live shows, and Ebcioğlu  composed Turkey’s entry in the 1975 Eurovision Song Contest. Unfortunately, “Seninle Bir Dakika” (“A Minute With You”) came in last place, but the nostalgic song has remained popular in Turkey, especially among the older segments of the population.
• David Cope - EMI
• (есть правда спорный момент, что это скорее алгоритмическое семплирование; хотя такую же критику можно и к нейронкам пробовать применять)
• http://artsites.ucsc.edu/faculty/cope/mp3page.htm
• http://artsites.ucsc.edu/faculty/cope/Bibliography_page_2.htm
• + Algorithmic cycle: http://artsites.ucsc.edu/faculty/cope/works1.htm
• https://www.youtube.com/watch?v=guAQZ9T6owg
• todo:
• https://computerhistory.org/blog/algorithmic-music-david-cope-and-emi/
• Bach by Design (1997)
• Classical Music Composed by Computer
• Virtual Mozart
• Virtual Rachmaninov
• EMI even composed a complete symphony in the style of Mozart which was performed at the Santa Cruz Baroque Festival in 1997.
• Cope has produced thousands of other works in various styles using EMI, including 5,000 Bach chorales available on his website.
• http://artsites.ucsc.edu/faculty/cope/5000.html
• follow-up programs such as Alena and Emily Howell
• serialism, total serialism
• Milton Babbitt
• Pierre Boulez
• Karlheinz Stockhausen
• Tom Johnson
• Clarence Barlow
• Iamus
• https://en.wikipedia.org/wiki/Iamus_(album)
• Review - https://www.nature.com/articles/488458a

Контрапункт в JI

Здесь я попробую для себя переформулировать (и закодить) правила контрапункта, но не в 12tet системе, а в just intonation'е.

Но сначала надо узнать, что происходит в 12tet.

• coursera, курс approaching music theory: melodic forms and simple harmony
• в качестве нот выбран дорийский лад
• консонансы: 3rd (6/5), 5th (3/2), 6th (5/3), 8th (2/1)
• диссонансы: 2nd (9/8), 4th (4/3), 7th (9/5)
• очень диссонансно: тритон (sqrt(2)/1)
• фиксируем басовую линию, и пытаемся построить мелодию выше
• 1. первый интервал - октава, а дальше подгоняем ноты так, чтоб было консонантно и чтоб прыжки были как можно меньше
• 2. тут уже сложнее, добавляются разумные прыжки; добавляется voice exchange; в конце добавляется каденция; я пока что не понимаю, как это всё закодить

Немного сбоку - а как перевести лады в just intonation? Вот есть 5-limit схема:

•                          2nd    3rd    4th      5th     6th   7th
• lydian          1    9/8     5/4  25/18   3/2     5/3  15/8
• ionian          1    9/8     5/4    4/3     3/2     5/3  15/8
• mixolydian  1    9/8     5/4    4/3     3/2     5/3   9/5
• dorian          1    9/8     6/5    4/3     3/2     5/3   9/5
• aeolian         1    9/8     6/5    4/3     3/2     8/5   9/5
• phrygian      1  27/25   6/5    4/3     3/2     8/5   9/5
• locrian         1  27/25   6/5    4/3   36/25   8/5   9/5 (но в этом ладу почти никто не сочиняет)

Ещё один вопрос - если переходить из лада в лад - меняются ли ноты? Пока что не знаю.

L-системы (и другие штуки) в музыке

Что я узнал из статьи Besada про музыку Posadas'а (Math and Music, Models and Metaphors: Alberto Posadas' ‘Tree-Like Structures’), так это то, что уже много каких композиторов заиспользовало L-системы в своей музыке. Так что можно и в пост их единый собрать.

Чем примечательны:

• Self-similarity
• High incidence of ‘transitional’ macrostructures
• Fragmented, ‘cellular’ microstructure (depending on choice of symbol-music relationship)
• Lack of strong thematic material allows for less architectonic, more organic form

Hanspeter Kyburz - Cells (1994)

     L-system, Movement 1 of Cells by Hanspeter Kyburz

• 1. . 2
• 1223
• 12232334
• 1223233423343445 12232334233434452334344534454556 23343445344545563445455645565667 34454556455656674556566756676778 45565667566767785667677867787889 5667677867787889677878897889899...
• 1: pppp tremolo or sustained tone (starting in vc)
• 2: ppp fingered tremolo (vn)
• 3: pp va tritones, preceeded by fl colour trill
• 4: pp piano arabesque, oboe run demisemis, preceeded by vc harmonic gl 5: sfz short double-stop attack (strs)
• 6: p–f cresc. upward run in str, preceeded by vib octatonic scale
• 7: f melodic fragments harmonized in octatonic
• 8: mf–ff tremolo cresc.
• NB: some instruments start to operate independently of the system (e.g. pf/vib/ sax)

Enno Poppe:

• Wespe (2005)
• Stoff (2015)
• todo: есть ещё статья про генеративные штуки в композициях Herz (2002), Tier (2002), Wald (2010), Welt (2011/2012)

Philippe Manoury - Tensio (2010):

• музыка
• статья
• видео описание на французском, от ircam

Alberto Posadas:

• Arborescencias (2007)
• Glossopoeia (2009)
• Fúlgida niebla de sol blanquecino (2010)
• Del reflejo de la sombra (2010)
• La tentación de las sombras (2011)
• Tombeau (2014)
• Double (2014)
• интервью про предыдущие 2 работы
• Anklänge an François Couperin (2014)
• Anklänge an Robert Schumann (2015)

Michael Norris - Icons and Artifice (2007):

• описание и отрывок
• слайды с чуть большей информацией

Alejandro Reyes - целый альбом De Natura Organica (2016) треков на одной L-системе

2 примера L-систем для ритмов.

Renick Bell - Fractal Beats (2014-2016):

• музыка
• статья

Sean Cotterill/co34pt/theseanco - HSPTLFLDHS (2018):

• музыка
• код
• объяснение

Каноны Вузы (Vuza canons):

Fabien Lévy:

• Soliloque sur [X, X, X et X], commentaire par un ordinateur d’un concert mal compris de lui [SOLILOQUY on X, X, X and X—a computer’s commentary on a concert it misunderstood]
• Coïncidences
• Où niche l’hibou

Канон длины 72

Альбомы на Supercollider

ещё supercollider (типа todo):

http://ericfairbanks.org/music/supercollider/code/2012/08/26/space-with-supercollider.html

http://www.mcld.co.uk/supercollider/

http://backtrace.blogspot.com

github

https://github.com/codehearts/supercollider-music

https://github.com/jsiemasko/Schemawound

https://github.com/alikthename/Musical-Design-in-Supercollider

https://github.com/efairbanks/supercollider-scripts

Прошёлся по группе Supercollider в Facebook'е, поискал там слово "bandcamp", получилась такая подборка

( todo: http://sccode.org/search?q=bandcamp&type=0 )

sc140

https://supercollider.github.io/community/sc140

https://archive.org/details/sc140

https://ia800202.us.archive.org/29/items/sc140/sc140_sourcecode.txt

MCLD

http://www.mcld.co.uk/evolutionarysound/

with code! and explanations! вот то, что нужно

https://www.facebook.com/groups/supercollider/permalink/10151148269194533/

https://schemawound.bandcamp.com/album/they-want-to-make-your-body-move-i-want-to-hold-you-perfectly-still

http://sccode.org/schemawound

https://schemawound.com/post/31040026304/hello-world-i-am-lonely-too

https://schemawound.com/post/31739550457/please-hold-extended

more:

https://schemawound.com/post/136617212398/sctweet-2015-12-31

https://schemawound.com/post/105537034503/2014-12-12-tweet-deconstruction

https://schemawound.com/post/58980873930/bitwise

https://schemawound.com/post/53025350841/haiku

https://schemawound.com/post/45152975247/a-map-to-everything-disquiet0062-lifeofsine

https://schemawound.com/post/40100438722/11132012-tweet-explained

https://schemawound.com/post/38598262939/audiojournal

https://schemawound.com/post/38127291200/telegraph-incantation-disquiet0050-morsebeat

https://schemawound.com/post/37164766525/unloop-antidote-disquiet0048-libertederive

https://schemawound.com/post/29553190911/se7en-seconds-in-hea7en-by-the-throat

https://schemawound.com/post/29553173950/se7en-seconds-in-hea7en-please-hold

https://schemawound.com/post/29553156736/se7en-seconds-in-hea7en-sound-check

https://schemawound.com/post/29553130142/se7en-seconds-in-hea7en-the-crawl

https://schemawound.com/post/29070261257/signalvoid-3-the-piano

https://schemawound.com/post/29066507849/signalvoid-2-the-same-color-as-your-skin

https://schemawound.com/post/29065985817/signalvoid-1-burn-it-to-disc-then-burn-the-disc

https://schemawound.com/post/25710867587/supercollider-tweets-background-tips

https://schemawound.com/post/24520532915/sinusoid

https://schemawound.com/post/21065032399/the-tunnel

with code! and explanations!

https://xkatedral.bandcamp.com/album/daniel-m-karlsson-a-loss-of-self

http://danielmkarlsson.github.io

with code! and explanations!

https://www.facebook.com/groups/supercollider/permalink/10158844693434533/

https://jpdrecourt.bandcamp.com/album/in-between-longing-seed-0262

http://drecourt.com/writings/the-code-of-in-between-longing/

https://www.facebook.com/groups/supercollider/permalink/10157673145164533/

https://soundcloud.com/w_number/sets/stability-ep

with code!

https://modk.bandcamp.com/album/chaotic-behaviour-ep

https://github.com/mikmed14/SuperCollider/tree/master/ChaoticBehaviour

with code!

https://modk.bandcamp.com/album/looking-for-silence

https://github.com/mikmed14/SuperCollider/tree/master/LookingForSilence

https://www.facebook.com/groups/supercollider/permalink/10151395495664533/

https://schemawound.bandcamp.com/album/trancount

https://www.facebook.com/groups/supercollider/permalink/10151467993524533/

https://schemawound.bandcamp.com/album/body-movements

(это альбом ремиксов на самом деле)

https://www.facebook.com/groups/supercollider/permalink/10152483360074533/

https://schemawound.bandcamp.com/album/i-meant-to-break-your-will-never-your-heart

https://fancyyyyy.bandcamp.com/album/the-new-pulsar-generator-recordings-volume-1

есть такой пример, не с альбома

https://scsynth.org/t/pulsar-synthesis/1618/2

и такой

https://github.com/madskjeldgaard/scbookcode/tree/master/Ch%2016%20Microsound

и здесь есть

https://github.com/dkmayer/miSCellaneous_lib

https://www.facebook.com/groups/supercollider/permalink/10158780522414533/

https://pointsourcearts.bandcamp.com/track/triangular-cavity

https://www.facebook.com/groups/supercollider/permalink/10158586529619533/

https://chipmusik.bandcamp.com/album/voice-algorithm-chmr-c-d-051

https://www.facebook.com/groups/supercollider/permalink/10158666399754533/

https://auraumua.bandcamp.com/album/tonnebel

https://www.facebook.com/groups/supercollider/permalink/10158590855614533/

https://antenaartbr.bandcamp.com/album/077-metanoia

https://shalmaneser.bandcamp.com/album/the-new-prime

IDM

https://www.facebook.com/groups/supercollider/permalink/10157586805954533/

https://virtual440.bandcamp.com/

https://www.facebook.com/groups/supercollider/permalink/10158311833089533/

https://caserecordsofficial.bandcamp.com/album/minimal-remnant

https://www.facebook.com/groups/supercollider/permalink/10151660044959533/

https://liljedahl.bandcamp.com/album/abiogenesis

https://www.facebook.com/groups/supercollider/permalink/10157453348049533/

https://importantdronerecords.bandcamp.com/album/do-xa

JI

https://www.facebook.com/groups/supercollider/permalink/10158325656084533/

https://subnaught.bandcamp.com/album/gardens-2

https://www.facebook.com/groups/supercollider/permalink/10157581957564533/

https://neilcosgrove.bandcamp.com/album/open-k-flower

https://www.facebook.com/groups/supercollider/permalink/10151961605534533/

https://ifarmusiqueconcretecompilation.bandcamp.com/track/digitalworld

https://www.facebook.com/groups/supercollider/permalink/10151134249634533/

https://bjmklein.bandcamp.com/

https://www.facebook.com/groups/supercollider/permalink/10150919211314533/

https://emmanuelemaggiore.bandcamp.com/

with code!

https://www.facebook.com/groups/supercollider/permalink/10154829172624533/

https://frankchannel.bandcamp.com/track/fragment-vi

    Thanks :)) Most of this piece resulted from me experimenting with ideas in the book "Self-similar melodies" by Tom Johnson (http://ssm.editions75.com)

http://sccode.org/1-55Y

with code! но так себе идея

https://www.facebook.com/groups/supercollider/permalink/10157902391229533/

https://diegetic.bandcamp.com/

    I always publish all the code I use to create my music. I built the code for this data sonification project in the Jython Environment for Music (Glacial Mass was done in Supercollider). You can find my code at github.com/kevinswanberg.

https://github.com/diegeticmn

https://www.facebook.com/groups/supercollider/permalink/10155502637974533/

https://solitunesrecords.bandcamp.com/album/ultraxy

https://www.facebook.com/groups/supercollider/permalink/10153197610284533/

https://liew.bandcamp.com/album/whole

with code!

https://www.facebook.com/groups/supercollider/permalink/10156126144499533/

https://co34pt.bandcamp.com/album/hsptlfldhs

https://github.com/theseanco/hsptlfldhs

https://www.facebook.com/groups/supercollider/permalink/10152437448839533/

https://joshuayohe.bandcamp.com/releases

https://www.facebook.com/groups/supercollider/permalink/10156821347979533/

https://squib-box.bandcamp.com/album/her-high-noise

https://www.facebook.com/groups/supercollider/permalink/10155287404699533/

https://modula-t.bandcamp.com/album/rapjams

https://www.facebook.com/groups/supercollider/permalink/10154336700809533/

https://thesilenthowl.bandcamp.com/album/cort-ge-dalsaxy

(score in pdf, if buy)

https://www.facebook.com/groups/supercollider/permalink/10156398926034533/

https://rebell.bandcamp.com/

https://www.facebook.com/groups/supercollider/permalink/10158178066544533/

    I’m happy to share the following with you…

    Most of the work is done with SinOsc.ar which I bounce down to a soundfile. I then load that back into SC and mess around with it using GrainBuf.ar.

    Any sequencing is done using LNX_Studio.

    FX are FreeVerb, GVerb and delays with PV ugens in the feedback loop.

    Once each track is finished I glue them together in Ableton, mixing in the field recordings. I just like the album as 1 track rather than 10. I don’t do any sound design in Ableton just cross fades.

    I also used a Moog Sub 37 in 3 tracks, again sequenced in LNX_Studio + an external Spring reverb.

https://neilcosgrove.bandcamp.com/album/bravo-joyous-pudding

https://www.facebook.com/groups/supercollider/permalink/10157987965269533/

https://threearms.bandcamp.com/album/iiia021-scmute-noir-noir-split-cs

https://www.facebook.com/groups/supercollider/permalink/10158447655624533/

https://joowonpark.bandcamp.com/album/rmhs

http://joowonpark.net/rmhs/

https://www.facebook.com/groups/supercollider/permalink/10157781309079533/

https://thecoldvacuumofspace.bandcamp.com/album/neither-here-nor-there

https://www.facebook.com/groups/supercollider/permalink/10151852169319533/

https://luistamagnini.bandcamp.com/

https://www.facebook.com/groups/supercollider/permalink/10157820351819533/

https://crossfadetapes.bandcamp.com/album/crossfade-i

https://www.facebook.com/groups/supercollider/permalink/10156344361249533/

    It's a system for real-time sampling and processing called "Aquifer" and it several modules implemented in it. Basically it always needs some external source as input. I vintage organs (usually casio stuff) and some toys. Then I have two buffers which everything is recorded into them and there are several modules which interpret the sound from the buffers, each in a different way, and a mixer for summing everything up.

    Some of the modules are programmed to produce percussive sounds from whatever is in the buffer and some are for more granular / textures processing. There is also a feedback loop to feed the result back into the buffer.

    I also use pre-programmed sequences which trigger those patterns, which are in some cases based on speech rhythm.

    Btw here you can find some recordings from live concerts where it's really 100% supercollider:

    https://soundcloud.com/aquifr

    or here:

    https://www.youtube.com/watch?v=Kzwwv483r_8

https://sovnrecords.bandcamp.com/album/subterranean-currents

https://www.facebook.com/groups/supercollider/permalink/10156047983679533/

https://charleshutchins.bandcamp.com/album/christmawave

https://www.facebook.com/groups/supercollider/permalink/10157736574524533/

https://virtual440.bandcamp.com/track/turbo

https://edvirtual440.wordpress.com/2019/07/22/turbo/

https://www.facebook.com/groups/supercollider/permalink/10157136712339533/

https://lfsaw.bandcamp.com/album/virtual-shells

https://www.facebook.com/groups/supercollider/permalink/10154799992824533/

https://noremixes.bandcamp.com/album/modulationist

https://www.facebook.com/groups/supercollider/permalink/10154290263804533/

https://thevoidpointers.bandcamp.com/album/values

https://www.facebook.com/groups/supercollider/permalink/10155857279034533/

https://tonestrukt.bandcamp.com/album/n-3

link doesn't work

https://www.facebook.com/groups/supercollider/permalink/10157778146804533/

https://simakim.bandcamp.com/album/wysiwh-ebook

https://www.facebook.com/groups/supercollider/permalink/10155945123979533/

https://sampluta.bandcamp.com/album/broken-symmetries

https://www.facebook.com/groups/supercollider/permalink/10151619099574533/

https://liew.bandcamp.com/

https://www.facebook.com/groups/supercollider/permalink/10155227858789533/

https://reflectives.bandcamp.com/album/eight-gestures-for-eight-virtuals

https://www.facebook.com/groups/supercollider/permalink/10152116485159533/

https://orlandoferguson.bandcamp.com/

https://www.facebook.com/groups/supercollider/permalink/10153400570119533/

https://binarycanary.bandcamp.com/releases

https://www.facebook.com/groups/supercollider/permalink/10151864346869533/

https://bjmklein.bandcamp.com/album/opteryx

https://www.facebook.com/groups/supercollider/permalink/10154839332724533/

https://svdk.bandcamp.com/album/opus-iii

https://www.facebook.com/groups/supercollider/permalink/10151957059509533/

https://luistamagnini.bandcamp.com/album/pris

with code! (overtone)

https://www.facebook.com/groups/supercollider/permalink/10155577022644533/

https://paullucas.bandcamp.com/album/hollow

https://github.com/paullucas/overtone-works/blob/master/src/otworks/c15.clj

https://www.facebook.com/groups/supercollider/permalink/10152108256579533/

https://cryochamber.bandcamp.com/album/psychosis

https://www.facebook.com/groups/supercollider/permalink/10154950211324533/

https://auditionrecords.bandcamp.com/album/grammar

https://www.facebook.com/groups/supercollider/permalink/10151804729884533/

    Bjørn: Most of the music was originally intended for performance at the SC Symposium in Boulder, so it ended up being done entirely in SC. First, I wrote a crude framework to load patterns from files, store shared resources (such as audio files), determine when to start and stop them, and switch EWI settings (the melodies are played on EWI, except for track 4 where I used alto recorder). Then it was just a matter of writing the material, which I mostly did on the bus commuting to and from work. Generally, the code files use algorithms to generate the patterns, so the result is different (but recognizable) every time.

    When it came time to do the album, I rendered all the sequences to disk, imported them into Pro Tools, and mixed them like I would any other album. I added some new material as well, but not much. Other than having the luxury of multiple EWI takes, and more detailed mixing, it sounds pretty much like the live version.

https://shalmaneser.bandcamp.com/

https://www.facebook.com/groups/supercollider/permalink/10154526566849533/

https://awavepress.bandcamp.com/album/ingress

https://www.facebook.com/groups/supercollider/permalink/10157923222844533/

https://escrec.bandcamp.com/album/norwegian-disco

https://www.facebook.com/groups/supercollider/permalink/10153476423179533/

https://chipmusik.bandcamp.com/album/teiza-raizi-chmr-ep-021

https://www.facebook.com/groups/supercollider/permalink/10154611666614533/

https://feintrecords.bandcamp.com/album/derailing-single

https://www.facebook.com/groups/supercollider/permalink/10156641385984533/

https://acrerecordings.bandcamp.com/album/exsudate

https://www.facebook.com/groups/supercollider/permalink/10158081267479533/

https://soundcloud.com/w_number/sets/believe

https://www.facebook.com/groups/supercollider/permalink/10151643207589533/

https://voidmusic1.bandcamp.com/

https://www.facebook.com/groups/supercollider/permalink/10157200279499533/

https://liew.bandcamp.com/album/the-secret-of-mountains-and-tropical-fruits

https://www.facebook.com/groups/supercollider/permalink/10152281101219533/

https://liew.bandcamp.com/album/landing

https://www.facebook.com/groups/supercollider/permalink/10152472493044533/

https://fallopiantunes.bandcamp.com/album/new-awareness

https://www.facebook.com/groups/supercollider/permalink/10151927271699533/

https://cryochamber.bandcamp.com/album/psychosis

https://www.facebook.com/groups/supercollider/permalink/10151715005554533/

https://5kyeeidtal.bandcamp.com/track/fascist-in-dat-tree

https://www.facebook.com/groups/supercollider/permalink/10155951813044533/

https://noremixes.bandcamp.com/album/park-jones

https://www.facebook.com/groups/supercollider/permalink/10158636927164533/

https://micksussmanrosenberg.bandcamp.com/album/the-kressel-studies-vol-2

https://www.facebook.com/groups/supercollider/permalink/10152766699764533/

https://sombresoniks.bandcamp.com/track/anubis-house-of-dead-prince

https://www.facebook.com/groups/supercollider/permalink/10157790929849533/

https://joshuaadamacosta.bandcamp.com/album/revisiting-no-one-in-particular

https://www.facebook.com/groups/supercollider/permalink/10151744462754533/

https://xylemrecords.bandcamp.com/album/terraform-mars

https://www.facebook.com/groups/supercollider/permalink/10150608581354533/

https://ayankoko.bandcamp.com/album/bandwidth-feat-gerri-j-ger

https://www.facebook.com/groups/supercollider/permalink/10152488006994533/

https://workinklassnoize.bandcamp.com/album/unabomber

https://www.facebook.com/groups/supercollider/permalink/10152237150714533/

https://sebastianoldsmith.bandcamp.com/releases

https://www.facebook.com/groups/supercollider/permalink/10157255130194533/

https://soundcloud.com/w_number/sets/looking-for-silence

https://www.facebook.com/groups/supercollider/permalink/10152764750744533/

https://ifarmusiqueconcretecompilation.bandcamp.com/track/guitar-indication-5

https://www.facebook.com/groups/supercollider/permalink/10152883304554533/

https://5kyeeidtal.bandcamp.com/album/tidal-flow-unmastered-album

https://www.facebook.com/groups/supercollider/permalink/10151805617344533/

https://5kyeeidtal.bandcamp.com/track/skeyefourwern

https://www.facebook.com/groups/supercollider/permalink/10151786763294533/

https://5kyeeidtal.bandcamp.com/track/red-dirt-girl-rdg

with code

https://www.facebook.com/groups/supercollider/permalink/10156951894614533/

https://tefe.bandcamp.com/album/tefe-006-the-giants-of-chaos

https://archive.org/details/TEFE015Source

https://www.facebook.com/groups/supercollider/permalink/10152883286149533/

https://fractionstudiocompilation.bandcamp.com/track/sonologyst-a-scanner-darkly-italy

https://www.facebook.com/groups/supercollider/permalink/10151916064744533/

https://mutant-beatniks.bandcamp.com/album/animal-farm

Расширенная чистая интонация: гармонические ресурсы

 Какие есть гармонические техники (наверно я ограничусь 13-limit JI):

• tonality flux
• todo: построение JI шкал
• todo: классификация интервалов в центах
• otonal и utonal
• otonal-utonal chord progression:
• было 4:5:6:7 (или 1:5/4:3/2:7/4)
• стало 4:x:6:y (или 1:6/5:3/2:12/7, а если mod 2: 3/3:3/5:3/2:3/7 или 1/3:1/5:1/2:1/7)
• то есть x = 24/5, y = 48/7
• harmony area
• microtonal modulations
• более хитрый тоннетц
• ...

Складки, Делёз и математика

 Запостил недавно на реддите ответ, изначально вопрос был про соприкосновения art и math, но также в вопросе просили разъяснений по книжке Делёза про складки и Лейбница, решил сюда тоже продублировать, чего добру пропадать:

If we talk about maths, you could better check out these pages:

• https://en.wikipedia.org/wiki/Mathematics_of_paper_folding (so, it's about origami)

• There' also a PBS documentary about origami called "Between the Folds" - https://www.youtube.com/watch?v=aFrDN5eYPOQ

• https://arxiv.org/abs/1608.00152 - The mathematics of taffy pullers (btw, try to search on youtube "taffy pullers")

• Folding also appears in strange or chaotic attractors - http://sprott.physics.wisc.edu/stretch.htm and http://www.scholarpedia.org/article/Chaos_topology#Four_different_templates

• Folds appear in the theory of singularities ("catastrophe theory"): Fold catastrophe - https://en.wikipedia.org/wiki/Catastrophe_theory#Potential_functions_of_one_active_variable; there's actually another singularity named "swallowtail", and it inspired Dali to make a painting https://en.wikipedia.org/wiki/The_Swallow%27s_Tail

(далее - более слабая часть ответа)

If we talk about philosophy and Deleuze:

1. Deluze is not the easiest philosopher to understand. The terminology that he uses is never static, it's always changing, from book to book, from interpreter to interpreter.

2. You can try to search for Deleuze dictionaries, e. g. - https://www.simonosullivan.net/articles/deleuze-dictionary.pdf - here's an overview of the fold.

3. There's a seminar by Deleuze, translated into English - https://deleuze.cla.purdue.edu/seminars/foucault/lecture-25 - where he also speaks about avant-garde music by Boulez "Pli selon pli" (Fold by fold)

4. If you are interested in contemporary philosophy discussion of this concept, then you can try to check out this seminars about the fold - https://www.youtube.com/watch?v=3spbzZz4nf8, also including discussions of Simondon's works (this first one is for free, others are hidden by paywall)

Алгоритмы в музыку (сонификация алгоритмов)

Есть мысль - придумать несколько музыкальных моделей, для того, чтоб применить для них всевозможные алгоритмы, известные по литературе или спортивному программированию.

То есть, такая извращённая идея - знаю, что есть интересный любопытный алгоритм. Хочу подогнать что-нибудь музыкальное, какую-то музыкальную структуру представить в виде какой-то структуры данных, а дальше для данной структуры глянуть алгоритмы и что мы вытащим в результате работы этого алгоритма.

• hidden markov models
• не знаю почему, но меня постоянно привлекает эта структура
• типа это же про analysis, про bayesian inference
• heavy-light декомпозиция

Связь тембра и звуковысотности

 Примеры композиций, которые используют эту связь:

• John Moriarty - The UnWarrior Suite - https://www.youtube.com/watch?v=i_eQ4ZfaWH4&list=PLF6XElTgRwmMYiXOsm0Ewk-iorFKd61jj&index=1
• John Chowning "Stria" (может правда речь о другом треке, или их несколько; первый FM-синтез)
• William Sethares "C2ShiningC"
• The timbre progresses from a harmonic timbre (with partials following the harmonic series) to a 'pseudo-harmonic' timbre (with partials adjusted to align with the notes of the current tuning) and back again.
• Twice as rapidly, the tuning progresses (via polyphonic tuning bends), within the syntonic temperament, from an initial tuning in which the tempered perfect fifth (P5) is 695 cents wide (19-tone equal temperament, 19-tet) to a second tuning in which the P5 is 720 cents wide (5-tet), and back again.
• всякие разные Bohlen-Pierce произведения на кларнетах
• https://en.xen.wiki/w/Kai-metalbar-nonoct - не знаю что это, но тут "tonality diamond" of a metal bar
• вообще пример из статьи https://en.xen.wiki/w/Generalized_overtone_tuning
• в ней же упомянут - Hyperpiano - https://www.youtube.com/watch?v=xcSod-sj2CE
• и ещё примеры для Hyperpiano - https://www.youtube.com/watch?v=83ZUpzs5MaI
• также есть альбомы от того же Сетареса - Xentonality и Exomusicology
• https://sethares.engr.wisc.edu/xentone.html
• https://sethares.engr.wisc.edu/exo.html
• и кажется музыка на гамеланах связана с их тембрами явным образом
• есть ещё статья "Relating Spectrum and Tuning of the Classical Thai Renat Ek"
• и ещё одна тема - https://en.xen.wiki/w/Mixed_timbre

Расширенная чистая интонация: алмазы тональностей

Так, в особенности мне стали интересны всякие разные варианты про tonality diamonds. Потому что там есть extended JI.

Какие есть примеры:

• Partch diamond
• https://www.chrysalis-foundation.org/musical-mathematics-pages/partch-diamond/
• 11-limit
• подходит для 2d-клавиатуры
• а вот как Erv Wilson разместил на клавиатуре 43 pitches Партча (несколько вариантов)
• http://anaphoria.com/Partchpapers.pdf
• Partch included four illustrations based on Meyer’s tonality diamond: a 5-limit Incipient Tonality Diamond, an 11-limit Expanded Tonality Diamond, an 11-limit Block Plan of the Diamond Marimba, and The Tonality Diamond on a 13-Limit. (Genesis of a Music (1949), p. 110, 158, 213, and 327, respectively. Genesis of a Music (1974), p. 110, 159, 261, and 454, respectively)
• Meyer diamond
• https://www.chrysalis-foundation.org/musical-mathematics-pages/meyer-diamond/
• 7-limit
• Forster diamond
• https://www.chrysalis-foundation.org/musical-mathematics-pages/forster-diamond/
• 13-limit
• вроде тоже подходит для 2d-клавиатуры
• 15-limit tonality diamond
• https://en.xen.wiki/w/15-limit_tonality_diamond
• Hendrix diamond
• https://en.xen.wiki/w/13-Hendrix_diamond
• в каком-то смысле это 15-limit
• есть ощущение, что 33 pitches Кайла Ганна тоже можно организовать в виде diamond'а
• https://www.kylegann.com/RomancePostmoderne.html
• тогда это будет 15-limit
• Fabio Costa - не знаю diamond или нет, но видны зато решения для 2d-клавиатуры
• https://www.youtube.com/watch?v=VswBvmpnS1c
• 31:31, (25:15?), 63:15 (кажется их две с этим limit'ом)
• логика размещения нот такая же как у Эрва Вильсона с 41 нотой Партча (2 ноты были смёржены друг с другом)
• http://ripnread.com/sample-page/tuning/
• 15-limit
• у Erv'а Wilson'а есть ещё всякие разные diamond'ы
• http://anaphoria.com/partchcomment.html
• 'Eckland' is the first in a set of tunings that extend the diamond properties to 2 points instead of a single 1/1 mapped to a 31 tone keyboard pattern. Wilson commented that multiple diamonds offers both great modulatory resources but also yield many more repeatable tetrachords. Andrus adds 4 filler notes to Eckland and is mapped to a 41 tone keyboard. Andromeda shows mapping of two full diamonds with Cassandra showing the ogdoadic 13 and 15 limit diamond onto a 53 tone keyboard with charts of the closely related scales.

Теории микротонов

 Какие есть теории микротонов и как я их сейчас понимаю:

• регулярные темперации - берём какой-то n-limit JI, задаём коммы, и дальше можно маппить все доступные дроби до темперированных. Заодно мы можем получить генератор (например квинту) и период (обычно это октава, иногда это часть октавы, но наверно бывают и тритавы, и прочее). Иногда это может быть даже будет MOS, тогда у нас получается конечное число нот в октаве, но вообще необязательно.
• если генератор один - то это линейная темперация
• впрочем никто не мешает тут попробовать взять и более высокие гармоники, но считать, что они просто грубо приближены
• todo: в том месте, когда это конечный набор нот - то в случае 5-limit кажется мы получаем параллелограмм, тор, эллиптическую кривую - так ведь?
• MOS звукоряды - тут вроде всё просто; иногда они совпадают с регулярными темперациями
• todo: интересно, бывают ли MOS звукоряды, не пересекающиеся с регулярными темперациями?
• todo: отдельный момент, видимо, MOS подмножества из EDO нот
• просто всякие разные EDO, типа 15 или 22
• combination product sets (cps) - такие конечные подсетки из произведений (нечётных) гармоник, обладают интересной геометрией, а ещё у них отсутствует тональный центр
• extended JI - такая громадная решётка из рациональных чисел (типа не только 3 и 5, но и 7, 11, 13, 17, 19, а иногда даже и выше)
• Метатональности Сошинского и Олейникова - https://www.youtube.com/watch?v=seqasXk7ahs
• всякие другие примеры (Bohlen-Pierce, 833, система Кайла Ганна,  ...)
• есть ещё такие игры с тембром:
• Golden non-Meantone:
• From article by Walter O'Connell in Xenharmonikon 15 (1993), but written in the 60s. Period golden ratio of 1.618.., generator: 2:1 ratio. Works with 7, 9, 11 and 14 note scales. Intended for a timbre consisting of golden-section partials.

Дальше - вопрос - как этим пользоваться?

• один из вариантов, который я сейчас прорабатываю - это путешествие по tonnetz'у
• ещё один вариант - маппинг MOS звукорядов на двумерную Hex клавиатуру
• todo: cps вроде бы тоже можно смаппить на эту клавиатуру

какие ещё свойства нужно изучить:

• мелодические
• гармонические
• comma pumps
• спектралистские
• формообразующие
• ...

Программирую микротона

Здесь собираю разные мини-эксперименты по микротонам (раньше этот текст был в посте про диаграмму Фарея, но там стало тесно и вообще превратилось в оффтопик)

• хотел взять scale labyrinth, и переделать его для варианта с q-деформацией. В итоге вроде разобрался как устроен этот лабиринт. Это некоторая модификация дерева Штерна-Броко. Каждое кольцо соответствует знаменателю дроби d, например 7. Каждый горизонтальный сегмент - несократимой дроби n/d, например 4/7. Каждый сегмент ограничен своими "родителями", например, 4/7 = 1/2 + 3/5 (в смысле как медианта). Вот. Ага. Дошло, как применить q-деформацию - теперь, скажем, дроби 3/7 и 4/7 разъедутся ещё и по разным кругам. План:
• добавить q-деформацию
• кстати, у меня теперь есть свой scale labyrinth! Что можно в него допилить:
• выписать названия для MOS шкал, а именно:
• meantone/syntonic (81/80, todo: надо разобраться в разнице между ними)
• сюда входит привычная темперация
• наверно тут нет очевидного генератора, но можно взять например чистую квинту, тогда будет Pythagorean diatonic tuning (701.955 центов)
• число шагов - 2, 3, 5, 7, 12; 17, 29, 41, 53 (есть другой вариант с 19, 31, 43, 55 - надо разобраться, возможно надо взять другой генератор (типа меньше 700))
• mavila (135/128, ещё называется как pelogic family)
• чем интересен - the mavila diatonic scale is similar to the normal diatonic scale - except interval classes are flipped. Wherever there was a major third, you'll find a minor third, and vice versa.
• optimal 5-limit tuning is 679.8 cents
• число шагов - 2, 3, 5, 7, 9, 16, 23, 30, 53
• TODO:
• заценить xen.wiki: In addition to the 7-note anti-diatonic scale described, Mavila also has a 9 note "superdiatonic" MOS, the "super-Ionian" mode of which looks LLLsLLLLs. This is the basis for the [Armodue_theory]
• и вот эти поинты тоже дико интересны
• TODO:
• Mavila generates a 16 tone chromatic MOS. In a certain sense, much of mavila makes sense if viewed within the lens of a 16-tone chromatic gamut, similarly to how much of meantone is thought of in the setting of a 12-tone chromatic gamut.
• TODO:
• After the 16 tone chromatic scale is the 23 tone enharmonic MOS, which can be thought of as an "extended mavila" analogous to the "extended meantone" 19-tone enharmonic scale. If the mavila fifth is flatter than that of 16-EDO (675 cents), it will instead generate an MOS at 25 notes. This is similar to how if the meantone fifth is tuned sharper than 12-EDO, it will instead generate a 17-tone MOS rather than a 19-tone one.
• porcupine (250/243)
• Lastly, porcupine temperament is extremely notable as being, in a certain sense, the simplest 5-limit temperament that is not supported by 12-EDO (and is thus inherently xenharmonic), but is at least as good intonationally.
• тут нет какого-то единственного генератора, возьмём 162.75 (POTE)
• число шагов - 7, 8, 15, 22; 37, 59
• интересно, нашёл такое на xen.wiki:
• In the 7-limit, that become Meantone⋎Porcupine = <JI>, Meantone⋏Porcupine = <1>; hence, we may consider 7-limit meantone and porcupine to be totally unrelated.
• miracle (225/224, todo: marvel temperament)
• While not particularly striking as a 5-limit temperament, it is one of the very best linear temperaments in the 7 and 11 limits
• здесь есть конкретный генератор - cекор - 116.63 (ну хотя он тоже плавает)
• число шагов - 10, 11, 21, 31, 41
• кстати, не в тему, но раз уж начал тут писать. Попробую ещё эксперимент по приближению к combination product set'ам через MOS'ы (правда пока что перебираю не все варианты)
• hexany
• 2)4 cps - 6 нот
• для [1, 3, 5, 7] нашлось такое - генератор 697, 19 шагов,
• LL'sLLs'L'LsLs'LLsL'Ls'Ls
• dekany
• 2)5 или 3)5 cps - 10 нот
• чего-то хорошего не нашёл
• pentadekany
• 2)6 или 4)6 cps - 15 нот
• чего-то хорошего не нашёл
• eikosany
• 3)6 cps - 20 нот
• нашлось приближение в 76 нот, генератор 426.1 или 773.7
• стоит попробовать полный вариант перебора, где cps строится от одной тоники (которая всё равно виртуальная), а MOS - от другой
• при tolerance = 6 и шаге сдвигов в 10 центов:
• для hexany нашлись варианты в 15 нот, генераторы 310 (сдвиг в 350) или 890 (сдвиг в 200)
• sssLsssLsssLssL или LssLsssLsssLsss
• 
  
• для dekany есть варианты в 26 нот, например, генератор 234 (сдвиг 970)
• LssssLssssLssssLssssLsssss
• 
  
• pentadekany - generator: 233.899999999991 , shift: 150
• 36 LssssssLssssssLssssssLssssssLsssssss
• 
  
• eikosany - generator: 351.50000000002046 , shift: 200
• 41 LsLsLssLsLssLsLsLssLsLssLsLsLssLsLssLsLss
• и ещё можно попробовать ослабить критерий близости нот (сейчас было выбрано 5 центов)

For example, the fractions 1/2 and 3/5 have a mediant 4/7, so the interval between

1/2 and 4/7 is the valid tuning range of an MOS with 2 large and 5 small tones (the

anti-diatonic), while the interval between 4/7 and 3/5 is the valid tuning range of its

inverse - the diatonic 5L, 2s. At precisely 4/7, the large and small steps are equally

sized and the two scales meet. Each triple of fractions made from two adjacent

fractions and the fraction between them on the row below corresponds, therefore,

to the three landmark tunings:

    - the central value gives the generator/period ratio where the MOS scale meets its inverse,

    - the outer values give the tunings at which the small steps of the MOS, or its inverse shrink to zero.

The tuning range over which bL, ds, and its inverse dL, bs, are coherent is

bounded by the tunings at which their respective lowest cardinality embedding

scales are equally tuned. That is, at (2a+c)/(2b+d) and (a+2c)/(b+2d).

For instance, the diatonic has boundary tunings of 4/7 and 3/5,

and their mediant (embedding scale) is 7/12, so the range over which

the diatonic scale is coherent is 4/7 to 7/12; similarly,

the anti-diatonic has boundary tunings at 1/2 and 4/7 with a mediant (embedding scale) of 5/9,

so the range over which it is coherent is 5/9 to 4/7, which can be gleaned from [5].

любопытно, надо попробовать ещё и эти границы нарисовать

то есть есть:

- valid tuning range

- coherent tuning range

интересное замечание:

Taking mediants is a common step in such enumeration sequences (including the Farey sequence, a source of much beautiful mathematics on QP1), but is best understood not as an algebraic operation on Q but rather as a geometric operation on QP1.

Сложность и формообразование

 Здесь буду собирать различные примеры сложностей (complexity) и формообразований (что-то типа growth of form). Всякая классика и не очень. Примеров конечно бесконечное число можно привести, но мне это нужно для вдохновения и ещё раз для.

глоссарий - https://en.wikipedia.org/wiki/Glossary_of_systems_theory

много моделей:

• https://www.complexity-explorables.org/explorables/
• https://github.com/r03ert0/interesting

классика:

• https://en.wikipedia.org/wiki/Metabolic_pathway
• листья растений
• http://algorithmicbotany.org/papers/Leaves2017/index.html
• Физики объяснили морщинистую форму листьев водных растений - https://nplus1.ru/news/2020/01/27/aquatic-plant
• L-systems
• physarum
• https://github.com/fogleman/physarum
• https://sagejenson.com/physarum
• swarms
• https://en.wikipedia.org/wiki/Swarm_behaviour#Mathematical_models
• Collective behavior: How animals work together - https://knowablemagazine.org/article/living-world/2020/collective-behavior-how-animals-work-together
• https://theweek.com/articles/932005/learning-from-collective-wisdom-animals
• The Ecology of Collective Behavior in Ants - https://www.annualreviews.org/doi/10.1146/annurev-ento-011118-111923
• муравьи
• Ant Foraging Revisited - https://www.cc.gatech.edu/~turk/bio_sim/articles/ant_foraging_revisited.pdf
• physarum
• https://github.com/fogleman/physarum
• https://sagejenson.com/physarum
• Turing patterns
• https://www.quantamagazine.org/a-math-theory-for-why-people-hallucinate-20180730/
• статья Turing instabilities in biology, culture, and consciousness? On the enactive origins of symbolic material culture
• New theory deepens understanding of Turing patterns in biology - https://phys.org/news/2018-06-theory-deepens-turing-patterns-biology.html
• Biologists Home In on Turing Patterns - https://www.quantamagazine.org/biologists-home-in-on-turing-patterns-20130325/
• Ancient Turing Pattern Builds Feathers, Hair — and Now, Shark Skin - https://www.quantamagazine.org/ancient-turing-pattern-builds-feathers-hair-and-now-shark-skin-20190102/
• repeated blur and sharpen - https://twitter.com/matthen2/status/1327449422532263936
• Universal model for the skin colouration patterns of neotropical catfishes of the genus Pseudoplatystoma - https://www.nature.com/articles/s41598-020-68700-0
• bird/starling flocking/murmurations
• Agent based simulation fo a flock of starlings
• Hybrid Projection Model
• https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4115545/#s2title
• https://theweek.com/articles/932005/learning-from-collective-wisdom-animals
• Information transfer and behavioural inertia in starling flocks
• https://www.sciencemag.org/news/2014/07/how-bird-flocks-are-liquid-helium
• https://www.nature.com/articles/nphys3035
• картинка - https://meduza.io/short/2021/03/07/tysyachi-skvortsov-sbilis-v-stayu-v-forme-ogromnoy-ptitsy-chtoby-otpugnut-hischnikov-fotografiya

что-то менее классическое:

• primordial particle system
• How a life-like system emerges from a simplistic particle motion law - https://www.nature.com/articles/srep37969
• https://www.youtube.com/watch?v=makaJpLvbow
• crystal growth and Voronoi diagrams
• Термиты построили гнезда в соответствии с кривизной поверхности
• https://nplus1.ru/news/2020/07/28/termite-nest
• Schlieren texture of liquid crystal nematic phase
• https://en.wikipedia.org/wiki/Liquid_crystal
• heliosphere
• бассейны рек
• наука синхронизации
• аттрактор Лоренца (и модулярные формы)
• турбулентность
• Mathematicians Prove Universal Law of Turbulence - https://www.quantamagazine.org/mathematicians-prove-batchelors-law-of-turbulence-20200204/
• An Unexpected Twist Lights Up the Secrets of Turbulence - https://www.quantamagazine.org/an-unexpected-twist-lights-up-the-secrets-of-turbulence-20200903/
• Self‐adjusting systems avoid chaos
• todo: надо примеры поискать
• dual-phase evolution
• Non-equilibrium thermodynamics
• metastability
• todo: для начала - https://www.youtube.com/watch?v=1Ji93A2T9JI
• Stochastic resonance
• https://en.wikipedia.org/wiki/Stochastic_resonance
• https://en.wikipedia.org/wiki/Stochastic_resonance_(sensory_neurobiology)
• http://www.scholarpedia.org/article/Stochastic_resonance
• и в целом это называется Noise-Induced Phenomena
• В падающей на переохлажденную поверхность капле увидели волны замерзания - https://nplus1.ru/news/2020/01/28/ice-wave
• Физики научились управлять следами от капель - https://nplus1.ru/news/2020/02/18/liquid-deposition-pattern
• северные сияния
• Eco-evolutionary significance of “loners”
• https://journals.plos.org/plosbiology/article?id=10.1371/journal.pbio.3000642
• https://www.quantamagazine.org/out-of-sync-loners-may-secretly-protect-orderly-swarms-20200521/
• Lévy walk Wired Into Animals May Help Them Hunt - https://www.quantamagazine.org/random-search-wired-into-animals-may-help-them-hunt-20200611/
• The Hidden Magnetic Universe Begins to Come Into View - https://www.quantamagazine.org/the-hidden-magnetic-universe-begins-to-come-into-view-20200702/
• стекло
• The Physics of Glass Opens a Window Into Biology - https://www.quantamagazine.org/the-physics-of-glass-opens-a-window-into-biology-20180611/
• Ideal Glass Would Explain Why Glass Exists at All - https://www.quantamagazine.org/ideal-glass-would-explain-why-glass-exists-at-all-20200311/
• To Make the Perfect Mirror, Physicists Confront the Mystery of Glass - https://www.quantamagazine.org/to-make-the-perfect-mirror-physicists-confront-the-mystery-of-glass-20200402/
• Why Is Glass Rigid? Signs of Its Secret Structure Emerge. - https://www.quantamagazine.org/why-is-glass-rigid-signs-of-its-secret-structure-emerge-20200707/
• Global Wave Discovery Ends 220-Year Search - https://www.quantamagazine.org/weather-data-reveals-long-predicted-pressure-waves-20200813/
• Why Are Plants Green? To Reduce the Noise in Photosynthesis. - https://www.quantamagazine.org/why-are-plants-green-to-reduce-the-noise-in-photosynthesis-20200730/
• Math of the Penguins - https://www.quantamagazine.org/math-of-the-penguins-20200817/
• The Mathematical Structure of Particle Collisions Comes Into View - https://www.quantamagazine.org/new-particle-collision-math-may-offer-quantum-clues-20200820/
• The self-made beauty of the centriole - https://knowablemagazine.org/article/living-world/2018/self-made-beauty-centriole
• Gerrymandering (?)
• Renormalization
• How Mathematical ‘Hocus-Pocus’ Saved Particle Physics - https://www.quantamagazine.org/how-renormalization-saved-particle-physics-20200917/
• Complexity Scientist Beats Traffic Jams Through Adaptation - https://www.quantamagazine.org/complexity-scientist-beats-traffic-jams-through-adaptation-20200928/
• A Physicist’s Approach to Biology Brings Ecological Insights - https://www.quantamagazine.org/a-physicists-approach-to-biology-brings-ecological-insights-20201013/
• Scientists Uncover the Universal Geometry of Geology - https://www.quantamagazine.org/geometry-reveals-how-the-world-is-assembled-from-cubes-20201119/
• predator-prey in chemical signaling
• https://twitter.com/calebhuwm/status/1328748309272539138
• https://www.nature.com/articles/s41557-020-00575-0
• Emergent Road Rules In Multi-Agent Driving Environments
• https://fidler-lab.github.io/social-driving/
• https://github.com/fidler-lab/social-driving
• https://arxiv.org/abs/2011.10753
• What Is an Individual? Biology Seeks Clues in Information Theory - https://www.quantamagazine.org/what-is-an-individual-biology-seeks-clues-in-information-theory-20200716/
• Some Proteins Change Their Folds to Perform Different Jobs - https://www.quantamagazine.org/metamorphic-proteins-change-their-folds-for-different-jobs-20210203/
• What Is Life? Its Vast Diversity Defies Easy Definition - https://www.quantamagazine.org/what-is-life-its-vast-diversity-defies-easy-definition-20210309/

Gene Ward Smith

Решил сделать бекап данных по нему.

Вот его музыка - https://archive.org/search.php?query=creator%3A%22Gene+Ward+Smith%22

Вот ссылки о нём

https://en.xen.wiki/w/Gene_Ward_Smith

https://microtonal.miraheze.org/wiki/Gene_Ward_Smith

https://en.xen.wiki/w/Microtonal_Music_by_Gene_Ward_Smith

Его посты в рассылках:

Subject: Wedge products and the torsion mess

http://www.robertinventor.com/tuning-math/s___3/msg_2075-2099.html#2098

Subject: Unlocking the mysteries of the wedge invariant

https://robertinventor.com/tuning-math/s___3/msg_2100-2124.html#2122

Вот то, что смог вытащить с его сайта (todo - значит, что не нашёл, но когда-то этот материал был на его сайте):

https://web.archive.org/web/20190317163716/http://www.tonalsoft.com/gws/home.aspx

Circulating Temperaments

https://web.archive.org/web/20121025200638/http://tonalsoft.com/gws/circ.html

It is a sad fact that common practice Western music is based on triads, and the thirds of these triads are woefully out of tune. The major third in 12-equal is 13.69 cents sharp, and the minor third is 15.64 cents flat. Considering that anything beyond ten cents over or under a just intonation is entering the region of major cheeziness, we can see why 12-equal was not in general use during much of the common practice period, and that other systems held sway instead. One plan for dealing with the problem of the thirds is a circulating, or well-temperament. This makes the thirds which are most likely to be used a little bit better, at the expense of other thirds, which are made a little bit worse. The circulating temperments used on this website employ a new and bolder plan: the thirds are made a lot better, and instead of trying to make up the difference with out-of-tune versions of 5/4 or 6/5, we use instead in-tune versions of xenharmonic thirds: the supermajor third 9/7, the subminor third 7/6, along with 14/11 and 13/11. The resulting chords sound strange but not repulsive. Temperaments adopting this plan are [grail], [bifrost], and [cauldron]. Another approach leading to similar results is to well-tune a scale in just intonation, where a well-tuning is a regular tuning, meaning using fifths and thirds of the same size, whicb is chosen with an eye to tempering the scale to a circulating temperament. An example of this approach is [duowell].

It is also possible to simply use a single meantone fifth for twelve notes, and bite the bullet of the resulting very sharp "wolf" fifth. One way to do this is to use a mild version of meantone, such as 1/6-comma. A way I think works better is to take a wolf fifth near or at 20/13. Systems adopting this approach are [ratwolf] and wilwolf.

todo: grail

http://tonalsoft.com/gws/grail.html

Bifrost

This has six meantone fifths, leading to three pure thirds; on each side of the chain of meantones we put two pure 3/2's, so we have four pure fifths in total; we then round out with two sharp fifths of size sqrt(2048/2025 sqrt(5)), of size 706.355 cents. This leads to three sharp major thirds which are within a cent of being pure 14/11's, being of size 416.619 cents; and two more sharp thirds of size 406.843 at E and Ab, which are the only real problems with this temperament.

! bifrost.scl

! [45/64*5^(1/4), 1/2*5^(1/2), 16/45*5^(3/4), 5/4, 2/5*5^(3/4), 15/16*5^(1/4), 5^(1/4),

! 1/2*10^(1/2), 1/2*5^(3/4), 8/15*5^(3/4), 5/4*5^(1/4), 2]

!

Six meantone fifths, four pure fifths, two on each side, two fifths of sqrt(2048/2025 sqrt(5))

12

!

86.802144

193.156856

299.511569

386.313714

503.421572

584.847143

696.578428

793.156857

889.735285

1001.466571

1082.892142

1200.000000

todo: cauldron

http://tonalsoft.com/gws/cauldron.html

todo: duowell

http://tonalsoft.com/gws/duowell.htm

todo: ratwolf

http://tonalsoft.com/gws/ratwolf.html

Linear Temperaments

https://web.archive.org/web/20121025200628/http://tonalsoft.com/gws/linear.htm

A rank two temperament is a [regular temperament] with two generators. If it is possible for one of the generators to be an octave, such a temperament is called a linear temperament (in the strict sense.) However, rank two temperaments in general are often called linear. The most common choice for generators is for one generator to be an octave, or some nth part of an octave for some integer n; in this case this generator is called the period and the other the generator.

A rank two temperament may be uniquely defined in various ways; one is by means of the [wedgie], another by way of the [comma sequence], and still another by means of the mapping, or [icon], for a reduced set of generators. Here we are calling a pair of generators reduced if one generator is the period, and the other is the unique generator greater than one and less than the square root of the period (or less than half the period in logarithmic terms) which together with the period gives a generator pair for the temperament. This last definition depends on the exact tuning, and hence in theory may not be uniquely determined; in practice this seldom matters but for this and other reasons when working with temperaments using computer programs the wedgie is preferable as a means of defining the temperament. The icon, or tuning map, for the reduced generator pair, listing the period first and the generator second, we may call the standard icon.

[Here is a list of seven-limit linear temperaments.]

Listed below are some of the important families of linear temperaments.

[Meantone]

[Miracle]

[Orwell]

todo:

regular temperament

http://tonalsoft.com/gws/regular.html

wedgie

http://tonalsoft.com/gws/wedgie.html

comma sequence

http://tonalsoft.com/gws/commaseq.htm

list of seven-limit linear temperaments

http://tonalsoft.com/gws/sevnames.htm

meantone

http://tonalsoft.com/gws/meantone.htm

miracle

http://tonalsoft.com/gws/miracle.htm

orwell

http://tonalsoft.com/gws/orwell.html

Planar Temperaments

http://tonalsoft.com/gws/planar.htm

A rank three temperament is a [regular temperament] with three generators. If one of the generators can be an octave, it is called a planar temperament, though the word is sometimes applied to any rank three temperament. The octave classes of notes of a planar temperament can be embedded in a plane as a lattice, hence the name.

The most elegant way to put a Euclidean metric, and hence a lattice structure, on the pitch classes of a planar temperament is to start from a lattice of pitch classes for just intonation, and orthogonally project onto the subspace perpendicular to the space determined by the commas of the temperament. For instance, 7-limit just intonation has a [symmetrical lattice structure] and a 7-limit planar temperament is defined by a single comma. If u = |* a b c> is the octave class of this comma, then v = u/||u|| is the corresponding unit vector. Then if g1 and g2 are the two generators, j1 = g1 - (g1.v)v and j2 = g2 - (g2.v)v will be projected versions of the generators, and (j1.j1)a^2 + 2(j1.j2)ab + (j2.j2)b^2 the projected quadratic form defining the metric stucture of the planar temperament lattice. Here the dot product is defined by the [bilinear form] giving the metric structure. One good, and canonical, choice for generators are the generators found by using [Hermite reduction] with the proviso that if the generators so obtained are less than 1, we take their reciprocal.

The Hermite generators for marvel (225/224 planar) temperament are {2,3,5}, and the projected lattice strucuture is defined by the norm sqrt(11a^2-14ab+11b^2), where "a" is the exponent of 3 and "b" of five. The Hermite generators for breed (2401/2400 planar) temperament are {2,49/40,10/7}, and the projected lattice structure has 49/40 perpendicular to 10/7, so it has orthogonal axes, with a norm given by sqrt(11a^2+8b^2), where now "a" is the exponent of 49/40, and "b" the exponent of 10/7.

todo:

symmetrical lattice structure/7-limit lattice

http://tonalsoft.com/gws/sevlat.htm

Symmetric Bilinear Form

http://mathworld.wolfram.com/SymmetricBilinearForm.html

Hermite Normal Form

http://mathworld.wolfram.com/HermiteNormalForm.html

Crystal Balls

http://tonalsoft.com/gws/crystal.htm

We may define the nth q-limit Hahn shell as the octave classes at exactly Hahn distance n from the unison in terms of the q-odd-limit [Hahn norm]. The number of notes in the 5-limit Hahn shell is (for n>0) 6n, and in the 7-limit Hahn shell n has 10n^2+2 notes. If we take the union of the Hahn shells up to shell n we obtain the q-limit crystal ball; the reason behind that name is that the number of notes in the 7-limit crystal balls are called crystal ball numbers or magic numbers in some chemical and crystallographic contexts. The number of notes in the nth 5-limit crystal ball is 3n^2 + 3n + 1 and in the nth 7-limit crystal ball is (2n + 1)(5n^2 + 5n + 3)/3. Because of the way they are formed crystal balls are not especially regular as scales, but they are abundently supplied with chords.

Here are the first few 5-limit Hahn shells:

Shell 0

[1]

Shell 1 -- the 5-limit consonances

[6/5, 5/4, 4/3, 3/2, 8/5, 5/3]

Shell 2

[25/24, 16/15, 10/9, 9/8, 32/25, 25/18, 36/25, 25/16, 16/9, 9/5, 15/8, 48/25]

Shell 3

[128/125, 27/25, 144/125, 125/108, 75/64, 32/27, 125/96, 27/20, 45/32, 64/45,

40/27, 192/125, 27/16, 128/75, 216/125, 125/72, 50/27, 125/64]

Shell 4

[81/80, 648/625, 135/128, 625/576, 256/225, 625/512, 768/625, 100/81, 81/64,

162/125, 512/375, 864/625, 625/432, 375/256, 125/81, 128/81, 81/50, 625/384,

1024/625, 225/128, 1152/625, 256/135, 625/324, 160/81]

Here are the first three 7-limit Hahn shells:

Shell 0

[1]

Shell 1 -- the 7-limit consonances

[8/7, 7/6, 6/5, 5/4, 4/3, 7/5, 10/7, 3/2, 8/5, 5/3, 12/7, 7/4]

Shell 2

[50/49, 49/48, 36/35, 25/24, 21/20, 16/15, 15/14, 35/32, 10/9, 28/25,

9/8, 25/21, 60/49, 49/40, 32/25, 9/7, 64/49, 21/16, 49/36, 48/35,

25/18, 36/25,35/24, 72/49, 32/21, 49/32, 14/9, 25/16, 80/49, 49/30,

42/25, 16/9, 25/14, 9/5, 64/35, 28/15, 15/8, 40/21, 48/25, 35/18,

96/49, 49/25]

Here are the first two 7-limit crystal ball scales:

Crystal ball 1 13 notes -- the 7-limit Tonality Diamond

[1, 8/7, 7/6, 6/5, 5/4, 4/3, 7/5, 10/7, 3/2, 8/5, 5/3, 12/7, 7/4]

Crystal ball 2 55 notes

[1, 50/49, 49/48, 36/35, 25/24, 21/20, 16/15, 15/14, 35/32, 10/9,

28/25,9/8, 8/7, 7/6, 25/21, 6/5, 60/49, 49/40, 5/4, 32/25, 9/7, 64/49,

21/16, 4/3, 49/36, 48/35, 25/18, 7/5, 10/7, 36/25, 35/24, 72/49, 3/2,

32/21, 49/32, 14/9, 25/16, 8/5, 80/49, 49/30, 5/3, 42/25, 12/7, 7/4,

16/9, 25/14, 9/5, 64/35, 28/15, 15/8, 40/21, 48/25, 35/18, 96/49, 49/25]

Crystal ball one can also be described as Cube[2], the 2x2x2 [cube scale], which consists of the notes of the eight chords [i, j, k] with i, j, and k either -1 or 0. Crystal ball two consists of Cube[4], the 4x4x4 cube with i, j, and k from -2 to 1, minus the eight chords [-2 -2 1], [-2 1 -2], [-2 1 1], [1 -2 -2], [1 -2 1], [-2 -2 -2], [1 1 -2], [1 1 1].

The first two crystal balls can also equally well be described as Euclidean ball scales; they began to diverge with the third crystal ball. If we take everything within a radius of one of the unison, we get crystal ball one; if we take everything within a radius of two, we get crystal ball two. This means we also have two intermediate scales, Euclidean balls of radius sqrt(2) and sqrt(3).

Euclid 2 19 notes

[1, 21/20, 15/14, 8/7, 7/6, 6/5, 5/4, 4/3, 48/35, 7/5,

10/7, 35/24, 3/2, 8/5, 5/3, 12/7, 7/4, 28/15, 40/21]

Euclid 3 43 notes

[1, 49/48, 36/35, 25/24, 21/20, 16/15, 15/14, 35/32, 10/9, 28/25, 8/7,

7/6, 25/21, 6/5, 60/49, 49/40, 5/4, 9/7, 21/16, 4/3, 48/35, 7/5, 10/7, 35/24,

3/2, 32/21, 14/9, 8/5, 80/49, 49/30, 5/3, 42/25, 12/7, 7/4, 25/14, 9/5, 64/35,

28/15, 15/8, 40/21, 48/25, 35/18, 96/49]

todo:

Hahn norm/Hahn Distance

http://tonalsoft.com/gws/hahn.htm

Chord cubes/cube scale

http://tonalsoft.com/gws/cube.htm

A cube scale is a 7-limit scale (we might also consider the 9 odd limit) whose notes are derived from a cube in the [7-limit lattice] of chords. For odd n, we will call the octave-reduced set of notes deriving from all chords of the form [i, j, k], (1-n)/2 <= i, j, k <= (n-1)/2, Cube[n]. If n is even, we will use Cube[n] to refer to the notes of [i, j, k] with 1-n/2 <= i, j, k < n/2. If n is odd, Cube[n] has (n+1)^3/2 notes to it; if n is even, its growth is more complicated but still approximately cubic. For odd n, the inversion of the scale gives another scale, centered around a minor rather than a major tetrad. Here are the first three cube scales:

Cube[2] -- the stellated hexany, 14 notes

[1, 21/20, 15/14, 35/32, 9/8, 5/4, 21/16, 35/24, 3/2, 49/32, 25/16, 105/64, 7/4, 15/8]

Cube[3] 32 notes

[1, 49/48, 25/24, 21/20, 15/14, 35/32, 9/8, 8/7, 7/6, 6/5, 60/49, 49/40, 5/4, 9/7, 21/16, 4/3, 7/5, 10/7, 35/24, 3/2, 49/32, 25/16, 8/5, 105/64, 5/3, 42/25, 12/7, 7/4, 25/14, 9/5, 15/8, 35/18]

Cube[4] 63 notes

[1, 50/49, 49/48, 36/35, 25/24, 21/20, 16/15, 15/14, 35/32, 10/9, 28/25, 9/8, 8/7, 7/6, 25/21, 6/5, 128/105, 60/49, 49/40, 5/4, 32/25, 9/7, 35/27, 64/49, 21/16, 4/3, 168/125, 49/36, 48/35, 25/18, 480/343, 7/5, 10/7, 343/240, 36/25, 35/24, 72/49, 125/84, 3/2, 32/21, 49/32, 54/35, 14/9, 25/16, 8/5, 80/49, 49/30, 105/64, 5/3, 42/25, 12/7, 7/4, 16/9, 25/14, 9/5, 64/35, 28/15, 15/8, 40/21, 48/25, 35/18, 96/49, 49/25]

Dwarves

http://tonalsoft.com/gws/dwarf.htm

Dwarves

Suppose v is a val such that the gcd of the coefficients is 1, v(2)>0 and for every odd prime p, v(p)>=0. According to my original definition of val at any rate, vals are defined for all positive rational numbers, but for all but a finite number of primes they map the prime to 0. Bearing the precise definition in mind, we can define the dwarf scale for v, dwarf(v), as the reduction to the octave of the integers n of minimal Tenney height which form a complete set of residues v(n) mod v(2). The reason for the name "dwarf" is that height is as small as possible. The results are not extremely sensitive to this exact definition, as sorting by Hahn distance from unity and using Tenney height to break ties seems to usually lead to the same result.

Because v(2n) mod v(2) = v(n) mod v(2), the integers n will always be odd. Because if v(q) = 0 then v(qn) = v(n), the integers n will always be in the p-limit, where p is the largest prime for which v(p)>0. If r is any prime for which v(r)=0, then r does not appear in the factorization of the integers n, so this definition also covers subgroup situations, such as {2,3,7}-scales, so long as 2 is in the picture and we are using octave equivalence.

Because of the way they are constructed, dwarves are always permutation epimorphic and have a bias towards otonality over utonality. Here are some examples:

<7 11 16| 1, 9/8, 5/4, 45/32, 3/2, 27/16, 15/8

transposes to: 1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8

<7 11 16 20| 1, 9/8, 5/4, 21/16, 3/2, 27/16, 7/4

transposes to: 1, 9/8, 7/6, 4/3, 3/2, 5/3, 7/4

<10 16 23 28| 1, 35/32, 9/8, 5/4, 21/16 45/32, 3/2, 105/64, 7/4, 15/8

transposes to: 1, 16/15, 7/6, 6/5, 4/3, 7/5, 3/2, 8/5, 7/4, 15/8  

Marvelous dwarves

A marvelous dwarf is a scale with the following attributes:

(1) It is a marvel tempering of a 5-limit dwarf.

(2) It has the same number n of otonal tetrads, otonal pentads, utonal tetrads and utonal pentads. As a consequence of this it also has n subminor and n supermajor tetrads.

(3) It is covered by its pentads--that is, every note is harmonized by a pentad, and the scale is the union of its pentads.

(4) It has more 5-limit triads than pentads.

(5) It has no approximate tetrads deriving from anything but marvel; in the 5-limit scale which is tempered the smallest comma which produces approximate tetrads is 225/224.

If every condition but the third--the covering condition--is satisfied, I call it a semimarvelous dwarf. Why there are these scales exhibiting such regularity as a result of finding the 5-limit dwarf is an interesting question. Whatever the reason for it, the marvelous dwarves--of size 12, 15, 18, 19, 20, 21, and 25--seem like excellent scales for instrumentalists and composers interested in 9-limit harmony and scales in this size range. The 25-note scale, whose 5-limit preimage is the Euler genus of 15^4, is particularly striking from the point of view of the quantity of pentads it supplies.

There is a marvelous or semimarvelous dwarf for each scale size from 11 to 22, and then the 25 note scale. So far as I know this is the complete list.

Here is a brief description; the numbers are pentad number, numbers of major-minor triads, and size/pentad ratio.

<11 17 26| 1 6-6 11 semimarvelous

<12 19 28| 2 6-6 6 marvelous

<13 20 30| 1 7-6 13 semimarvelous

<14 22 33| 2 7-6 7 semimarvelous

<15 24 35| 3 8-8 5 marvelous

<16 25 37| 2 7-6 8 semimarvelous

<17 27 40| 4 10-9 4.25 semimarvelous

<18 29 42| 4 10-10 4.5 marvelous

<19 30 44| 5 12-11 3.8 marvelous

<20 32 47| 6 12-12 3.333 marvelous

<21 33 49| 5 12-12 4.2 marvelous

<22 35 51| 6 14-13 3.667 semimarvelous

<25 40 58| 9 16-16 2.778 marvelous

todo:

Composers

http://tonalsoft.com/gws/composers.html

Colleagues

http://tonalsoft.com/gws/coll.htm

Meantone Music

http://tonalsoft.com/gws/meanmus.htm

Modern Masters of Meantone

http://tonalsoft.com/gws/mmm.htm

Popular

http://tonalsoft.com/gws/pops.html

Mad Science Tuning

http://tonalsoft.com/gws/mad.html

Intervals and Vals

http://tonalsoft.com/gws/intval.html

The Wedge Product

http://tonalsoft.com/gws/wedge.html

The Brat

http://tonalsoft.com/gws/brat.html

Bosanquet Lattices

http://tonalsoft.com/gws/bosanquet.html

TOP and Tenney Space

http://tonalsoft.com/gws/top.htm

Kees space and Kees tuning

http://tonalsoft.com/gws/kees.htm

The Hermite Basis

http://tonalsoft.com/gws/hermbas.htm

Composing in Meantone

http://tonalsoft.com/gws/composing.htm

Names of seven-limit commas

http://tonalsoft.com/gws/commalist.htm