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I am a 23 year old man from regional New South Wales, Australia.
This is the user page of '''Budjarn Lambeth''', the wiki contributor and amateur music theorist.


I have a casual interest in xenharmonic/microtonal music. My favourite artists are Brendan Byrnes, Sevish and Greystar. I am nervous, timid, and care a lot about what other people think of me, but I want to be as helpful as I can be.
Not to be confused with ''[[lambeth chords]]'' nor ''[[1573/1568]] (the lambeth comma)''. Those are unrelated to me.


Frontiers have always fascinated me: The unexplored depths of the Earth's ancient history, lost to deep time. The unexplored far reaches of the solar system, the galaxy and the universe. And, the unexplored depths of the infinite pitch spectrum, an infinite expanse, hiding in plain sight between the keys of the piano!
== Personal information ==
* Location: New South Wales, Australia
* Born: 1999
* Pronouns: he/him/his or she/her/hers
* Contactable at: https://www.facebook.com/budjarnlambeth/ or on my [[User talk:BudjarnLambeth|talk page]]
<br><br>


I am prone to making careless mistakes, especially when calculating things with numbers, so apologies in advance for any of those. Please correct them if you see them.
== Music ==
My favourite tracks I have made :)
# [https://m.youtube.com/watch?v=scCuGXnj5IY 33edo track Feb 2024] (bright, lush, mellow and colourful synthwave)
# [https://www.youtube.com/watch?v=_NepcrTf9ng Recording in my "Rocket temperament"] (melancholic, wistful dungeon synth; loosely inspired by [[Greystar]])
# [https://www.youtube.com/watch?v=6N_8QM2UK5I Compressed 8ve 80edo track Aug 2025] (contemporary classical improv on synthesizer)
# [https://www.youtube.com/watch?v=tIwfO1lZWAc 21edo track May 2025] (dark dungeon synth; loosely inspired by Indonesian [[gamelan]] and Japanese music)
# [https://www.youtube.com/watch?v=iZI9x2Fto-A 60afdo track Dec 2024] (bright, spacious, tropical dungeon synth; loosely inspired by Greystar)


=== Music I've made ===
https://www.youtube.com/playlist?list=PLVHnByvMeRTohfmDPkc0EfOByWaX4xXAF


https://budjarnlambeth.bandcamp.com/
YouTube discography:  


=== Contact information ===
https://youtube.com/playlist?list=PLVHnByvMeRTohfmDPkc0EfOByWaX4xXAF
https://www.facebook.com/budjarnlambeth/


=== My favourite scales ===


==== Overtone scales ====
Bandcamp discography:


Dozenal Cliffedge
https://budjarnlambeth.bandcamp.com/
14/12
16/12
18/12
23/12
24/12


Dozenal Dungeon
15/12
16/12
18/12
19/12
24/12


Dozenal Freeway
''Most tracks on Bandcamp have not yet been uploaded to YouTube and vice versa.''
14/12
<br><br><br><br>
16/12
18/12
20/12
22/12
24/12


Dozenal Geode
==My favourite microtonal composers==
14/12
*'''[[Brendan Byrnes]]'''
16/12
*[[Greystar]]
18/12
*[[JUMBLE]]
22/12
*[[Benyamind]]
24/12
*[[Sevish]]
<br><br><br>


Spectral Mechanical
==Wiki contributions==
19/16
===Discoveries===
20/16
====Invented approaches to constructing tunings ====
24/16
These are the same category of thing as “equal temperaments”.
28/16
32/16


Spectral Mirage
''Procedural approaches'':
18/16
* [[Basal subgroup]]s
21/16
* [[Half-prime subgroup|Half-basis, third-basis, quarter-basis, etc. subgroups]]
24/16
* [[Equalizer subgroup]]s
25/16
* [[Sin(x) tuning#Generalizations|Function(x) tuning]]s
30/16
* [[Polytriadic]] tunings
32/16
* [[Substitute harmonic]]s
* [[Tritavesque]] intervals
* [[Sin(x) tuning#Generalizations|Waveform(x) tuning]]s
* [[Ed255/128|Equal divisions of 255/128]]
* [[Ed257/128|Equal divisions of 257/128]]


Spectral Moonbeam
''Sculptural approaches'':
18/16
* [[Breuddwyd scale]]s
19/16
* [[Breuddwyd scale|Sonhar tuning]]s
24/16
* [[Sooty fox scale]]s
30/16
* [[Breuddwyd scale|Wijzerplaat scale⁠]]s
32/16
* [[Gallery of arithmetic pitch sequences|APS#farab scales]]
* [[Gallery of arithmetic pitch sequences|APS#gene scales]]
* [[Gallery of arithmetic pitch sequences|APS#hekt scales]]
* [[Gallery of arithmetic pitch sequences|APS#jot scales]]
* [[Gallery of arithmetic pitch sequences|APS#méride scales]]
* [[Gallery of arithmetic pitch sequences|APS#tina scales]]


Spectral Springwater
====Invented approaches to classifying tunings ====
18/16
These are the same category of thing as “pentatonic scales”.
20/16
* [[the Riemann zeta function and tuning#Non-record edos|Anti-zeta EDOs]]
24/16
* [[the Riemann zeta function and tuning#Non-record edos|Indecisive EDOs]]
28/16
* [[the Riemann zeta function and tuning#Non-record edos|Local zeta EDOs]]
32/16
* [[the Riemann zeta function and tuning#''k''-ary peak EDOs|Parker EDOs]]
* [[Human EDO size categorization]] (HUECAT)
* [[Metascale]]s
* [[Runoff]] scales
* [[Structural beating]]


Spectral Volcanic
====Invented tunings (scales)====
17/16
These are the same category of thing as “the chromatic scale”.
20/16
* The [[Shoal17#Scales|algal bloom]] scale
24/16
* The [[25edo#Scales|amulet]] scale
28/16
* The [[anisopteran scale]]
32/16
* The [[antipental blues]] scale
* The [[14afdo#Scales|apex]] scale
* The [[5- to 10-tone scales from the modes of the harmonic series#Macrotonal NEJI edos|apple mint]] scale
* The [[Gallery of arithmetic pitch sequences#APS of tinas|Argentina]] scale
* The [[30edo#Scales|bathypelagic]] scale
* The [[breuddwyd inverse]] scale
* The [[breuddwyd6|breuddwyd-6]] scale
* The [[blackened skies]] scale
* The [[5- to 10-tone scales from the modes of the harmonic series#Macrotonal NEJI edos|bubblegum]] scale
* The [[5- to 10-tone scales from the modes of the harmonic series#Macrotonal NEJI edos|candycorn]] scale
* The [[30edo#Scales|carousel]] scale
* The [[9edf#Scales|cloudscape]] scale
* The [[5- to 10-tone scales from the modes of the harmonic series#Macrotonal NEJI edos|cola]] scale
* The [[9edf#Scales|corrugated]] scale
* The [[32afdo#Scales|cosmic]] scale
* The [[dalmatian scale]]
** The Dante, Da Vinci, Dawkins, Deepak, Deja Vu, Delgado, Dolly, Dylan, Fergus & Hansel [[101afdo#Scales|scales]]
* The [[30edo#Scales|dewdrops]] scale
* The [[Shoal17#Scales|diffractive scale]]
* The [[Shoal17#Scales|diffractive tetrad]]
* The [[27edo#Scales|direct sunlight]] scale
* The [[Shoal17#Scales|driftwood]] scale
* The [[Peppermint-24#Salt and pepper scale and its subsets|evacuated planet]] scale
* The [[firedance]] scale
* The [[5- to 10-tone scales in 60edo|fossa tetratonic]] scale
* The [[5- to 10-tone scales in 60edo|fossa pentatonic]] scale
* The [[56edo#Scales|frankincense]] scale
* The [[goldfish]] scale
* The [[27edo#Scales|hypersakura]] scale
* The [[Peppermint-24#Ketchup and mustard scale and its subsets|ketchup and mustard]] scale
* The [[5- to 10-tone scales from the modes of the harmonic series#Macrotonal NEJI edos|lemonade]] scale
* The [[5- to 10-tone scales from the modes of the harmonic series#Macrotonal NEJI edos|liquorice]] scale
* The [[9edf#Scales|livewire]] scale
* The [[lost spirit]] scale
* The [[magnetosphere scale]]
* The [[16afdo#Scales|mechanical]] scale
* The [[9edf#Scales|molten pelog]] scale
* The [[9edf#Scales|molten slendro]] scale
* The [[5- to 10-tone scales in 47zpi|monsoon]] scale
* The [[16afdo#Scales|moonbeam]] scale
* The [[30edo#Scales|nightdrive]] scale
* The [[oclock]] scale
* The [[User:BudjarnLambeth/Over the Hedge|Over the Hedge]] scale
** The H.ammy, H.eather, O.zzie, R.J, S.tella, T.iger & V.erne [[User:BudjarnLambeth/Over the Hedge|scales]]
* The [[29edo#Scales|palace]] scale
* The [[30edo#Scales|pelagic]] scale
* The [[Peppermint-24#Salt_and_pepper_scale_and_its_subsets|pepperbass]] scale
* The [[9edf#Scales|purgatory]] scale
* The [[9edf#Scales|pylon]] scale
* The [[47edo#Scales|quasi-equiheptatonic minor pentatonic]] scale
* The [[9edf#Scales|quest]] scale
* The [[Shoal17#Scales|refractive scale]]
* The [[Shoal17#Scales|refractive tetrad]]
* The [[5- to 10-tone scales in 47zpi|rockpool]] scale
* The [[47edo#Scales|sabertooth hexatonic]] scale
* The [[47edo#Scales|sabertooth neutral]] scale
* The [[47edo#Scales|sabertooth pentatonic]] scale
* The [[Peppermint-24#Salt and pepper scale and its subsets|salt and pepper]] scale
* The [[Shoal17#Scales|shoal17 over-9 tetrad]]
* The [[breuddwyd6|snow braid]] scale
* The [[9edf#Scales|snowcone]] scale
* The [[5- to 10-tone scales from the modes of the harmonic series#Macrotonal NEJI edos|sourgummy]] scale
* The [[5- to 10-tone scales from the modes of the harmonic series#Macrotonal NEJI edos|spearmint]] scale
* The [[spectral blues]] scale
* The [[9edf#Scales|swan]] scale
* The [[5- to 10-tone scales from the modes of the harmonic series#Macrotonal NEJI edos|trailmix]] scale
* The [[Batch 89 temperaments#Example inflected MOS scale: UFO scale|UFO scale]]
* The [[breuddwyd inverse|vinyl braid]] scale
* The [[16afdo#Scales|volcanic]] scale
* The [[5- to 10-tone scales from the modes of the harmonic series#Macrotonal NEJI edos|whitechocolate]] scale
* The [[26ed5#13ed5plus|13ed5plus]] scale
* The [[80edo#Scales|16-tone 5&16edo scale]]
* The [[80edo#Scales|12-tone 5&16edo scale]]
* The [[80edo#Scales|14-tone 8&10edo scale]]
* The [[80edo#Scales|12-tone 8&10edo scale]]
* The [[80edo#Scales|14-tone 8&20edo scale]]
* The [[80edo#Scales|12-tone 8&20edo scale]]
* [[1ed148.5c|APS148.5¢]]
* [[1ed237.8c|APS237.8¢]]
* [[Gallery of arithmetic pitch sequences#APS of farabs|APS3.21farab]]
* [[Gallery of arithmetic pitch sequences#APS of hekts|APS7hekt]]
* [[1ed28.5c|APS715jot]]
* [[1ed28.7c|APS720jot]]
* [[Gallery of arithmetic pitch sequences#APS of mérides|APS4/5méride]]
* [[Gallery of arithmetic pitch sequences#APS of mérides|APS5/8méride]]
* [[Gallery of arithmetic pitch sequences#APS of mérides|APS7/12méride]]
* [[Gallery of arithmetic pitch sequences#APS of tinas|APS70tina]]


Subharm Starship
====Invented tunings (regular temperaments)====
42/40
These are the same category of thing as “quarter-comma meantone temperament”.
42/34
* [[7921/7776|Attic]] temperament
42/32
* [[979/972|Basement]] temperament
42/28
* [[Subgroup temperaments#Blacktetra|Blacktetra]] temperament
42/24
* [[1000/979|Canine]] temperament
42/21
* [[Substitute harmonic#Daggerminished|Daggerminished]] temperament
* [[Substitute harmonic#Narrowed compton|Dotcom]] temperament
* [[Vavoom family|Doubloon]] temperaments
* [[Meantone family#Dreamtone|Dreamtone]] temperament
* [[Equalizer subgroup|Ectophotia]] temperament
* [[Equalizer subgroup|Fifigeist]] temperament
* [[Batch 89 temperaments#Floor|Floor]] temperament
* [[No-twos subgroup temperaments#Fomalhaut|Fomalhaut]] temperament
* [[Subgroup temperaments#Fourwar|Fourwar]] temperament
* [[Substitute harmonic#Narrowed compton|Glycome]] temperament
* [[No-sevens subgroup temperaments#Hypnotone|Hypnotone]] temperament
* [[8000/7921|Incisor]] temperament
* [[Substitute harmonic#Narrowed compton|Income]] temperament
* [[Substitute harmonic#Narrowed compton|Intercom]] temperament
* [[Vavoom family|Kaboom]] temperament
* [[Equalizer subgroup|Lieerie]] temperament
* [[Lightyear comma|Lightyear]] temperament
* [[2883/2848|Lilac]] temperament
* [[8277/8192|Lilly pilly]] temperament
* [[7936/7921|Lily]] temperament
* [[No-fives subgroup temperaments#Lipsett|Lipsett]] temperament
* [[Batch 89 temperaments#Lylla|Lylla]] temperament
* [[Substitute harmonic#Narrowed compton|Minicom]] temperament
* [[Molar comma|Molar]] temperament
* [[Substitute harmonic#Narrowed compton|Newcome]] temperament
* [[Nimona]] temperament
* [[Substitute harmonic#Narrowed compton|Outcome]] temperament
* [[729/712|Oxidizer]] temperament
* [[Substitute harmonic#Pajaraxe|Pajaraxe]] temperament
* [[Equalizer subgroup|Phantadecimal]] temperament
* [[Equalizer subgroup|Phantasrutal]] temperament
* [[Equalizer subgroup|Poltercompton]] temperament
* [[Batch 89 temperaments#Rocket|Rocket]] temperament
* [[Substitute harmonic#Narrowed compton|Romcom]] temperament
* [[Substitute harmonic#Sahara|Sahara]] temperament
* [[Substitute harmonic#Narrowed compton|Satcom]] temperament
* [[Equalizer subgroup|Semihaunt]] temperament
* [[Equalizer subgroup|Shadecimal]] temperament
* [[Equalizer subgroup|Shrutarnatural]] temperament
* [[Substitute harmonic#Narrowed compton|Sitcom]] temperament
* [[Subgroup temperaments#Sixscared|Sixscared]] temperament
* [[Equalizer subgroup|Spectrephore]] temperament
* [[89/88|Tailwind]] temperament
* [[Batch 89 temperaments#Teefs|Teefs]] temperament
* [[Substitute harmonic#Narrowed compton|Telecom]] temperament
* [[Subgroup temperaments#Tetramagic|Tetramagic]] temperament
* [[Vavoom family#Vacuum|Vacuum]] temperament
* [[Vavoom family#Void|Void]] temperament
* [[Vavoom family#Vortex|Vortex]] temperament
* [[Equalizer subgroup|Werecatler]] temperament
<br><br>


===Improvements===
====Significant page creations====
* Authored the [[survey of efficient temperaments by subgroup]] to help people navigate quickly to their favourite temperament pages
* Authored the [[Gallery of arithmetic pitch sequences]] & [[Gallery of combination product sets]] pages to help people find musical [[scale]]s
* Authored the pages about [[equipentatonic]] and [[equiheptatonic]] scales
* Created the [[Xenharmonic Wiki:Cross-platform dialogue]] page to aid collaboration between editors who are mostly active on Facebook and those who are mostly active on Discord
* Authored pages answering the [[FAQ]] questions:
** “[[What are this wiki's strengths and weaknesses?]] (How does it compare to other resources?)”
** “[[I'm a new editor, what is everything I can do to help?]]”
** “[[Systematic comma names explained|How do I decode those strange looking comma names?]]”
* Authored dozens of wanted pages about [[:Category:Composers|composers]]
* Authored several extensive [[:Category:Lists of scales|lists of scales]]


====Bulk page improvements====
* Constructed the 5-, 6-, 7-, 8-, 9-, 10-, 11- and 12-period tables on the [[map of rank-2 temperaments]]
* Heavily added to the 2-, 3- and 4-period tables on the [[map of rank-2 temperaments]]
* Added dozens of sources and inline citations to the [[historical temperaments]] page
* Added [[Template:Temp gallery]] to all '[[User:BudjarnLambeth/Prominent temperament gallery pages|very highly prominent temperament gallery pages]]'
* Added [[Template:Temp gallery]] to all '[[User:BudjarnLambeth/Prominent temperament gallery pages|highly prominent temperament gallery pages]]'
* Added [[Template:Temp gallery]] to all '[[User:BudjarnLambeth/Prominent temperament gallery pages|moderately prominent temperament gallery pages]]'
* Added [[Template:Temp gallery]] to all '[[User:BudjarnLambeth/Prominent temperament gallery pages|less prominent temperament]] families and clans'
* Merged “useful tools” & “software” into [[Software|one unified page]]
* Added infobox, intro, intervals, modes, subheadings & todo categories to dozens of previously close-to-empty [[MOS scale]] pages
* Added intro, intervals table and harmonics table to several hundred [[equal-step tuning]] pages (which used to just be blank pages with a single infobox)


====Structural wiki improvements====
* Added categories & formatting tags to several hundred [[:Category:Pages with Scala files|pages with Scala files]]
* Added links around the wiki to over 150 previously [[Special:LonelyPages|orphaned pages]], bringing the number down from above 250 to below 70 (as of 4 Dec 2024)
* Created [[Template:Navbox Lumatone]] with very extensive help from [[User:ArrowHead294]]
** With ArrowHead294’s extensive help, incorporated the template into over 50 pages for [[:Category:Lumatone mappings]]
* Ran a survey of editors, created the [[Xenharmonic Wiki:Optional guidelines for interval page todo categories]] (XW:IntTodo) from the results, and ensured all interval pages were categorized in line with XW:IntTodo


15 EDO
===== • List and table creations=====
Constructed:
* [[List of taxicab-2 intervals]]
* [[User:BudjarnLambeth/Table of n-comma meantone generators]]
* [[Harmonic entropy of just intervals]]
* [[Table of zeta-stretched edos]]


Atoll
===== • Comma documentation and organisation=====
79.716
* Added all missing commas from “[[:Category:Large commas]]” to the table on the page “[[Large commas]]”
398.578
* Added all missing commas from “[[:Category:Medium commas]]” to the table on the page “[[Medium commas]]”
478.294
* Added all missing commas from “[[:Category:Small commas]]” to the table on the page “[[Small comma]]”
717.441
* Added all missing commas from “[[:Category:Unnoticeable commas]]” to the table on the page “[[Unnoticeable commas]]”
956.588
* Constructed “[[:Category:Commas by name]]” and its many subcategories
1195.735
* Categorised all 5, 7, 11, 13, 17, 19, 23 and >23 limit large commas into ’Commas by name’ subcategories
* Categorised all 5, 7, 11, 13, 17, 19, 23 and >23 limit medium commas into ’Commas by name’ subcategories
* Categorised all 5, 7, 11, 13, 17, 19, 23 and >23 limit small commas into ’Commas by name’ subcategories
* Categorised all 5, 7, 11, 13, 17, 19, 23 and >23 limit unnoticeable commas into ’Commas by name’ subcategories
* Conducted the [[User:BudjarnLambeth/Survey of named and unnamed commas]]


Augmented
====Name changes I was involved in====
318.863
* There was an old temperament that pre-dated my participation in the wiki. Its old, original name was “septimal temperament”. When I discovered the name, I disliked it because of its potential for confusion with other uses of the term “septimal”. I decided I wanted to change it. Many people on [[Discord]], the Wiki Workgroup on [[XA Facebook]], and the wiki talk page agreed. I coined the new name “'''[[7th-octave temperaments#Austinpowers|austinpowers temperament]]'''” which has now replaced it.
398.578
* There was an old temperament that pre-dated my participation in the wiki. Its old, original name was “catcall temperament”. When I discovered the name, I disliked it because it was demeaning to women. I decided I wanted to change it. Many people on [[Discord]] and on the wiki talk page agreed. I coined the new name “'''[[catnip|catnip temperament]]'''” which has now replaced it.
717.441
* There was an old temperament that pre-dated my participation in the wiki. Its old, original name was “decades temperament”. Some people on the XA Discord said that they disliked it due to its convoluted etymology. I coined the new name “'''[[gamelstearn|gamelstearn temperament]]'''” based on its two defining commas. After a few days, not many people commented or reacted to the proposal, but all were in favour and none were against, so the name "gamelstearn" was adopted.
797.157
* There was an old comma that pre-dated my participation in the wiki. Its old, original name was “the Buddhisma”. There was nothing wrong with that name in and of itself, the problem is that it was explicitly stated to be named after the sectarian group the 969 Movement, and the comma number was 969/968. I thought this was inappropriate, so I proposed a few replacement names on [[Discord]]. Of the names I coined “'''[[969/968|the kingfisher comma]]'''” was the clear favourite, so I renamed the comma to the kingfisher comma.
1116.020
* There used to be two temperaments named “semidim temperament”. To resolve the confusion, after discussion on the XA Discord, myself and [[Lériendil]] came up with new names for each of the two temperaments. Lériendil coined “[[ammon|ammon temperament]]” for the first, and I coined “'''[[octonion|octonion temperament]]'''” for the second.
1195.735
* There was an old comma that pre-dated my participation in the wiki. Its old, original name was “the erogluisma”. Its original namer posted on [[Discord]] that he no longer liked it, and would like someone to please rename it. I coined the new name “'''[[1682/1681|the shaftesburisma]]'''” which has now replaced it.
* Someone else described a new comma and named it “the shikanokonokonokokoshitantanma”. This name was kept, but it was too long for some use cases, so I invented an alternate short version - “'''[[nokotan|the nokotan]]'''” - for use cases where a shorter name is needed.
* The comma “68630377364883 / 68630356164608” was too long to be referred to by its number alone, it needed a name instead. So I named it “'''[[zudilisma|the zudilisma]]'''”.
<br><br>


Equipentatonic
=== Full list of contributions ===
239.147
* [[Special:Contributions/BudjarnLambeth]]
478.294
<br><br>
717.441
956.5881
1195.735


Firedance
=== (To-do) ===
239.147
Things I [[Wikifuture|plan to do]] on the wiki in the near future. Feel free to go ahead and do these any of these things first if you’re interested in doing so. I feel overwhelmed by the sheer amount of things I want to do and am very happy for any help I receive getting through it all :)
558.010
717.441
956.588
1195.735


Monsoon
==== Structural wiki improvements ====
239.147
* Make “Undertone scales” page that exists in mutually supportive conversation with “[[IFDO]]” in a similar way how “[[Overtone scale]]” does with “[[AFDO]]”
478.294
* Add “[[:Category:Temperaments]]” to all temperament [[special:ListRedirects|redirects]]
717.441
1036.304
1195.735


Rainbow
==== Construct diagrams or lists ====
398.578
* For every temperament on the page [[Survey of efficient temperaments by subgroup]], generate a [[Scala]] file of whichever of its MOS scales is closest to (2 x its [[graham complexity]]). List all the Scala files in a wiki page along the lines of “Scala files for efficient subgroup temperaments” or something like that and link it from the Survey page
558.010
* Make all the wanted tables listed in [[:Category:Todo:add rank 2 temperaments table]]
717.441
* Make manual, curated tables of [[interval]]s for:<small>
876.872
** [[8ed5/4]] to 9ed5/4
1036.304
** [[4ed4/3]] to 8ed4/3
1195.735
** [[4ed5/3]] to 6ed5/3
** [[11ed5/3]]
** [[5ed7/4]] to 12ed7/4
** [[5ed9/5]] to 7ed9/5
** [[9ed11/5]]
** [[11ed9/4]]
** [[5ed7/3]]
** [[8ed7/3]]
** [[5ed5/2]] to 6ed5/2
** [[8ed5/2]]
** [[12edt]]
** [[3ed4]] to 6ed4
** [[8ed4]] to 15ed4
** [[3ed5]] to 5ed5
** [[4ed6]]</small>
* Make all the wanted mappings listed in [[:Category:Todo:add lumatone mapping]]
* Construct a table of the sharpness or flatness of every prime harmonic in every edo up to 53.
** Each edo will be a row, and the columns will be from -60%, -55%, -50%… -5%, 0, +5%… +50%, +55%, +60%. And I will list each prime in whichever bin it is closest too (so if prime 7/1 had +23.9% error in Nedo, it goes in column +25% in that edo’s row).
** Make a separate absolute error table in cents for EDOs 1 to 30, and relative error table for all EDOs 1 to 53 with the possibility of expanding both to higher edos later.
* Construct a version of an “[[odd limit]]” but for each “[[equalizer subgroup]]” and document them on the equalizer subgroup page
* Construct a [[tonality diamond]] or something like one for each “[[equalizer subgroup]]” and document them on the equalizer subgroup page


Rockpool
==== Improve pages I made ====
398.578
* On my page [[5- to 10-tone scales in 60edo]], change the scales to be in the more compact format that is used on edo pages instead of a ‘Scala-like’ format
478.294
* On my page [[5- to 10-tone scales in 72edo]], change the scales to be in the more compact format that is used on edo pages instead of a ‘Scala-like’ format
717.441
* On my page [[5- to 10-tone scales in 84edo]], change the scales to be in the more compact format that is used on edo pages instead of a ‘Scala-like’ format
956.588
* On my page [[5- to 10-tone scales in 91edo]], change the scales to be in the more compact format that is used on edo pages instead of a ‘Scala-like’ format
1195.735
* On my page [[5- to 10-tone scales in 96edo]], change the scales to be in the more compact format that is used on edo pages instead of a ‘Scala-like’ format


==== 21edo ====
==== Explore, discover, document ====
* Find nice 5 to 7 tone subsets of [[Shaka10]], [[Starling12]], [[Thrush12]] and document them under a “subsets” heading on those scales’ pages
* Find more unique scales in [[Catnip]][36] in [[60edo]] that can’t be replicated in [[Compton]][36] and document them on [[5- to 10-tone scales in 60edo]]
* Find 4 to 7 tone subsets for every 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 or 19-tone scale page


Pelog Lima
==== Approximate x in y ====
114.28571428571429
* Approximate the following scales in modes 34 to 101 of the harmonic series and document the results in “[[5- to 10-tone scales from the modes of the harmonic series]]”:
285.7142857142857
<pre>
685.7142857142858
Tobago6
800.
101.53846
350.76923
600.
701.53846
950.76923
1200.
1200.


==== 22edo ====
Zeus1
157.57576
387.87879
545.45455
703.03030
1042.42424
2/1


Manohari
Oceanfront5
272.72727272727275
227.82
490.90909090909093
486.09
709.0909090909091
713.91
927.2727272727273
972.18
981.8181818181819
1200.
1200.


Shailaja
Oceanfront Ionian Pent.
272.72727272727275
455.64
709.0909090909091
486.09
763.6363636363636
713.91
981.8181818181819
1169.55
1200.
1200.


Subminor Hexatonic
Oceanfront Minor Harmonic Pent.
218.1818181818182
258.27
272.72727272727275
486.09
490.90909090909093
713.91
709.0909090909091
1169.55
981.8181818181819
1200.
1200.


Subminor Pentatonic
Oceanfront Mixolydian Pent.
272.72727272727275
455.64
490.90909090909093
486.09
709.0909090909091
713.91
981.8181818181819
972.18
1200.
1200.


==== 43edo ====
Oceanfront Adrift Pent.
197.37
258.27
683.46
713.91
1200.
</pre>


Blues Aeolian Hexatonic
* Approximate the following scales in modes 102 to 200 of the harmonic series and document the results in “[[5- to 10-tone scales from the modes of the harmonic series]]”:
306.9767441860465
<pre>
502.3255813953488
5edo
613.953488372093
1\5
697.6744186046511
2\5
809.3023255813954
3\5
4\5
5\5
 
6edo
1\6
2\6
3\6
4\6
5\6
6\6
 
7edo
1\7
2\7
3\7
4\7
5\7
6\7
 
Slendric6
233.68421
467.36842
701.05263
934.73684
966.31579
1200.
1200.
 
Blues Aeolian Pentatonic I
Tobago6
306.9767441860465
101.53846
502.3255813953488
350.76923
697.6744186046511
600.
809.3023255813954
701.53846
950.76923
1200.
1200.
 
Blues Aeolian Pentatonic II
Zeus1
306.9767441860465
157.57576
697.6744186046511
387.87879
809.3023255813954
545.45455
1004.6511627906976
703.03030
1042.42424
1200.
1200.
 
Blues Bright Double Harmonic
Oceanfront5
111.62790697674419
227.82
390.69767441860466
486.09
502.3255813953488
713.91
697.6744186046511
972.18
809.3023255813954
1004.6511627906976
1088.3720930232557
1200.
1200.
 
Blues Dark Double Harmonic
Oceanfront Ionian Pent.
195.34883720930233
455.64
306.9767441860465
486.09
502.3255813953488
713.91
613.953488372093
1169.55
697.6744186046511
809.3023255813954
1088.3720930232557
1200.
1200.
 
Blues Dorian Hexatonic
Oceanfront Minor Harmonic Pent.
306.9767441860465
258.27
502.3255813953488
486.09
697.6744186046511
713.91
893.0232558139535
1169.55
1004.6511627906976
1200.
1200.
 
Blues Dorian Pentatonic
Oceanfront Mixolydian Pent.
306.9767441860465
455.64
697.6744186046511
486.09
893.0232558139535
713.91
1004.6511627906976
972.18
1200.
1200.
 
Blues Dorian Septatonic
Oceanfront Adrift Pent.
306.9767441860465
197.37
502.3255813953488
258.27
613.953488372093
683.46
697.6744186046511
713.91
893.0232558139535
1004.6511627906976
1200.
Blues Harmonic Hexatonic
195.34883720930233
306.9767441860465
502.3255813953488
697.6744186046511
1088.3720930232557
1200.
Blues Harmonic Septatonic
306.9767441860465
502.3255813953488
613.953488372093
697.6744186046511
809.3023255813954
1088.3720930232557
1200.
Blues Leading
306.9767441860465
502.3255813953488
613.953488372093
697.6744186046511
1004.6511627906976
1088.3720930232557
1200.
Blues Minor
306.9767441860465
502.3255813953488
613.953488372093
697.6744186046511
1004.6511627906976
1200.
Blues Minor Maj7
306.9767441860465
502.3255813953488
613.953488372093
697.6744186046511
1088.3720930232557
1200.
Blues Pentachordal
195.34883720930233
306.9767441860465
502.3255813953488
613.953488372093
697.6744186046511
1200.
Dominant Pentatonic
195.34883720930233
390.69767441860466
697.6744186046511
1004.6511627906976
1200.
Dorian
195.34883720930233
306.9767441860465
502.3255813953488
697.6744186046511
893.0232558139535
1004.6511627906976
1200.
Double Harmonic
111.62790697674419
390.69767441860466
502.3255813953488
697.6744186046511
809.3023255813954
1088.3720930232557
1200.
Hirajoshi
195.34883720930233
306.9767441860465
697.6744186046511
809.3023255813954
1200.
Ionian Pentatonic
390.69767441860466
502.3255813953488
697.6744186046511
1088.3720930232557
1200.
Javanese Pentachordal
111.62790697674419
306.9767441860465
613.953488372093
697.6744186046511
1200.
Kokin-Joshi
195.34883720930233
306.9767441860465
697.6744186046511
893.0232558139535
1200.
Locrian
111.62790697674419
306.9767441860465
502.3255813953488
613.953488372093
809.3023255813954
1004.6511627906976
1200.
Lydian
195.34883720930233
390.69767441860466
613.953488372093
697.6744186046511
893.0232558139535
1088.3720930232557
1200.
Major
195.34883720930233
390.69767441860466
502.3255813953488
697.6744186046511
893.0232558139535
1088.3720930232557
1200.
Major Pentatonic
195.34883720930233
390.69767441860466
697.6744186046511
893.0232558139535
1200.
Minor
195.34883720930233
306.9767441860465
502.3255813953488
697.6744186046511
809.3023255813954
1004.6511627906976
1200.
Minor Harmonic
195.34883720930233
306.9767441860465
502.3255813953488
697.6744186046511
809.3023255813954
1088.3720930232557
1200.
Minor Harmonic Pentatonic
195.34883720930233
306.9767441860465
697.6744186046511
1088.3720930232557
1200.
Minor Hexatonic
195.34883720930233
306.9767441860465
502.3255813953488
697.6744186046511
1004.6511627906976
1200.
Minor Melodic
195.34883720930233
306.9767441860465
502.3255813953488
697.6744186046511
893.0232558139535
1088.3720930232557
1200.
Minor Pentatonic
306.9767441860465
502.3255813953488
697.6744186046511
1004.6511627906976
1200.
Mixolydian
195.34883720930233
390.69767441860466
502.3255813953488
697.6744186046511
893.0232558139535
1004.6511627906976
1200.
Mixolydian Harmonic
390.69767441860466
502.3255813953488
697.6744186046511
809.3023255813954
1004.6511627906976
1200.
Mixolydian Pentatonic
390.69767441860466
502.3255813953488
697.6744186046511
1004.6511627906976
1200.
Phrygian
111.62790697674419
306.9767441860465
502.3255813953488
697.6744186046511
809.3023255813954
1004.6511627906976
1200.
Phrygian Dominant
111.62790697674419
390.69767441860466
502.3255813953488
697.6744186046511
809.3023255813954
1004.6511627906976
1200.
Phrygian Dominant Hexatonic
111.62790697674419
390.69767441860466
502.3255813953488
697.6744186046511
1004.6511627906976
1200.
Phrygian Dominant Pentatonic
390.69767441860466
502.3255813953488
697.6744186046511
809.3023255813954
1200.
Phrygian Pentatonic
111.62790697674419
306.9767441860465
697.6744186046511
809.3023255813954
1200.
Picardy Hexatonic
195.34883720930233
390.69767441860466
502.3255813953488
697.6744186046511
809.3023255813954
1200.
Picardy Pentatonic
195.34883720930233
390.69767441860466
697.6744186046511
809.3023255813954
1200.
1200.
</pre>
* Approximate the following scales in BOTH prime modes 75-100 of the harmonic series AND sqfr semiprm modes 75-100 and document the results in “[[5- to 10-tone scales from the modes of the harmonic series]]”:
<pre>
Antipental Blues
7/6
4/3
11/8
3/2
7/4
2/1
Evened Minor Hexatonic
187.500
312.500
500.000
700.000
987.500
1200.000
Evened Phrygian Dominant Hexatonic
112.500
387.500
500.000
712.500
987.500
1200.000
Flattened Blues Aeolian Pentatonic
287.500
487.500
687.500
787.500
1200.000
Flattened Blues Minor Maj7
287.500
500.000
600.000
700.000
1100.000
1200.000
Roughened Blues Dorian Pentatonic
300.000
700.000
912.500
987.500
1200.000
Sharpened Blues Aeolian Pentatonic
312.500
712.500
800.000
1000.000
1200.000
Sharpened Blues Dorian Hexatonic
312.500
512.500
712.500
900.000
1012.500
1200.000
Sharpened Blues Harmonic Septatonic
312.500
512.500
612.500
712.500
812.500
1112.500
1200.000
Sharpened Dorian Harmonic
212.500
312.500
612.500
712.500
912.500
1012.500
1200.000
Sharpened Hirajoshi
200.000
312.500
700.000
812.500
1200.000
Sharpened Kokin-Joshi
200.000
312.500
700.000
900.000
1200.000
Sharpened Minor Pentatonic
312.500
500.000
700.000
1012.500
1200.000
</pre>


==== 53edo ====
==== Xenharmonify 12edo scales ====
Approximate [[12edo]] subsets with these tunings, test them by noodling, and list all good-sounding ones in a big page:
* [[August]] in EDOs: ................. {{EDOs|21, 33, 45, 58, 69}}
* [[Augene]] in EDOs: ................. {{EDOs|15, 27, 39, 66}}
* [[Bidia]] in EDOs: ................. {{EDOs|56, 68, 80}}
* [[Diaschismic]] in EDOs: ................. {{EDOs|46, 58, 104}}
* [[Dimcomp]] in EDOs: ................. {{EDOs|44, 48, 56, 60, 68, 72}}
* [[Diminished]] in EDOs: ................. {{EDOs|20, 28, 32, 44}}
* [[Grackle]] in EDOs: ................. {{EDOs|53, 65, 77, 89}}
* [[Helenus]]/[[Photia]] in EDOs: ................. {{EDOs|41, 53, 65, 118}}
* [[Hexe]] in EDOs: ................. {{EDOs|30, 42}}
* [[Injera]] in EDOs: ................. {{EDOs|14, 26, 38, 40, 90, 102}}
* [[Misty]] in EDOs: ................. {{EDOs|39, 51, 63, 75, 87, 99}}
* [[Nickel]] in EDOs: ................. {{EDOs|16, 21, 28}}
* [[Pajara]] in EDOs: ................. {{EDOs|22, 34, 56}}
* [[Pajarous]] in EDOs: ................. {{EDOs|22, 32, 54, 76, 120}}
* [[Passion]] in EDOs: ................. {{EDOs|25, 37, 49, 61, 73, 110}}
* [[Ripple]] in EDOs: ................. {{EDOs|71, 83, 95, 107, 119}}
* [[Semimeantone]] in EDOs: ................. {{EDOs|38, 50, 62}}
* [[Srutal]]/[[Srutal archagall]] in EDOs: ................. {{EDOs|22, 34, 46, 80, 114, 126}}
* [[Wronecki]] in EDOs: ................. {{EDOs|66, 78, 90, 102, 114, 126}}
<br><br>


Dozenal Blues
==Random stuff I’d like to share==
271.6981132075472
498.1132075471698
611.3207547169811
701.8867924528302
1200.


Dozenal Cliffedge
===List of equal divisions including novelties===
271.6981132075472
* [[User:BudjarnLambeth/List of equal divisions including novelties]]
498.1132075471698
<br>
701.8867924528302
1132.0754716981132
1200.


Dozenal Dungeon
===Hot takes===
384.90566037735846
'''Everything in this section is opinion, not facts!'''
498.1132075471698
701.8867924528302
792.4528301886792
1200.


Dozenal Geode
'''Do not give any of it too much weight.'''
271.6981132075472
498.1132075471698
701.8867924528302
1041.509433962264
1200.


Spectral Arcade
* '''Silliness:''' We should take ourselves less seriously: we should have more fun, we should be more silly 🎉
384.90566037735846
* '''Arbitrary numbers:''' It’s fine to explore the 2.3.5.101 [[subgroup]] if you want to. It doesn’t matter that it’s less [[concordant]] than 2.3.5.7. Explore it anyway and see what happens. Do some stuff with [[:Category:Novelties|arbitrary numbers]] that don’t make logical sense and just see what comes of it. That’s where the fun is! (Though do keep a “2.3” or “3.5” in there just to smooth over the gnarliest edges, heehee.)
475.47169811320754
* '''Importance of common names:''' I would prefer to use a temperament [[Temperament naming|named]] something fun like “waterslide” or “jinglebells” even if it has lots of error, over one named something dry and bland like “countertrihexakleismatic” even if it’s super [[damage|accurate]] and [[badness|technically better]] - a bland name can kill a temperament’s appeal, a fun name can create appeal out of nothing.
701.8867924528302
* '''My opinion on zeta peak indexes:''' My view on to what degree [[zeta peak index]]es are useful or not.
973.5849056603773
1200.


Spectral Mechanical
::: [[User:BudjarnLambeth/My opinion on zeta peak indexes]]
294.33962264150944
* '''Misrepresentation of traditions:''' A cheeky little opinion piece I wrote criticising the negative and celebrating the positive ways Western tuning theory has borrowed from the world’s traditions:
384.90566037735846
701.8867924528302
973.5849056603773
1200.


Spectral Moonbeam
::: [[User:BudjarnLambeth/Cultural appropriation-o-meter]]
203.77358490566036
* '''Comma naming debate brainstorm:''' A futile attempt I made to try to begin resolving one of the big behind the scenes debates among wiki editors:
294.33962264150944
701.8867924528302
1086.7924528301887
1200.


Spectral Springwater
::: [[User:BudjarnLambeth/Brainstorming mostly bad ideas to solve the comma naming debate]]
203.77358490566036
<br><br>
384.90566037735846
701.8867924528302
973.5849056603773
1200.


Spectral Starship
===Approximations of gamelan scales in edos up to 60===
90.56603773584905
* [[User:BudjarnLambeth/Edo approximations of gamelan scales]]
384.90566037735846
<br><br>
475.47169811320754
701.8867924528302
973.5849056603773
1200.


Spectral Volcanic
===Eagle JI subsets===
113.2075471698113
* [[User:BudjarnLambeth/Eagle JI subsets]]
384.90566037735846
<br><br>
701.8867924528302
973.5849056603773
1200.


=== Families of scales and temperaments ===
===12edo interpreted as a 2.3.5.17.19 subgroup temperament===
This is a personal tool to help me navigate the wiki.          
* [[User:BudjarnLambeth/12edo as a 2.3.5.17.19 tuning]]
<br><br>


==== Just intonation scales ====
===Tunings for each subgroup===
* [[Just intonation|Just intonation scales]]
* [[User:BudjarnLambeth/Tunings for each subgroup]]
**[[Subharmonic series|Aliquot scales (modes of the subharmonic series)]] / [[EDL|EDLs]]
<br><br>
**[[Chord cubes]]
**[[Crystal balls]]
**[[Highschool scales]]
**[[Homothetic just intonation|Homothetic JI scales]]
**[[Tritriadic scale|Tritriadic scales]]
**[[Tonality diamond|Tonality diamonds]]
***[[Breed diamonds]]
***[[Diaconv scales]]
**[[Combination product sets]]
***[[Dekany|Dekanies]]
***[[Eikosany|Eikosanies]]
***[[Hexany|Hexanies]]
***[[Pentadekany|Pentadekanies]]
***[[Pentany|Pentanies]]
**[[Overtone scale|Overtone scales (modes of the harmonic series)]] / [[AFDO|AFDOs]]
***Modes of the [[oddharmonic series]]
***Modes of the [[prime harmonic series]]
***[[Primodality|Primodal scales]]
***[[Neji|NEJI scales]]
***Modes of the [[triangulharmonic series]]


==== Tempered scales ====
===Gallery of xenharmonic word painting (you can contribute!)===
*[[Temperament|Temperaments]]
* [[User:BudjarnLambeth/Gallery of xenharmonic word painting]]
**[[Clippers]]
<br><br>
**[[Elves]]
**[[List of essentially tempered scales|Essentially tempered scales]]
**[[Lesfip scales]]
**[[Well temperament|Well temperaments]]
***[[Kartvelian scales]]
**[[Dwarves]]
***[[Especially tempered dwarves]]
***[[Marvelous dwarves]]
**[[Hobbits]]
***[[MOS scale|Moment of symmetry scales]]
****[[MOS Cradle|MOS cradles]]
**[[Regular temperament|Regular temperaments]]
***[[Rank-4 temperament|Rank-4 temperaments]]
****[[Keenanismic temperaments]]
****[[Werckismic temperaments]]
****[[Swetismic temperaments]]
****[[Lehmerismic temperaments]]
****[[Kalismic temperaments]]
*** [[Rank-3 temperament|Rank-3 temperaments]]
**** Families defined by a 2.3.5 comma
*****''see [[Tour of regular temperaments]]''
**** Families defined by a 2.3.7 comma
*****''see [[Tour of regular temperaments]]''
**** Families defined by a 2.3.5.7 comma
*****''see [[Tour of regular temperaments]]''
**** Temperament groups defined by an 11-limit comma
*****''see [[Tour of regular temperaments]]''
*** [[Rank-2 temperament|Rank-2 temperaments]] ([[Map of rank-2 temperaments|map]])
****[[Map of linear temperaments|Linear temperaments]]
****[[Linear temperament|Moment of symmetry scales]]
*****[[MOS Cradle|MOS cradles]]
****Families defined by a 2.3 comma
*****''see [[Tour of regular temperaments]]''
****Families defined by a 2.3.5 comma
*****''see [[Tour of regular temperaments]]''
****Clans defined by a 2.3.7 comma
*****''see [[Tour of regular temperaments]]''
****Clans defined by a 2.3.11 comma
*****''see [[Tour of regular temperaments]]''
****Clans defined by a 2.3.13 comma
*****''see [[Tour of regular temperaments]]''
****Clans defined by a 2.5.7 comma
*****''see [[Tour of regular temperaments]]''
****Clans defined by a 3.5.7 comma
*****''see [[Tour of regular temperaments]]''
****Temperament groups defined by a 2.3.5.7 comma
*****''see [[Tour of regular temperaments]]''
*** [[Equal-step tuning|Rank-1 temperaments / Equal-step tunings]]
****Equal multiplications
*****[[Ambitonal sequence|Ambitonal sequences (AS) / Equal multiplications of a rational interval]]
*****[[Arithmetic pitch sequence|Arithmetic pitch sequences (APS) / Equal multiplications of an irrational interval]]
****Equal divisions
*****[[EDO|Equal divisions of the octave (EDOs or ED2s)]]
*****[[The Riemann zeta function and tuning|Zeta-stretched equal divisions]]
*****[[Edonoi|Edonois]]
******[[EDT|Equal divisions of the third harmonic (EDTs or ED3s)]]
******[[Ed4|Equal divisions of the fourth harmonic (ED4s)]]
******[[Ed5|Equal divisions of the fifth harmonic (ED5s)]]
******[[Ed6|Equal divisions of the sixth harmonic (ED6s)]]
******[[Ed7|Equal divisions of the seventh harmonic (ED7s)]]
******[[Ed8|Equal divisions of the eighth harmonic (ED8s)]]
******[[Ed9|Equal divisions of the ninth harmonic (ED9s)]]
******[[Ed10|Equal divisions of the tenth harmonic (ED10s)]]
******[[Ed11|Equal divisions of the eleventh harmonic (ED11s)]]
******[[:Category:Edonoi|...]]
******[[Ed3/2|Equal divisions of the perfect fifth (EDFs or ED3/2s)]]
******[[Ed4/3|Equal divisions of the perfect fourth (ED4/3s)]]
******[[Ed5/3|Equal divisions of the major sixth (ED5/3s)]]
******[[Ed5/4|Equal divisions of the major third (ED5/4s)]]
******[[:Category:Edonoi|...]]
******[[Ed5/2|Equal divisions of 5/2 (EDXs or ED5/2s)]]
******[[Ed7/2|Equal divisions of 7/2 (ED7/2s)]]
******[[:Category:Edonoi|...]]
******[[Ed7/3|Equal divisions of 7/3 (ED7/3s)]]
******[[Ed8/3|Equal divisions of 8/3 (ED8/3s)]]
******[[Ed10/3|Equal divisions of 10/3 (ED10/3s)]]
******[[:Category:Edonoi|...]]
******[[Ed9/4|Equal divisions of 9/4 (EDIXs or ED9/4s)]]
******[[Ed11/5|Equal divisions of 11/5 (ED11/5s)]]
******[[Edφ|Equal divisions of Phi (Edφs)]]
******[[CET|''n''-cent equal tunings (''n''-cETs)]]


==== Fokker blocks ====
===What I look for in a good xenharmonic piece===
The tuning sounds consonant, all the combinations of intervals you used sound within the same range of consonant-ness as all the combinations available in 12edo. There are more dissonant and less dissonant sections, but none that sound more dissonant than the maximum limit of dissonance I'm willing to tolerate (ie, the maximum amount of dissonance achievable with 12edo). All that is a long winded way of saying nothing sounds "out of tune".


* [[Fokker block|Fokker blocks]]
The timbre also sounds good. Despite being midi instruments, the samples used and whatever effects were applied to them (reverb, volume levels, etc.) all come together well into something that is more than the sum of its parts. The timbre works well with the tuning to create a kind of pleasant, warm glow.
**[[Linear temperament|Moment of symmetry scales]]
***[[MOS Cradle|MOS cradles]]
**[[Gallery of wakalixes|Superwakalixes]]
**[[Gallery of wakalixes|Wakalixes]]
***r-wakalixes
****Triple wakalixes
****Quadruple wakalixes
****Pentuple wakalixes
****...


==== Metascales ====
The piece has enough dynamic range, sense of harmonic movement, and interaction between melody lines to hold my interest, to not sound bland, but instead sound captivating.
* [[Metascales]]
<br><br><br>
**[[Compound scale|Compound scales]]
** [[Scale products and scale powers]]
** [[MOS Cradle|MOS cradles]]
** [[Muddle|MOS muddles]]
** [[Polymicrotonality|Polysystemic scales / Polymicrotonal scales]]


==== Periodic scales ====
===Ratios as planets===
* [[Periodic scale|Periodic scales]]
The ratios of the octave as celestial bodies of the solar system:
**[[Generator-offset property|Generator-offset scales]]
* '''[[2/1]], the sun''', defining the overall structure of the entire octave.
**[[Maximal harmony epimorphic scales]]
* '''[[3/2]], Jupiter''', the biggest pull aside from the sun itself. Marks out a rough middle of the solar system.
**[[SN scale|Step-nested scales]]
* '''[[4/3]], Saturn''', a kind of echo of Jupiter. Vast and mighty in its own right, similar to Jupiter, but colder and subtler.
***[[Linear temperament|Moment of symmetry scales]]
* '''[[5/3]], Uranus''', has a strong enough pull to be one of the big guys, but markedly smaller than Jupiter and Saturn. Has a gentle appearance.
****[[MOS Cradle|MOS cradles]]
* '''[[5/4]], Neptune''', the last of the giant planets, with their huge gravitational pull. Has an energetic, lively surface with the fastest winds in the solar system. Defines the boundary of the Kuiper Belt (like the boundary of major and minor tonality).
**[[Harmonotonic tuning|Harmonotonic / step-monotonic tunings]]
* '''[[6/5]], Earth''', a small planet that punches above its weight. It manages to hold an unreasonably large moon, given the small gravitational influence it has. It even developed life and sent probes to explore the other planets. It has far more influence than it seems like it should.
***[[Xenharmonic series]]
* '''[[7/4]], Venus''', a kind of parallel version of Earth. Its diameter and gravitational pull are almost identical to Earth, but its character is completely different. Home to a completely different temperature, chemical composition and volcanic system.
****[[Ambitonal sequence|Ambitonal sequences (AS) / Equal multiplications of a rational interval]]
* '''[[7/5]], Mars''', appears barren and frozen upon first glance, but looking closer reveals a rich and storied history, and potential wonders and microcosms hidden in its underground caves.
****[[Subharmonic series|Aliquot scales (modes of the subharmonic series)]] / [[EDL|EDLs]]
* '''[[8/5]], Mercury'''. Caught in a region of the solar system so wild that much of its mass has been stripped away by the solar wind while it was still forming. Still, it holds firm, a foothold in difficult territory.
****[[Arithmetic pitch sequence|Arithmetic pitch sequences (APS) / Equal multiplications of an irrational interval]]
* '''[[7/6]], the moon'''. Caught in Earth’s orbit, it’s difficult to view separately from the Earth’s influence. If the Earth wasn’t there, it’d be a whole planet in its own right.
****[[Powharmonic series]]
* '''[[9/5]], Ceres'''. It doesn’t pull much weight on its own, but in the context of the wider solar system, it demarcates an important in-between point for the other planets. If you want to travel between the different planets, Ceres makes the journey much more pleasant.
****Modes of the [[prime harmonic series]]
* '''[[11/8]], Pluto''', often dismissed outright due to having such a tiny gravitational pull. But it is a whole other type of ‘planet’, the first of a whole new family. When approached on its own terms, it opens up a larger, more diverse solar system than we ever knew existed.
****[[Logharmonic series]]
<br><br>
****Modes of the [[metallic harmonic series]]
****Modes of the [[oddharmonic series]]
****[[Powharmonic series|Subpowharmonic series]]
****[[Logharmonic series|Sublogharmonic series]]
****Modes of the [[triangulharmonic series]]
***Non-arithmetic harmonotonic tunings
****[[Powharmonic series]]
****[[Logharmonic series]]
****[[Powharmonic series|Subpowharmonic series]]
****[[Logharmonic series|Sublogharmonic series]]
***[[Arithmetic tuning|Arithmetic tunings]]
****Equal length step tunings
*****[[Subharmonic series|Aliquot scales (modes of the subharmonic series)]] / [[EDL|EDLs]]
****Equal frequency step tunings
*****[[Overtone scale|Overtone scales (modes of the harmonic series)]] / [[AFDO|AFDOs]]
******Modes of the [[oddharmonic series]]
******Modes of the [[prime harmonic series]]
******[[Primodality|Primodal scales]]
******[[Neji|NEJI scales]]
******Modes of the [[triangulharmonic series]]
****Equal pitch step tunings
*****[[Equal-step tuning|Rank-1 temperaments / Equal-step tunings]]
******Equal multiplications
*******[[Ambitonal sequence|Ambitonal sequences (AS) / Equal multiplications of a rational interval]]
*******[[Arithmetic pitch sequence|Arithmetic pitch sequences (APS) / Equal multiplications of an irrational interval]]
******Equal divisions
*******[[EDO|Equal divisions of the octave (EDOs or ED2s)]]
*******[[The Riemann zeta function and tuning|Zeta-stretched equal divisions]]
*******[[Edonoi|Edonois]]
********[[EDT|Equal divisions of the third harmonic (EDTs or ED3s)]]
********[[Ed4|Equal divisions of the fourth harmonic (ED4s)]]
********[[Ed5|Equal divisions of the fifth harmonic (ED5s)]]
********[[Ed6|Equal divisions of the sixth harmonic (ED6s)]]
********[[Ed7|Equal divisions of the seventh harmonic (ED7s)]]
********[[Ed8|Equal divisions of the eighth harmonic (ED8s)]]
********[[Ed9|Equal divisions of the ninth harmonic (ED9s)]]
********[[Ed10|Equal divisions of the tenth harmonic (ED10s)]]
********[[Ed11|Equal divisions of the eleventh harmonic (ED11s)]]
********[[:Category:Edonoi|...]]
********[[Ed3/2|Equal divisions of the perfect fifth (EDFs or ED3/2s)]]
********[[Ed4/3|Equal divisions of the perfect fourth (ED4/3s)]]
********[[Ed5/3|Equal divisions of the major sixth (ED5/3s)]]
********[[Ed5/4|Equal divisions of the major third (ED5/4s)]]
********[[:Category:Edonoi|...]]
********[[Ed5/2|Equal divisions of 5/2 (EDXs or ED5/2s)]]
********[[Ed7/2|Equal divisions of 7/2 (ED7/2s)]]
********[[:Category:Edonoi|...]]
********[[Ed7/3|Equal divisions of 7/3 (ED7/3s)]]
********[[Ed8/3|Equal divisions of 8/3 (ED8/3s)]]
********[[Ed10/3|Equal divisions of 10/3 (ED10/3s)]]
********[[:Category:Edonoi|...]]
********[[Ed9/4|Equal divisions of 9/4 (EDIXs or ED9/4s)]]
********[[Ed11/5|Equal divisions of 11/5 (ED11/5s)]]
********[[Edφ|Equal divisions of Phi (Edφs)]]
********[[CET|Cent equal temperaments (cETs)]]


==== Miscellaneous scale structures ====
== Useful links for my own convenience ==
*[[Cartesian scale|Cartesian scales]]
<br>
*[[Euler-Fokker genus|Euler-Fokker genera]]
<big>[[User:BudjarnLambeth/Sandbox]]
*[[Omniconsonant scale|Omniconsonant scales]]
<br>
*[[Yantras]]
<br>
[[User:BudjarnLambeth/Sandbox2]]
<br>
<br>
[[Gallery of just intervals]]
<br>
<br>
[[Harmonic entropy of just intervals]]
<br>
<br>
[[User:Userminusone/Constructing a comma list using the Averiant]]
<br>
<br>
[[User:Ganaram inukshuk/Catalog of MOS Scales]]
</big>

Latest revision as of 09:06, 21 August 2025

This is the user page of Budjarn Lambeth, the wiki contributor and amateur music theorist.

Not to be confused with lambeth chords nor 1573/1568 (the lambeth comma). Those are unrelated to me.

Personal information



Music

My favourite tracks I have made :)

  1. 33edo track Feb 2024 (bright, lush, mellow and colourful synthwave)
  2. Recording in my "Rocket temperament" (melancholic, wistful dungeon synth; loosely inspired by Greystar)
  3. Compressed 8ve 80edo track Aug 2025 (contemporary classical improv on synthesizer)
  4. 21edo track May 2025 (dark dungeon synth; loosely inspired by Indonesian gamelan and Japanese music)
  5. 60afdo track Dec 2024 (bright, spacious, tropical dungeon synth; loosely inspired by Greystar)


YouTube discography:

https://youtube.com/playlist?list=PLVHnByvMeRTohfmDPkc0EfOByWaX4xXAF


Bandcamp discography:

https://budjarnlambeth.bandcamp.com/


Most tracks on Bandcamp have not yet been uploaded to YouTube and vice versa.



My favourite microtonal composers




Wiki contributions

Discoveries

Invented approaches to constructing tunings

These are the same category of thing as “equal temperaments”.

Procedural approaches:

Sculptural approaches:

Invented approaches to classifying tunings

These are the same category of thing as “pentatonic scales”.

Invented tunings (scales)

These are the same category of thing as “the chromatic scale”.

Invented tunings (regular temperaments)

These are the same category of thing as “quarter-comma meantone temperament”.



Improvements

Significant page creations

Bulk page improvements

Structural wiki improvements

• List and table creations

Constructed:

• Comma documentation and organisation

Name changes I was involved in

  • There was an old temperament that pre-dated my participation in the wiki. Its old, original name was “septimal temperament”. When I discovered the name, I disliked it because of its potential for confusion with other uses of the term “septimal”. I decided I wanted to change it. Many people on Discord, the Wiki Workgroup on XA Facebook, and the wiki talk page agreed. I coined the new name “austinpowers temperament” which has now replaced it.
  • There was an old temperament that pre-dated my participation in the wiki. Its old, original name was “catcall temperament”. When I discovered the name, I disliked it because it was demeaning to women. I decided I wanted to change it. Many people on Discord and on the wiki talk page agreed. I coined the new name “catnip temperament” which has now replaced it.
  • There was an old temperament that pre-dated my participation in the wiki. Its old, original name was “decades temperament”. Some people on the XA Discord said that they disliked it due to its convoluted etymology. I coined the new name “gamelstearn temperament” based on its two defining commas. After a few days, not many people commented or reacted to the proposal, but all were in favour and none were against, so the name "gamelstearn" was adopted.
  • There was an old comma that pre-dated my participation in the wiki. Its old, original name was “the Buddhisma”. There was nothing wrong with that name in and of itself, the problem is that it was explicitly stated to be named after the sectarian group the 969 Movement, and the comma number was 969/968. I thought this was inappropriate, so I proposed a few replacement names on Discord. Of the names I coined “the kingfisher comma” was the clear favourite, so I renamed the comma to the kingfisher comma.
  • There used to be two temperaments named “semidim temperament”. To resolve the confusion, after discussion on the XA Discord, myself and Lériendil came up with new names for each of the two temperaments. Lériendil coined “ammon temperament” for the first, and I coined “octonion temperament” for the second.
  • There was an old comma that pre-dated my participation in the wiki. Its old, original name was “the erogluisma”. Its original namer posted on Discord that he no longer liked it, and would like someone to please rename it. I coined the new name “the shaftesburisma” which has now replaced it.
  • Someone else described a new comma and named it “the shikanokonokonokokoshitantanma”. This name was kept, but it was too long for some use cases, so I invented an alternate short version - “the nokotan” - for use cases where a shorter name is needed.
  • The comma “68630377364883 / 68630356164608” was too long to be referred to by its number alone, it needed a name instead. So I named it “the zudilisma”.



Full list of contributions



(To-do)

Things I plan to do on the wiki in the near future. Feel free to go ahead and do these any of these things first if you’re interested in doing so. I feel overwhelmed by the sheer amount of things I want to do and am very happy for any help I receive getting through it all :)

Structural wiki improvements

Construct diagrams or lists

Improve pages I made

  • On my page 5- to 10-tone scales in 60edo, change the scales to be in the more compact format that is used on edo pages instead of a ‘Scala-like’ format
  • On my page 5- to 10-tone scales in 72edo, change the scales to be in the more compact format that is used on edo pages instead of a ‘Scala-like’ format
  • On my page 5- to 10-tone scales in 84edo, change the scales to be in the more compact format that is used on edo pages instead of a ‘Scala-like’ format
  • On my page 5- to 10-tone scales in 91edo, change the scales to be in the more compact format that is used on edo pages instead of a ‘Scala-like’ format
  • On my page 5- to 10-tone scales in 96edo, change the scales to be in the more compact format that is used on edo pages instead of a ‘Scala-like’ format

Explore, discover, document

  • Find nice 5 to 7 tone subsets of Shaka10, Starling12, Thrush12 and document them under a “subsets” heading on those scales’ pages
  • Find more unique scales in Catnip[36] in 60edo that can’t be replicated in Compton[36] and document them on 5- to 10-tone scales in 60edo
  • Find 4 to 7 tone subsets for every 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 or 19-tone scale page

Approximate x in y

Tobago6
101.53846
350.76923
600.
701.53846
950.76923
1200.

Zeus1
157.57576
387.87879
545.45455
703.03030
1042.42424
2/1

Oceanfront5
227.82
486.09
713.91
972.18
1200.

Oceanfront Ionian Pent.
455.64
486.09
713.91
1169.55
1200.

Oceanfront Minor Harmonic Pent.
258.27
486.09
713.91
1169.55
1200.

Oceanfront Mixolydian Pent.
455.64
486.09
713.91
972.18
1200.

Oceanfront Adrift Pent.
197.37
258.27
683.46
713.91
1200.
5edo
1\5
2\5
3\5
4\5
5\5

6edo
1\6
2\6
3\6
4\6
5\6
6\6

7edo
1\7
2\7
3\7
4\7
5\7
6\7

Slendric6
233.68421
467.36842
701.05263
934.73684
966.31579
1200.

Tobago6
101.53846
350.76923
600.
701.53846
950.76923
1200.

Zeus1
157.57576
387.87879
545.45455
703.03030
1042.42424
1200.

Oceanfront5
227.82
486.09
713.91
972.18
1200.

Oceanfront Ionian Pent.
455.64
486.09
713.91
1169.55
1200.

Oceanfront Minor Harmonic Pent.
258.27
486.09
713.91
1169.55
1200.

Oceanfront Mixolydian Pent.
455.64
486.09
713.91
972.18
1200.

Oceanfront Adrift Pent.
197.37
258.27
683.46
713.91
1200.
Antipental Blues
7/6
4/3
11/8
3/2
7/4
2/1

Evened Minor Hexatonic
187.500
312.500
500.000
700.000
987.500
1200.000

Evened Phrygian Dominant Hexatonic
112.500
387.500
500.000
712.500
987.500
1200.000

Flattened Blues Aeolian Pentatonic
287.500
487.500
687.500
787.500
1200.000

Flattened Blues Minor Maj7
287.500
500.000
600.000
700.000
1100.000
1200.000

Roughened Blues Dorian Pentatonic
300.000
700.000
912.500
987.500
1200.000

Sharpened Blues Aeolian Pentatonic
312.500
712.500
800.000
1000.000
1200.000

Sharpened Blues Dorian Hexatonic
312.500
512.500
712.500
900.000
1012.500
1200.000

Sharpened Blues Harmonic Septatonic
312.500
512.500
612.500
712.500
812.500
1112.500
1200.000

Sharpened Dorian Harmonic
212.500
312.500
612.500
712.500
912.500
1012.500
1200.000

Sharpened Hirajoshi
200.000
312.500
700.000
812.500
1200.000

Sharpened Kokin-Joshi
200.000
312.500
700.000
900.000
1200.000

Sharpened Minor Pentatonic
312.500
500.000
700.000
1012.500
1200.000

Xenharmonify 12edo scales

Approximate 12edo subsets with these tunings, test them by noodling, and list all good-sounding ones in a big page:



Random stuff I’d like to share

List of equal divisions including novelties


Hot takes

Everything in this section is opinion, not facts!

Do not give any of it too much weight.

  • Silliness: We should take ourselves less seriously: we should have more fun, we should be more silly 🎉
  • Arbitrary numbers: It’s fine to explore the 2.3.5.101 subgroup if you want to. It doesn’t matter that it’s less concordant than 2.3.5.7. Explore it anyway and see what happens. Do some stuff with arbitrary numbers that don’t make logical sense and just see what comes of it. That’s where the fun is! (Though do keep a “2.3” or “3.5” in there just to smooth over the gnarliest edges, heehee.)
  • Importance of common names: I would prefer to use a temperament named something fun like “waterslide” or “jinglebells” even if it has lots of error, over one named something dry and bland like “countertrihexakleismatic” even if it’s super accurate and technically better - a bland name can kill a temperament’s appeal, a fun name can create appeal out of nothing.
  • My opinion on zeta peak indexes: My view on to what degree zeta peak indexes are useful or not.
User:BudjarnLambeth/My opinion on zeta peak indexes
  • Misrepresentation of traditions: A cheeky little opinion piece I wrote criticising the negative and celebrating the positive ways Western tuning theory has borrowed from the world’s traditions:
User:BudjarnLambeth/Cultural appropriation-o-meter
  • Comma naming debate brainstorm: A futile attempt I made to try to begin resolving one of the big behind the scenes debates among wiki editors:
User:BudjarnLambeth/Brainstorming mostly bad ideas to solve the comma naming debate



Approximations of gamelan scales in edos up to 60



Eagle JI subsets



12edo interpreted as a 2.3.5.17.19 subgroup temperament



Tunings for each subgroup



Gallery of xenharmonic word painting (you can contribute!)



What I look for in a good xenharmonic piece

The tuning sounds consonant, all the combinations of intervals you used sound within the same range of consonant-ness as all the combinations available in 12edo. There are more dissonant and less dissonant sections, but none that sound more dissonant than the maximum limit of dissonance I'm willing to tolerate (ie, the maximum amount of dissonance achievable with 12edo). All that is a long winded way of saying nothing sounds "out of tune".

The timbre also sounds good. Despite being midi instruments, the samples used and whatever effects were applied to them (reverb, volume levels, etc.) all come together well into something that is more than the sum of its parts. The timbre works well with the tuning to create a kind of pleasant, warm glow.

The piece has enough dynamic range, sense of harmonic movement, and interaction between melody lines to hold my interest, to not sound bland, but instead sound captivating.


Ratios as planets

The ratios of the octave as celestial bodies of the solar system:

  • 2/1, the sun, defining the overall structure of the entire octave.
  • 3/2, Jupiter, the biggest pull aside from the sun itself. Marks out a rough middle of the solar system.
  • 4/3, Saturn, a kind of echo of Jupiter. Vast and mighty in its own right, similar to Jupiter, but colder and subtler.
  • 5/3, Uranus, has a strong enough pull to be one of the big guys, but markedly smaller than Jupiter and Saturn. Has a gentle appearance.
  • 5/4, Neptune, the last of the giant planets, with their huge gravitational pull. Has an energetic, lively surface with the fastest winds in the solar system. Defines the boundary of the Kuiper Belt (like the boundary of major and minor tonality).
  • 6/5, Earth, a small planet that punches above its weight. It manages to hold an unreasonably large moon, given the small gravitational influence it has. It even developed life and sent probes to explore the other planets. It has far more influence than it seems like it should.
  • 7/4, Venus, a kind of parallel version of Earth. Its diameter and gravitational pull are almost identical to Earth, but its character is completely different. Home to a completely different temperature, chemical composition and volcanic system.
  • 7/5, Mars, appears barren and frozen upon first glance, but looking closer reveals a rich and storied history, and potential wonders and microcosms hidden in its underground caves.
  • 8/5, Mercury. Caught in a region of the solar system so wild that much of its mass has been stripped away by the solar wind while it was still forming. Still, it holds firm, a foothold in difficult territory.
  • 7/6, the moon. Caught in Earth’s orbit, it’s difficult to view separately from the Earth’s influence. If the Earth wasn’t there, it’d be a whole planet in its own right.
  • 9/5, Ceres. It doesn’t pull much weight on its own, but in the context of the wider solar system, it demarcates an important in-between point for the other planets. If you want to travel between the different planets, Ceres makes the journey much more pleasant.
  • 11/8, Pluto, often dismissed outright due to having such a tiny gravitational pull. But it is a whole other type of ‘planet’, the first of a whole new family. When approached on its own terms, it opens up a larger, more diverse solar system than we ever knew existed.



Useful links for my own convenience


User:BudjarnLambeth/Sandbox

User:BudjarnLambeth/Sandbox2

Gallery of just intervals

Harmonic entropy of just intervals

User:Userminusone/Constructing a comma list using the Averiant

User:Ganaram inukshuk/Catalog of MOS Scales