31-EDO Music Theory: The Overtone Scale & Harmonic Approximation in 31edo
Вставка
- Опубліковано 29 тра 2024
- One of the most appealing features about 31edo is its ability to approximate the first 16 harmonics fairly well. While there is some error, given the small amount of 31 notes, it performs very well.
#musictheory #31edo #Xenharmonic
00:00 Intro
00:10 8-16 Overtone Scale
00:48 The Vibes
01:46 Basic Construction
02:48 EDO Step vs Overtone
04:56 Voicing the 14th and 15th Harmonic Clash
05:40 Associated Chord Symbols
06:56 Harmonic Approximation and Relative Error of Harmonics 8-16
10:13 Just Intonation vs 31EDO
11:35 Harmonics 16-32 Error Check
12:19 Useful Chord Voicings
15:25 Conclusion
![Zhea Erose - WXTCHCRXFT [37-tone Just Intonation] EP](http://i.ytimg.com/vi/a63V_fAPNaA/mqdefault.jpg)
Me thinking harmony 4 at school was hard... beautiful theory as always Z.
I remember learning about and learning to listen for overtones from my grandpa back in the mid 90's whose dad was a piano technician. He taught me about how the 5ths are flat from just in the 12tet scale and which harmonics to listen for to tune certain intervals to temper the scale. Of course, I didn't go so far as to actually be able to properly temper a 12edo scale on the piano. You have to learn to estimate the speed of the beats between the harmonics which takes a lot of training. But I was able to listen in and hear up to the 11th or 12th harmonic of any mid to low range note. And I immediately had that sense of...these sounds are beautiful, why can't we USE them?!
The lumatone is brilliant but I also think if the Haaken Continuum could be upgraded so that you could have a digital visual display that can be altered in virtually any way you wish would be ideal. You could set it to any temperament or what I'd love to do is have guidelines on top of the 12edo red/black strips that can be adjusted in real time so that you can move through endless pitch lattices throughout a completely extended JI piece.
i love this! i would do anything for a haken continuum that could update the display. the 12-edo black and red is ok but if you could alter the layout while playing that would be game changing.
that's a really awesome experience. indeed it is strange we left behind the 7th, 13th, and 11th harmonics in the western music tradition.
This is the video that got me into xenharmony. Hearing that subminor third changed everything for me, as I realized that microtones weren't all just about weird alien sounds. This stuff is truly beautiful, and it's what I was made to explore. It's a shame I didn't discover it earlier.
My head has been in Gregorian Modes 3&4 for so long that when I went off to try some modal variants of this scale, the mode based off of the Maj7 just sounded like home!
god I will never get tired of hearing harmonic sevenths
please make more of these theory videos btw, they're actually interesting to some of my musician friends who previously dismissed microtonality entirely!
scratches the brain real nice
How I wish these series would continue… 😬 These are gold! 🤩
My favorite types of videos are ones where I can tell the speaker is spitting facts, but I have little to no clue what they're talking about. It's so engaging. It encourages me to take a deeper look to gain a fuller understanding which is super fulfilling. Thanks for sharing.
Say it don’t spray it, I always say.
Totally loving your channel! So useful for me as I program a DAW to make odd music. Love your descriptions of the mood these chords make. Many thanks for the information.)
I regret discovering this UA-cam channel so late.
I just discovered that 31 notes is a thing through a fantastic piano performance of Sweet Lorraine. These videos are very helpful!
Mum did not tell me that when she taught me to play the piano, I learnt about Crotchets and clefts, mostly to entertain old church members, learning to play sweet And Low which left me in tears, so did Debussy’s Clare De Lune, but Row, Row, Row The Boat cheered me up really Amelia, ❤ from Petra Brown
really fascinating stuff, I could barely get my head around regular old music theory and this seems quite challenging
after a bit, you don't really notice the extra notes. think of all notes as just adding points of resolution to the octave spectrum. for most new tunings, first step is just listening to all the intervals and playing them for yourself.
i should've started with that on this series but that's not as fun and i know i wouldn't have kept up the series if i started with it lol. i'll do more basic vids on 31 soonish.
I really like what Zhea said about just playing them and hearing them for yourself, because with 12 edo theory, that's really the best way to learn it. It helps most to have a strong personal connection to the sounds to better understand new concepts, which you then incorporate by playing with those. Hear them and experiment with them! Play around freely and strictly and see what you can find. Sometimes you find patterns that didn't show up in the source you learned it from.
I haven't really gotten into xenharmony yet as a composer/ songwriter, but having a super strong 12 edo foundation is really helping me make sense of these 31 edo concepts.
I think what I appreciate most about your tutorials is the way you are able to provide timbrel descriptions of how these particular frequencies interact within this system. What are your thoughts on orchestration using some of these harmonic structures? The particular descriptions seem to hold true in the context of a sine wave but when you bring more chaotic overtone properties of diverse instrumentation (acoustic or synthesized) how does that impact the timbrel affect of micro tonality?
Wow! More overtone scale blues please ! I'm not a hardcore musician, but I like to just use this scale with a major 6th replacing the neutral 6th. I know it's scrubby or whatever, but 31edo's approximation of harmonic 13th is sooo weird. M6 is a lot more idiomatic to 31 edo, giving a chain of 5 fifths, and cool chords like a neutral II chord.
Agreed! The 13th harmonic approx is really fucked up in 31edo. It sounds like a hybrid between maj6 and n6. I don't really like it much. But it's useful to have for other scales.
i just try to absorb what i can, and your voice in and of itself is melodic to my tympanic membranes
P.S. your teaching of music is so different, a different approach to sound, ❤ you Amelia
To someone who is just starting the 31-EDO journey, this video is nothing less than perfection!
The diagrams clearly comparing harmonics, EDO steps, intervals, cents and their combinations has addressed so much of my frustration and confusion in regards to using my Lumatone for what it does best!
Thank you so much! 🙂🙏
6:38 Yes, definitely !!! 7th, 13th and particularly 11th harmonics are everywhere present in traditionnal fiddle music, e.g. in central France.
11th harmonics are in Scandinavian fiddle/folk music as well
For overtones 7 to 14, if the overtone number is n, the interval n/4 is a type of nth interval. For example, 11/4 is a type of 11th.
I think octave stretching, which is important for string instruments like the piano or the guitar, and with which 31edo is naturally compatible, accounts pretty well for the flat 3/2. As for 13/8, I agree, it's not really there, and in isolation it resembles 105/64 more. In the context of the harmonic series as a scale however the harmonic identity as a 13 shines strong for me. Thank you for another great video :)
awesome video as always, zhea! :D
your content is fantastic!!! Grateful for you!
Nice to have admission of the limitations of 31EDO in the video as well. But it would also be good to have a list of what alternative ratios it gives when it is off from the ones you want. For instance, this is what makes 12EDO work so well: Every note in it(*) falls close some ratio that has a largest prime number of 19 or smaller, so apart from the minor second, the intervals all sound good even when they are fairly off from the just intonation interval they nominally represent (like 7/4 getting substituted with 16/9). I haven't gone through this with 31EDO yet, but I did go through it with 17EDO, and most of the notes in it have the same property, which is why probably it still manages to sound decent even though it is quite far off from nominal just intonation in many cases. It is a little worse in this department, because some of the closest ratios have 23 as their largest prime number, and thus sound noticeably rough.
(*)Although I am suspecting that in some cases, falling on a perfect root and thus an irrational number has its own virtue, especially the tritone (= square root of 2) that gets so much use in modern jazz and to a significant extent also in Baroque music. This may also be true for the major and minor thirds that everybody in the microtonal community seems to love to hate, although those fall very close to rational numbers as well (but less so for the major third). So unjust intonation may have its place as well.
I don’t think that’s how harmony works. Prime limits are meaningless outside of the lattice and even within 31edo I don’t view it in a regular lattice. Instead, voicing specific combinations align with pure vertical structures. For example, 31edo contains an essentially perfect 13:17:19 or 31:53:58. Gorgeous /13 and /31 vertical map. Referencing the prime harmonics like you are doing positions them all /2 but most low complexity vertical structures arent /2 except the conventional ones. Lattices and regular views arent aligned with timbre. The divergent, irregular vertical concordance map is.
see: primodality
@@ZheannaErose I wouldn't swear that prime limits are the whole story, but judging from which intervals sound roughest, they seem to be a significant part of it.
That said, I seem to have dual optima in hearing preference, being attracted to both just intervals with simple ratios like 3/2 and to certain irrational numbers like square root of 2.
One of the best videos of Music Theory I have seen
Great Stuff👍🏻
Really excellent summary! “Well done,” and thanks!
I also like the fact that you’re exploring and demonstrating with a variety of timbres. Pitch ranges also make a big difference, especially for the more complex harmonies: In lower pitch ranges, even comparatively simple harmonies like a dominant 7th chord can sound a little muddled, although using the 7:4 sm7 (harmonic seventh) helps a lot.
I personally perceive that, with ratios involving integers larger than 25 or so, it _starts_ getting a bit iffy as to whether our ears attribute any really-distinctive meaning to those ratios. They become increasingly likely to be perceived as “off” variants of simpler ratios. For example, a 33:17 ratio, I think most listeners will perceive as an off-octave than a anything in its own right. That, since the octave is just sooo basic to our ears, and 33:17 is a much more complex harmony.
Rad. So glad I found this channel. Thanks for sharing and doing these.
You're the COOLEST. I loved your presentation!!
Thank you, you are a very point of inspiration!
72edo composer here, it's fascinating that a prime- numbered equal division of the octave can have such an in depth usage. What software/instruments do you use to write in this system?
I've found that prime numbered systems actually perform better than composite ones, with the exceptions of 12, 15(to a degree), 22, 27, 34 and 46
I think it's because of the fact that with primes you don't necessarily get any mediocre approximations like 1\3 for 5/4, 1\4 for 6/5, 3\5 and 4\7 for 3/2
72edo is a based tuning system
Great series.
Beautiful sounds and awesome theoretical teaching, as always! What software are you using to compare the overtones and their approximation in 31 as seen at 8:20? Is that a feature in tonescape?
sangat membantu ! Matur suksma
I started experimenting with different tet systems and found that 32, 16, 8, and 4 (and anything that follows the 2^n pattern) tet systems have notes that fit perfectly in a harmonic series. While mathematically, physically, and harmonically satisfying, it really takes some of the life away from the sound as lots notes glob into one uniform tone. With things like 31, having many mismatches gives it uniqueness and life. It sounds beautiful, not perfect, and that is perfect within it's self.
(Edit) 16 sounds beautiful and resonate for things with like 4 part harmony, but too many notes just push a single note out at you
Hi. You’re mistaken. 2, 4, 8, 16, 32edo are actually some of the least harmonically accurate edos which exist in respect to the harmonic series’s primary shape. There are few accurate points within that set and no where close to the accuracy of 31. 31edo is multiple times closer to harmonic approximation than 32.
I think I see the issue, I spaced my notes by hertz, not cents.
Beautiful
Is anyone able to recode the built in kontact dynamic just intonation to this scale? or others??
Back here for another listen, after having heard more examples of microtonal music, including 31EDO. While I don't disown what I said in my previous posts, I think I can now say that the bad taste I was left with for 31EDO stemmed from clunky progressions(*) in 12EDO music hastily converted to 31EDO, which through (I guess) bad luck were overrepresented in the first several samples of 31EDO music that I heard. Upon later hearing compositions either written from the ground up in 31EDO or converted from other-EDO by somebody who actually went to the trouble to avoid such conversion problems, it actually sounds pretty good overall.
(*)From memory, since I am not particularly inclined to go back through all the clunkers, I think much of this results from running afoul of 31EDO NOT tempering out both types of 5-limit diesis, which 12EDO tempers out. This extra tempering in 12 notes per octave (EDO or well temperament) removes harmonic opportunities, but it also acts as a pair of guard rails to keep you from shooting yourself in the foot with common progressions made by stacking thirds.
I still need to go through 31EDO and look at what just ratios it comes up with when it misses a nominal just ratio. (Yes, lists like this exist, but all the ones I have seen leave out some just ratios or even whole intervals, or have several just ratios scrambled together for a given interval with no easy way to see which one is closer to a given micro-step, or both.)
Edit: I think 31 notes per octave also has some wiggle room in it for experimentation with well tempered (instead of exactly equal) versions to try to get better approximations of just intonation in parts of the gamut, but I haven't heard much of anyone trying this.
Amazing, but who wrote the nomenclature for the sonortie names? Is there an institution that officially has declared the names for such sonorties or is it more of what the composers desire?
What programs are you using to play in these tunings? Also do you know if it’s possible to play microtones through Finale’s playback?
Thank you
Where did you get those glyphs for 53, 59, and 61? I’d like to start using them instead of bracketed accidentals
This is extremely interesting. Questions: You are using the unit of a cent, which I have always known as 1/1200 of an octave, per 12-EDO. Is the cent that you reference 1/3100 of an octave or 1/1200 of an octave? Also, what software are you using for demonstrating these intervals? I'd like to try that software, myself.
it’s 1/1200
If you're wondering why specifically 31 notes, it's because it does a better job at approximating the harmonic series than pretty much every other EDO of about the same size. Other EDOs that do well are 22, 41, 46, 53, and 72.
Or if you hate yourself, you can use 270edo. The entire 8-16 harmonic scale is approximated very well, even for its size.
thats not why i use it
@@ZheannaErose I think I meant to say "better job approximating the harmonic series" instead of just "better job". But I know you also like the way the notes beat together in 31 compared to other tunings
What song is at the end?
Sola, from my channel here: ua-cam.com/video/qW3VOVh4rlg/v-deo.html
2:40 Wouldn't the harmonic 7th above A be Gd instead of Gdb?
yea, good catch. thx. was writing fast and loose lol
@@ZheannaErose Thanks for clearing it up. I'm really enjoying this series so far by the way. Can't wait to see what future videos you'll be making.
2:54 Is the second harmonic *approximated* by the octave, or can we say the second harmonic *is* the octave?
the 2nd harmonic *is* the octave (2/1 ratio). in equally divided tunings (like 12edo and 31edo), they all have perfect octaves since thats being equally divided. EDOs are actually “ED2s” n equal divisions of a 2/1. :)
you can equally divide anything though so there are some equal tunings which divided the 3rd harmonic! these are called EDT or ED3 (equal division of the twelfth or 3/1). This creates a new “octave” called the tritave. :)
Bohlen-Pierce for example is a famous scale which is 13 equal divisions of the tritave. It doesnt support octaves. :)
Those errors sound massive with pure synth sounds, but when you sing or use acoustic instruments it gets smoothed down or naturally adjusted by ear. One day you or someone just as smart will figure out how to get the best of both worlds.
the acoustic ji version of the 8:16 though still sounds much clearer than the actual 31 version irl though. I doubt actual performing musicians with free pitch control will be playing real 31 though. I assume since its so close to JI in some areas most will adaptively adjust.
wanted to make sure i had a very obvious continuous sound quality for the a/b comparison. the fifths honestly sound pretty fuzzy in 31. its my only complaint -- that and the 13th harmonic is non existent really
those errors are absolutely massive to me. even when there is more noise in the tuning accuracy.
The tone of the midi that Scala uses isn't that great. I wonder if there's some way to route the player to a daw and actually play the midi through a soft synth. I've tried 11/8 on electric guitar with open strings, and I've preferred 11/4 an octave higher. I feel like these higher harmonics sound better in higher voicings, mixed with lower intervals like octaves and fifths. Somehow Jon Catler makes 11th and 13th harmonics sound good, but it's higher on the guitar neck: ua-cam.com/video/O2eukIoSsKM/v-deo.html
@@epiphoney there is. yea the timbre is harsh. i picked it because it has the most overtones and shows the JI lock rhythm more than another timbre there. I usually use patch 52 when I just wanna make it sound ok. But the harsher sound reveals the ji lock better.
it's easy to move all these intervals into ji for a different instrument but I was a bit lazy and had scala on hand
Can you have the first sound without any voice over
Is there a downloadable scl somewhere to use with vsts..?
here is 31equal and the 8-note subset exportable to whatever format you want (hit read more)
sevish.com/scaleworkshop/?name=31%20equal%20divisions%20of%202%2F1&data=38.709677%0A77.419355%0A116.129032%0A154.83871%0A193.548387%0A232.258065%0A270.967742%0A309.677419%0A348.387097%0A387.096774%0A425.806452%0A464.516129%0A503.225806%0A541.935484%0A580.645161%0A619.354839%0A658.064516%0A696.774194%0A735.483871%0A774.193548%0A812.903226%0A851.612903%0A890.322581%0A929.032258%0A967.741935%0A1006.451613%0A1045.16129%0A1083.870968%0A1122.580645%0A1161.290323%0A1200.&freq=440&midi=69&vert=5&horiz=1&colors=white%20black%20white%20white%20black%20white%20black%20white%20white%20black%20white%20black&waveform=semisine&env=organ
here is the 8-note subset: sevish.com/scaleworkshop/?name=harmonic%20scale%20as%208%20notes&data=193.548387%0A387.096774%0A541.935484%0A696.774194%0A851.612903%0A967.741935%0A1083.870968%0A1200.&freq=440&midi=69&vert=5&horiz=1&colors=white%20black%20white%20white%20black%20white%20black%20white%20white%20black%20white%20black&waveform=semisine&env=organ
@@ZheannaErose Thank you.. Can't wait to try it.. Are you choosing the 8 based on their proximity to the regular 12 tet notes or is there some other criteria?
@@MreenalMams it’s discussed in this video. They are notes chosen from 31-EDO based on their proximity and closeness to the first 8-16 overtone in the harmonic series.
@@ZheannaErose I have to be honest, I don't fully understand everything you say in the videos.. I don't know what 'first 8-16 overtone in the harmonic series' means.. but I love watching and trying to learn as much as I can.. And I love playing with these tunings.. Thank you for the videos..
@@MreenalMams the harmonic series is a naturally occurring scale that the majority of pitched instruments produce. every single time you sing or speak a pitch, you produce the harmonic series. when you arrange pitch in a similar way to what naturally already exists, interesting effects occur. this is called just intonation. its basically using nature's pre-existing scale and making music with it. our normal tuning system with 12 notes doesn't approximate this natural scale very closely. some tuning systems like 31edo give you much closer approximations to "natures scale"
What are the chords at 1:18 and 1:26?
😵💫😵💫😵💫
This is like university level material
What program are you using? This is great
The DAW she uses is FL Studio I think
thanks for ur reply! i actually was referring to the program shes using to generate the sounds of the microtones. @tristanbay
If you want to get good harmonic approximation out of a meantone system, IMO 43-TET is the best at this, since it is meantone like 12-TET and 31-TET but does an even better job than 31-TET at approximating the 8:9:10:11:12:13:14:15 ratio. In fact, 43-TET has unambiguous mappings for every single prime harmonic up to and including 113, missing only 23, 71, 89, and 103.
It should be noted that the accuracy you can get out of a meantone system is inherently limited by the requirement for the first two steps to be equal in size (since meantone tunings map 9:8 and 10:9 to the same interval by definition).
50-TET completely misses 9 and has a flat 7, while 55-TET completely misses 13 and 7.
That's an interesting view of 43edo. I tend to prefer 31 or 19 since 43 has an inconsistent 9/7 and 14/9, and the 6/5 is quite flat (by about 9 cents). But 19edo is definitely not as good at things like harmonic 7 and 11 as 31edo. Maybe I should try 43 sometime
12 edo approximation would be the mixolydian bebop scale!
Close, it would be Lydian Dominant Bebop. The 11th harmonic is 551c so its directly between a p4 and #4. Technically closer to 600c :)
What about 32 edo?
what about it? its nothing like 31edo. 32 has a lot of cool features but 31 is particularly valuable to me for what it has. nothing else is similar to 31edo except 55edo, edos higher, or just intonation
@@ZheannaErose Oh, okay. I was wondering if 32 would be any closer.
32edo has a wild pajara[10] scale and a very accurate 15th harmonic
so… up to the chord voicings part I was ok with most of the stuff, but you cannot talk about harmonic series when you move the notes around like that. There should be a specific ratio when you consider the spectrum, you cannot treat it like the modes. Composers like Grisey, Murail and Julian Anderson do operations on the harmonic series, such as compression, expansion and even combining two spectra but when you move the notes freely around the pitch space like this, what you are working with is not the harmonics anymore. The voicings work aurally not because they are generated from your harmonic scale, but because any set of notes can sound good given in the right context. This is also the reason why this approach works in your own compositions. You have a system and you apply it throughout the piece. But the system that you apply is not the “harmonic scale” anymore if you move the notes around like that, it is something else
This is an empty comment. The harmonic series does in fact work like this (modally). Further, this is a basic education video to help people learn these materials in 31 there is no JI other than the JI being sounded outside of 31.
I see your comment is about modality. The harmonic series is a hyperobject of concordance. It is full of modes that all vibrate with their own distinct timbral characteristics due to their unique pitch distributions while still being imbued with special concordance via the isodifferential nature of the entire harmonic series. Of course you can play it in different modes and of course they have special properties. Less so grant via alignment of a tones partials against another since the partials of acoustic timbres slopes off past 8 harmonics typically and more so from their isodifferential coherence which is ultimately the main contributor to why 4:5:6:7 sounds as it does. This is why you cite Grisey and other spectralists. They were concerned with aligning partials. I also refer to this partial alignment but that is NOT where all the concordance in the harmonic series comes from and this is a demonstrable fact. Even slightly tempered forms of primodal qualities significantly lose pure locking and irregular objects in tempered systems which align into verticality with a harmonic mode do resonate entirely differently.
Harmodality includes awareness of spectralism while the inverse is not true. In fact, what is traditional spectralism is simply a single localized perspective relative to the whole of potentiality.
This was a video for beginners to have an easier map of accessing this irregular combination in 31.
@@ZheannaErose as you are referring, alignment of the partials above the 8th harmonic series is not because you simply move the lower partials up and down an octave. I know it’s an educational video but the way you explain the harmonic series is just misleading. You are free to generate modes from the harmonic series. You can even root major and minor scales to the harmonics and subharmonics. But we call them major and minor scales, not harmonic scale, since the notes are not the degrees of the harmonic series anymore. I don’t understand what you mean by harmodality but that’s all I am saying
2:37 Pretty sure it's G half-flat, not G sesqui-flat.
Vrutal
you talk way too close to the mic, saliva noises all around. Content is amazing still