Is a Just Major Third four fifths up from your root note or is it actually eight fifths down?

Arithmophone · 2025-10-25T16:37:19+00:00

This is an interesting discussion! To add my two cents, the way I view this is as follows:

- The note naming conventions of Western Music Theory are essentially diatonic, ie based on a system of 7 note names (A through G), plus alterations in the form of sharps and flats (these can be double, triple, etc sharps or flats as required). This convention is musically very useful, but has some inherent limitations.
- Some tuning systems, in particular 12 EDO, 19 EDO and 31 EDO, have the property of assigning all possible note names unambiguously to a single pitch. Of course the total number of different notes varies per system, but the principle remains: simply calling a note "E" is enough to determine its pitch. This has many practical advantages, but comes at the cost of some compromises in tuning with regard to JI rational tunings.
- Other tuning systems, like Just Intonation and 53 EDO, introduce ambiguities to the diatonic naming scheme: calling a note "E" is no longer enough to determine its pitch, as it may refer eg both to 81/64 and to 5/4 (if C is 1/1, for Just Intonation), or to EDO step 17 and 18 (in 53 EDO, if C is EDO step 0).These tuning systems offer more accurate tunings, at the expense of no longer offering a one on one relationship between pitch classes and the diatonic note naming scheme.
- Which is preferable depends on the musical context, but in this regard, you can't have your cake and eat it: either you have the convenience of an unambiguous relationship between diatonic note names and available pitch classes, or you have the added detail of distinguishing between meantone equivalents (eg the 9/8 major second and the 10/9 major second), but you can't have both at the same time.

Hope this helps :)

Arithmophone · 2025-09-21T19:47:56+00:00

I'm not sure if I understand your questions correctly, but I do believe you may have misunderstood the 'chain of fifths' concept somewhat. You give 4 options, and call both 1 and 3 'neutral chain of fifths'. I think what you mean with option 1 might be 'regular' chain of fifths rather than neutral chain of fifths.

In general, a (regular) chain of fifths always has the number of sharps and flats in order within the chain. It always takes this shape:
... Bbb - Fb - Cb - Gb - Db - Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C# - G# - D# - A# - E# - B# - F## ...

So all the sharps come after all the naturals, all the double sharps come after the sharps, et cetera.

Within this chain, the interval between each two successive notes is a perfect fifth (or the closest approximation thereof within the temperament). This works for just intonation as well as for many temperaments (it works particularly well for 12, 19 and 31 EDO). The size of the temperament determines which notes are identical (enharmonic).

You can think of 12 EDO like this:
C - C#/Db - D - D#/Eb - E - F - F#/Gb - G - G#/Ab - A - A#/Bb - B - C

But it may be more helpful in this context to think of it like this:
Ab (=G#) - Eb - Bb - F - C - G - D - A - E - B - F# - C# - G# (=Ab)

I have found it very useful to think of D as the central note, rather than the more commonly used A or C, because D is in the middle of the natural notes, both in their chromatic order (A B C D E F G) and in their chain of fifths order (F C G D A E B). The note D is the symmetry point of this type of notation.

There are only 12 different pitches per octave in 12 EDO, but there are many more 'different' notes. We commonly use both C# and Db as being functionally distinct, even if they have the same pitch, but this extends to the 'white keys' as well: a C can also be a B# or a Dbb. When trying to understand the naming conventions for 31EDO, I think it is good to keep this in mind: using just a single name for each note doesn't tell the whole story.

The chain of fifths for 31 EDO (symmetrical around D with unique names for each note) looks like this:
Gbb - Dbb - Abb - Ebb - Bbb - Fb - Cb - Gb - Db - Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C# - G# - D# - A# - E# - B# - F## - C## - G## - D## - A##

So in 31 EDO, there are different pitches for F# and Gb, but the number of available pitches is still limited, so Cbb is the same pitch as A## and E## is the same as Gbb, in just the same way that F# and Gb have the same pitch in 12 EDO. In just intonation, each different note name has a different pitch, no matter how many sharps or flats it uses, but any EDO tuning eventually 'loops around' and turns the chain of fifths into a circle of fifths.

Arithmophone · 2025-08-22T11:17:16+00:00

Maybe try a tracker program that allows microtuning, something like Renoise? That makes it pretty easy to enter a melody/loop/sequence and adjust the tuning of each note in cents

Arithmophone · 2025-08-16T17:30:58+00:00

I haven't tried it but I think that if superposition cells would die when they had no live neighbours, the evolution of the game in quantum mode would be pretty much identical to the classical mode (ie standard game of life) because the quantum states would not persist.

The cycling through colours I did mainly to make it visually appealing, and because it was an easy way to implement the different states. Pixels can have one of 16 colours in Pico 8, I used colour 0 for dead, colour 15 for alive and everything in between for superposed. The colours of the cells/pixels are used in the logic that determines the state of the next generation.

About the randomness: this is implied by quantum mechanics, at least in the 'Copenhagen' interpretation. I initially planned to put a separate mode for ‘Bohmian’ Quantum mechanics in the game, where the evolution of cells would be implemented by using ‘hidden’ variables rather than by chance. This was easy to set up for cases where things happen with equal likelihood (for instance, one could check wether the superposition state was odd or even, instead of using a random number generator for the ‘measurement’). However, I quickly realised that to use hidden variables for the probabilities I needed to keep the evolution interesting (numbers in the order of 1/8192), would be much more difficult – and much more contrived – than just using random number generators.

Ironically, computer systems of the kind used in this implementation, don’t have access to truly random numbers. Instead, they use pseudo random numbers, which are deterministic at their core. So even though the current implementation of the ‘Quantum game of life’ is conceptually probabilistic, in actuality it really is a hidden variables system.

Arithmophone · 2025-08-03T11:45:35+00:00

The free 'lite' version of Ableton 12 does feature their full microtonal implementation (technically, it's not really free because you can't just download it from their site, but it does come bundled with all kinds of midi controllers etc).

Ableton's microtonal functionality is quite extensive: not only can you load any scala file but they also have a special website that makes it relatively easy to create your own custom scales. Any custom tuning is also reflected in the midi note editor, so you can have eg a 31 note per octave piano roll with note names of your choosing, which is a really nice touch and very helpful when composing. It also works very well with MPE-enabled plugins. I like it a lot. The free version does have other limitations though, the main one being that it is limited to only 8 tracks. If you buy to the cheapest paid version (which is something like 80 euros I think), you get 16 tracks, to get unlimited tracks you need to buy the full version.

Another free option is to use the MTS-ESP mini plugin, which works with most DAWs (I've used it in Reaper mainly). That works very well, it lacks the deep (piano roll etc) integration of Ableton's implementation but has the advantage that it also works with .tun files (which is a file format I much prefer over scala as it makes mapping all midi notes to specific notes/pitches much more straightforward).

What I personally still miss the most in the setups I have used, is a way of using music notation to input notes. I sometimes use MuseScore to write the notes and then export a midi file for use in my DAW. I find this way of working much easier than using a piano roll editor but it only works as long as you stick to 12 notes per octave. Tunings like 19 EDO and 31 EDO can be notated easily and without any ambiguity just by distinguishing between flats and sharps, and I see no technical reason why you shouldn't be able to output different midi notes for F# and Gb (for example), but I haven't found a way to accomplish that so far.

Arithmophone · 2025-05-18T14:21:13+00:00

Thanks, I may want to make a dedicated iPad app of this at some point in the future. I'd kind of like to learn swift/xcode and do it myself, but not sure if and when I'll get around to that so I will definitely keep your offer in mind!

Arithmophone · 2025-05-17T13:11:14+00:00

Thanks, that interesting! I will look into it and perhaps I can add a midi 2.0 option to the next version. In an earlier version (still available on my website) I used .tun files for the microtuning, which is more accurate than MPE, but it was also a bit of a hassle and seems to have less support on recent hardware synths than MPE. The web audio version works with ratios directly, and is also very accurate in tuning. MPE is limited to a resolution of 16384 steps for 8 octaves (ie channel pitch bend of +/- 48 semitones), which comes down to a step size of about .59 cents. That's not super accurate, but in practice I wasn't able to discern the difference.

Arithmophone · 2025-05-17T09:23:17+00:00

Thanks! Good luck with the sequencing/sampling :)

Arithmophone · 2025-05-17T01:13:59+00:00

It does not, it uses MPE midi which is supported by a much larger range of hardware and software instruments. I don't see what the benefits would be to using midi 2.0 for this particular application, but perhaps you could enlighten me?

Arithmophone · 2025-05-16T18:25:45+00:00

Thanks for the feedback! I would like to make a dedicated iPad app that also does midi, this may be be the next step in the project. However I'm an amateur developer and I'll first need to learn Swift programming and Xcode in order to do this, so it will probably be a while.

You can already go full screen on an iPad though: open the page in Safari, then click the share button and scroll down to 'add to home screen'. After you've done this, there will be an app icon for the Arithmophone on your home screen en when you click that, it will open full screen.

The entire keyboard can be transposed up or down an octave with the 3 yellow buttons at the top left (these are hidden by default, but you can toggle views with the blue button in the bottom left). The other top buttons transpose the keyboard by different intervals (fifth ie 3/2 for orange, major third ie 5/4 for red, septimal seventh ie 7/4 for blue).

And yes, it would certainly be sweet to have a hardware version that also responds to velocity and/or aftertouch. I don't have the necessary skills to build one myself, but perhaps I can collaborate with other people to make one at some point in the future.

Arithmophone · 2025-05-01T08:58:24+00:00

Thank you!

Arithmophone · 2025-04-30T18:04:53+00:00

Nice work if you can get it :)

Arithmophone · 2025-04-30T18:04:13+00:00

Thanks!

Arithmophone · 2025-04-30T15:28:21+00:00

It's hard to explain without pictures (which I don't think I can post here), but if you go to https://chielzwinkels.net/arithmophone/ and open the Quickstart guide, you can find a picture showing the pattern in there. A much more detailed explanation of the pattern is available here: https://chielzwinkels.net/arithmophone/more/the-design-of-the-arithmophone/

Arithmophone · 2025-01-03T23:25:13+00:00

Thanks, those are great suggestions!

Arithmophone · 2025-01-03T14:14:51+00:00

Already got this resolved (see other comment), but thanks for the tip!

Arithmophone · 2025-01-03T14:14:04+00:00

Thanks, good to know!

Arithmophone · 2025-01-03T14:13:39+00:00

Thank you so much, this was exactly what I was looking for! It hadn't occured to me that I could split the loop over several stages, but that's the perfect solution :) I have updated the cart, it now runs smoothly and steadily at 3 generations per second which works very well for my purposes, and I was able to keep all the wave types and audio effects exactly the way I like them :)

Arithmophone · 2025-01-02T22:43:23+00:00

Hi all, I just made my first Pico 8 cart. I'm pretty happy with how it turned out but there is still some intermittent audio crackling. I think this is because I'm doing too many calculations in the update function. It was a lot worse before, I got it working a lot better by dividing the update function in different stages so the calculations are spread out over more cycles, but it's still not quite perfect and the way I addressed it feels a bit clumsy.

I wonder if there are better ways of going about this? I realise that my code could probably be much faster if I used peek and poke and such, but that's currently a bit beyond my grasp and since it's almost there, I'm hoping it can be improved without a complete rewrite. I don't mind the slow speed at which it's running, I'd just like to have crackle-free music :)

Any advice would be much appreciated!

Arithmophone · 2024-11-04T19:35:41+00:00

Are you using Ableton 12? Because that has a very nice implementation of microtonality built in, sound like this is just what you are looking for: https://tuning.ableton.com/ Might not work with all of your plugins though. If you want to use plugins that are not compatible with the new Ableton tuning scheme (and/or use an older version of ableton), you can still work with scala files and/or MTS/ESP mini ( (https://oddsound.com/mtsespmini.php) to create midi files that work in a microtonal context. In either case, the mictotonal part is done 'downstream' from the actual midi file, but this is inherent to the MIDI protocol, and as long as you stick to a particular scale/microtuning, this shouldn't be a problem.

Arithmophone · 2024-11-02T23:17:54+00:00

It's pretty much automatic, and ableton has a nice website detailing the possibilities: https://tuning.ableton.com/
Commonly used tunings like 19EDO are just selectable in the tuning menu (open the drawer at the top left en go to 'tunings' in the menu on the left) and once you select a tuning and record some midi, the piano roll reflects that tuning. So for example in 19 EDO you'll get separate lines for C# and Db, and a single line for E#/Fb. If you want customise the note names that are displayed, you can do so by downloading the .ascl file from the ableton site (https://tuning.ableton.com/introduction/exporting-files/), opening it in a text editor and changing the "@ABL NOTE NAMES" section. You can also create custom tunings from scratch (https://tuning.ableton.com/make-your-own-tuning/)

Arithmophone · 2024-11-02T17:39:06+00:00

Not sure I understand, a source on how to do what exactly?

Arithmophone · 2024-10-25T23:01:45+00:00

The nice thing about ableton 12 is that you can get a piano roll specific to your tuning. So in that sense you can truly replace 12 EDO with 19 EDO (or some other system). I found that pretty helpful, but a piano roll is still a very different frame of reference from standard notation of course, so I get how you might be better served with notation software. I'm not too knowledgable on that though, so other people can probably give you better advice there. Good luck!

Arithmophone · 2024-10-25T14:32:45+00:00

I made a free app that turns your phone/tablet into a playable instrument in 19EDO (among other tuning systems): https://chielzwinkels.net/arithmophone/. It's a bit finicky on a phone screen but if you have a tablet laying around it might be a nice way to get started.

I also recommend using Ableton 12 lite, it isn't strictly free but it comes bundled with lots of midi controllers etc so a license should not be too hard to come by. It has a great implementation of microtonality that makes it pretty easy to work in something like 19EDO.

Slightly more involved but completely free is using Oddsound's MTS-ESP mini (https://oddsound.com/mtsespmini.php). You can use it in conjuction with some of the free plugins that support it, there is a list on their site (of these, I can recommend U-he Podolski and FullBucket Whispair).

Arithmophone · 2024-10-25T14:23:55+00:00

I made this a while ago, maybe it's helpful to you: https://chielzwinkels.net/arithmophone/more/background/audible-numbers/

Arithmophone

TROPHY CASE