JLMoriart

Here's the video: https://www.youtube.com/watch?v=M6mkyAtwrcw TLDW: If you use a circle of flat major-second-ish intervals to generate your notes (instead of a circle of fifths), you end up with some interesting scale structures and harmonies. You can play these new scales and harmonies in MuseScore using this plugin: https://musescore.org/en/project/porcupine-tunings Please feel free to let me know if you have any feedback, and to ask any questions you might have!

Any composers interested in tuning/microtonality may be interested to know that I've modified a MuseScore plugin originally made to retune your score to 31-tone equal temperament so that it can retune the score to any fifths-based tuning, including 19-TET, 31-TET, 17-TET, 22-TET, and many more. Just enter the size of the fifth, or click one of the buttons for preset values, and each note in your score will be retuned to that system. You can use the plugin to take advantage of specific features of a tuning (like how in 31 the doubly augmented third is a good approximation of the 11th harmonic, or how D# and Gb are enharmonically equivalent in 17), or write a piece of music like you always do and then see which tunings you think it sounds best in. (I recommend trying 31 in those cases!) The plugin is available here for anyone interested in giving it a try: https://musescore.org/en/project/fifths-based-tuning And here's a quick explainer video: https://www.youtube.com/watch?v=Uu5m7IN6Krs Please do let me know if you end up writing music with it, and also feel free to ask any questions about the theory or what valuable properties different fifths-based tunings have.

Our music notation system is built around the diatonic scale, but could we build it around a different scale, like the pentatonic scale? And what would that change? That's what I dive into in this video, if you'd like to check it out: The tl;dr here is that, when you change to a pentatonic notation system, the sharp/flat in that system ends up being the minor second from our diatonic system, and that leads to what I think is an inspiring new perspective for playing and composing music. In addition to providing a new perspective, it also makes a practical and audible difference when playing music in tunings where are enharmonic equivalents are no longer equivalent, since alterations of the pentatonic scale by the pentatonic sharp/flat (the diatonic minor second) will actually sound different than alterations of the scale by the diatonic sharp/flat (the diatonic augmented unison). In the video, there's a short boogie-woogie style composition that I wrote in 19-tone equal temperament with this pentatonic system in mind. What do you think about the results, and the idea of a pentatonic notation system itself? Would you try to write something using this notation system if a notation program made it available to you? I'd love to hear any thoughts, and please feel free to ask any questions about the video, and the initial theory I breeze through. (I had to gloss over that stuff to get to the meat of the video, and I do realize it ends up being a bit of a bombardment in the beginning there.)

Howdy composers! I just uploaded a video talking about "Pajara Temperament", a temperament and scale system generated by a "circle of minor seconds" (instead of a "circle of fifths") that repeats at the half-octave (instead of the octave). It's got its own unique scale structures (like the "symmetrical decatonic scale"), which can be played standard western tuning, which makes it easy to experiment with without actually going "microtonal" right away. https://youtu.be/Ze6qRCH9My8 Please feel free to ask any questions you might have and to let me know what you think =) Here's the script for the video: The scales we’ve used historically in western music (the pentatonic, diatonic, and chromatic scales) can be derived from continuous chains of perfect fifths. Unbroken chains of a single interval naturally arrive at melodically consistent scales at certain lengths, called “Moment of Symmetry” scales, and it’s probably no coincidence that, when that chain is made of fifths, or, more accurately, good approximations of the frequency ratio 3:2, MOS scales occur at lengths of 5, 7, and 12. As I’ve shown in previous tuning theory videos, that derivation also allows us to systematically adjust the tuning of those scales by changing the size of all instances of that generating interval. (For example, to better approximate the frequency ratio 5:4, we can slightly flat our fifths away from pure so that after stacking four fifths we land on a significantly flatter major third.) We can, however, stack intervals other than the fifth on top of each other and repeat them within larger “period” intervals (that may or may not be the octave) to derive other melodically consistent scales that can have their tuning changed regularly by adjusting the size of each instance of that single generating interval. One such system is called “Pajara Temperament”. Pajara’s period is the half-octave, and its generator is an approximation of the just diatonic semitone, with the frequency ratio 16:15. Pajara may be of special note to western musicians since standard western tuning, aka 12 equal divisions of the octave, supports Pajara temperament and its MOS scales. The MOS scales that Pajara generates are very different from our familiar western scales, however. One such a scale is the symmetrical decatonic scale (which is really just a 5-note MOS scale that repeats at the half octave). Similarly to how we can illustrate western scales as an array of fifths and octaves, we can illustrate Pajara as an array of half octaves and semitones, though our western note names lose their meaning in this context, and what’s important instead is each note’s location in the 2D array: A#-B -C -Db-D E -F -F#-G -Ab A#-B -C -Db-D The 5:4 frequency ratio (which is the bottom half of a major triad) is arrived at by going up one period and down two generators, and the 6:5 frequency ratio (which is the top half of a major triad) is arrived at by going up 3 generators. Given those facts, we can show that this MOS scale gives us access to 2 major chords, 2 minor chords, and 1 augmented chord in each half octave. Here are those chords in ascending order in standard western tuning up two half octaves: And here’s a short example piece using this scale tuned to standard western tuning: This piece can be rendered in any Pajara tuning, however, and retain it’s general melodic structure. It’s harmonic purity can also be improved upon by using a generator sharper than 100 cents. At approximately 109.1 cents, we end up with 22-tone equal temperament, which sounds like this: In between 12 and 22, a generator of approximately 105.9 cents generates 34 equal divisions of the octave, and is pretty close to optimal in terms of harmonic purity in Pajara temperament: We can tune the generator even sharper than the one that generates 22-edo to arrive at tunings where the large and small steps of the scale approach each other in size, leading to interesting melodic characteristics, at the cost of some harmonic accuracy. For instance, a generator of approximately 115.4 cents results in 52 equal divisions of the octave, and sounds like this: With a generator of 120 cents, the sizes of the large and small steps finally converge, resulting in 10 tone equal temperament with only a single “neutral” triad. Interestingly, after having written the piece in a more accurate Pajara tuning, I can’t help but still hear the original major and minor tonalities: And when rendered with a more forgiving timbre, I think it has an especially nice quality to it: Any piece of music written using this scale in western tuning can then be tuned to other Pajara tunings without losing its melodic structure, the same way any piece of music written in the pentatonic, diatonic, and chromatic scales can be tuned to other Syntonic tunings (aka Meantone tunings) without losing its melodic structure. I recommend giving this scale a shot, seeing what you come up with, and then, if you’re feeling adventurous, rendering it in other Pajara tunings!

Though it might seem counterintuitive at first, it turns out that you can notate 22-tone equal temperament using regular standard western notation. The intervals of each system simply correlate to the movements along the circle of fifths in either tuning. (For example, a minor third is found by going down three fifths and up two octaves. This is true using 12-tone equal temperament's fifth, or 22-tone equal temperament's fifth.) The fifth is different enough in size in 22 from standard western tuning's fifth, however, that, after going even a short distance in the circle of fifths, the tuning of the intervals derived therefrom shifts pretty dramatically. One crazy repercussion of that is how the augmented second is actually more "in tune" than the major third is, and so the purer C "major chord" is spelled C-D#-G! You could, of course, just treat that D# as a more in tune "kind of E" but, especially melodically, that can lead to funky unintended consequences. When actually treated melodically like an augmented second, however, I found that it worked exceedingly well. I wrote and notated a round (like "row row row your boat", where you can sing the song on top of itself) that I think demonstrates that feature of 22: https://www.youtube.com/watch?v=xONo-1OxSAw What do you think? Are you able to pick up the melody and sing/read along, with or without knowing how the notation lines up with 22 exactly? Any other composers interested in writing in 22-tone equal temperament can use the same free plugin for Sibelius that I did, from offtonic, here: http://offtonic.com/sib/plugins.html#ntet

The video also has a couple simple scale examples that composers curious about "microtonal" music may be interested in. Let me know if you have any questions about the theory!

You may be interested in a video I recently released that analyzes Gamelan Slendro Tunings and sees how well "moment of Symmetry" scales can approximate them. (MOS scales are cool because they can all be notated by a generalized western notation, though I don't go into that in the video.) I'm currently working on a second video for Pelog. Feel free to ask any questions about the theory in the video, or if you're curious about how MOS scales can be represented using a generalized western notation 😃

I'm not so much "worrying" about it as much as I am intrigued by the melodic nuances that show up in the tuning system (some of which are reflected in blues performance). If someone finds it melodically expressive to play their augmented seconds smaller, or finds it harmonically expressive to tune the "minor seventh" (aug-6) and "minor third" (aug-2) flat, then this is a relatively simple model that takes that into account and still maintains all of your standard western tonal relationships. If you'd like to hear the dramatic difference the tuning can make (which might make it worth "worrying" about), check out a piece I wrote that takes advantage of quarter-comma meantone's blues scale and higher-limit harmonies. Starts at 3:23:

Hey guys, please forgive the clickbait-ey title, I couldn't help myself =P In this new video I consider an alternative "enharmonic spelling" for the blues scale, and then show the differences between it and the original spelling in quarter-comma meantone tuning (where the enharmonic "equivalents" have different pitches). I'd love to hear what you all think about the two. Any questions please feel free to ask!

Thanks! It took about 1.5 tantrums worth of takes =P

A buddy of mine (Gareth Hearne) wrote this microtonal Sanctus in what's called "porcupine temperament", and I finally got around to making a recording of it: Similarly to how western music has a circle of fifths, Porcupine Temperament has a circle of small major seconds (approximately 160 cents wide). It approximates many intervals of the harmonic series as well or better than standard western tuning, especially the 11th harmonic. (Kind of like how barbershop singers sing their minor sevenths or augmented sixths flat to be in tune with the 7th note in the harmonic series, porcupine has a "fourth"-ish thing that lines up with the 11th note in the harmonic series and it, I think, blends very nicely!) If you have any questions on the theory I'd be happy to talk about it, thanks for listening =)

Actually, the stuff we're talking about here is not covered in most (any) theory books. Standard theory tells you that this ubiquitous, standard structure of music exists, and then gives this structure's components many names, but it doesn't deal at all with its derivation. It's these structures' derivations (and their implications and generalizations) that we're talking about. From my experience, theory books will deal more with *compositional tendencies* than anything more scientifically associated with the word "theory". This stuff will one day, I hope, be in theory books, when the term "MOS Scale" is as ubiquitously known as "diatonic" is today, but for now it is the fringe of experimental music making. Very cool stuff, but no one much cares for it because it requires stepping outside of some boxes with walls so tall that most don't know stepping outside is an option.

That's quite right, but my hope with the video I posted was to place emphasis on the *implications* of that difference, namely in the possibility for our enharmonics having different pitches and, therefor, audibly different functions.

Actually, though it may look daunting, it's quite easy to play. Much easier than a piano, IMO. Here's a video of Fur Elise played on one: Haha no, 12-tone equal temperament is no more or less valid than any other meantone tuning with any other size fifth, and other tunings even have different enharmonic equivalents. For example, 19-tone equal temperament has E#=Fb and B#=Cb. 12-TET It has it's benefits and its deficits, and tuning is just another dimension of expression that you can use to interpret a piece. Indeed, some pieces *require* our enharmonics to be equal through enharmonic modulation, while others may benefit for the more justly intoned triads of flatter fifths, while even still others may benefit melodically from sharping the fifth and achieving starker melodic contrasts. -J

This talks about old Halberstadt style keyboard instruments having different keys for the sharps and flats: http://en.wikipedia....iki/Split_sharp And the newer concept of a generalized keyboard also has different keys for sharps and flats, but laid out much more ergonomically: http://en.wikipedia....den_note_layout

Not sure what you mean by "you got to state outright your tuning system." The tuning system *is* stated outright the moment you state the size of the fifth you use because it defines all notes of the tuning. The moment you say your fifth is 700 cents, for example, you know your tuning is 12-tone equal temperament because stacking that fifth up and down the chain of fifths *directly results* in 12-tone equal temperament. -J

You change the size of the fifth when you play in other tunings (temperaments). Historically during the use of "meantone" the fifth ranged anywhere from 694 cents to 700 cents, the tunings in between 19-tone equal temperament ("1/3 comma meantone") and 31-tone equal temperament ("1/4 comma meantone"). Yes, the enharmonics are only different pitches when you use a tuning different from standard western tuning (12-tone equal temperament). The video was supposed to show, however, that they way they sound different in other tunings is what you imply when using one vs the other in 12-tone tuning where they have the same pitch. If you take a look at the video I posted again you'll see that the scale you're using (aka what key you're in) decides whether to use a sharp or flat, via the circle of fifths. Let me know if something about it is not clear. -J

The decision to make the unit for intervals have 1200 of them in every octave is arbitrary. You could make it 8 cents/octave (where the tritone would be four cents, and the minor third would be 2 cents), or something silly 2125. 1200 is just convenient so that you get 100 cents per minor second in standard western tuning, which makes it easier to deal with the math and compare other tunings to the original 12. -J

Actually there are, and were. New Instruments like generalized keyboards have different keys for the sharps and flats, and old organs had "split keys" where the enharmonics had different pitches. -J

The tunings in the video are not well tempered. Well tempered tunings use only 12 unequally spaced notes, where the pureness of some keys is maximized at the expense of others. The tunings I use are "meantone tunings", where all the interval sizes are regular and are defined by an (infinitely long) chain of fifths. That is, all the major seconds are the same size, all the perfect fifths, all the minor sixths, etc. If you choose a twelve note subset of the infinite chain of fifths you do end up with "wolf fifths", but that's only because they are something like C# to Ab, or a doubly diminished sixth, NOT a fifth. Because all the fifths are the same size, when you define its size the pitches of all other notes fall right into place (because their pitches are defined by their location in the circle of these regularly tempered fifths). So I *do* define which tuning I'm using, merely by the statement of the size of the fifth: http://en.wikipedia.org/wiki/Meantone_temperament Not sure if I follow the rest completely but I'll do my best to reply. The tuning with a fifth of 695 cents is about "1/3 comma meantone" to give it a name but, as I mentioned above, just by specifying the size of the fifth, the other notes are defined by their location in the circle of fifths. The reason you'd flat the fifth like that is then you end up with very pure thirds, and in fact a perfectly tuned minor third. Though the fifth is slightly flatter, over all the tuning is much "better" (or has less error). I subtracted the number 8400 because it is 7 octaves. What I mean is, I stacked up all those fifths, and then to compare it to the original note I started at I had to bring the new note down to the original octave. Originally it turned out that 12 fifths equalled 7 octaves (8400-8400=0), but the second time it turned out that 12 fifths did NOT equal 7 octaves (8340-8400=-60). So the 8400 came from octaves, not from the original fifth I used. This is very much so not true. Using the circle of fifths to define the rest of your notes provides you with good accuracy (consonance) over a wide range of values, as I showed in the video. Flatting the fifth to 695 cents gives you MUCH purer tuning than standard western tuning, and if you sharp it to 702 cents and then use C-Fb-G as your consonant triad it is *extremely* close to pure. -J

Quite right, I've corrected that in the video using annotations. Thanks! Keys based on those notes don't become unusable just because our enharmonics diverge. You just can't play a song that uses both on an interface like a piano that has only one key for the two of them.

Here is a YouTube video I put up recently describing the difference between sharps and flats: It requires a little prerequisite theory knowledge, though I hope it's accessible. Here is the video transcript: "What is the difference between F sharp and G flat?" It's a question asked often enough, but what comes along less often is a satisfactory answer. It is my hope that by the end of this video you will have not only an understanding of WHY there's a difference, but an understanding of how to use that difference to make, play, and talk about music more effectively. Let's start with a staple of music theory: the spiral of fifths. It shows that, starting from the naturals and using a chain of fifths, you can arrive at every note we could ever use in western music: The sharps, the flats, the double sharps, the double flats, and so on. Also, notice that we can derive any diatonic scale from a continuous, unbroken chain of seven fifths. For one example, the G major diatonic scale can be generated from a chain of fifths from C to F#, which rearranges to G-A-B-C-D-E-F#. Neither of those facts is coincidence. In fact it is by this chain of fifths that we DEFINE the notes that EXIST in western music, and it is by any 7 notes connected by an unbroken chain of fifths that we DEFINE our diatonic scales. So most simply, the difference between F# and Gb is each's different location in the spiral of fifths, and therefor the scales to which they do and do not belong. If you're using the G major scale, the right note is F#, and if you're using the Db major scale, Gb is the correct note. But let's delve a little further and ask "So what?" If they're the same pitch, why define things that way? First, know that intervals are measured in a unit called "cents" where there are 1200 in every octave, 100 cents per half step, and therefore 700 cents per fifth, at least in standard western tuning. Let's prove that F# equals Gb specifically because the fifth is 700 cents. From F# to Gb in fifths is F#-B-E-A-D-G-C-F-Bb-Eb-Ab-Db-Gb, or 12 fifths. Starting with Gb at 0 cents, and adding 12 fifths * 700 cents/fifth gives you 8400 cents. Dropping that note into the original octave by subtracting 6 octaves, 8400 cents - 1200 cents per octave times 6 octaves gives you 8400 cents minus 8400 cents, which comes out to 0 cents. Because 0 cents is a unison, F# is equal to Gb. The G major scale and I IV V I progression using these pitches sound like this: ~~~~~~~~~~~~~~ And the same pattern but replacing Gb for F# sounds exactly the same. The size of the fifth can change however, and when it is not exactly 700 cents then F# and Gb actually have DIFFERENT PITCHES. Motivations for adjusting the size of the fifth in the first place include better harmonic purity ("more in-tune") or melodic clarity (with a starker contrast between major and minor). Still, the over all musical structures stay pretty much the same. If we flat the fifth a nearly imperceptible 5 cents to 695 cents: From F# to Gb in fifths is still F#-B-E-A-D-G-C-F-Bb-Eb-Ab-Db-Gb, or 12 of these slightly flatter fifths. 12 fifths * 695 cents/fifth = 8340 cents Dropping that note down 6 octaves gives 8340 cents - 8400 cents = -60 cents. That means that, with a fifth of 695 cents, F# differs from Gb by 60 cents! The G major scale and I IV V I progression using these new pitches sound like this: ~~~~~~~~~~~~~~~ Different nuances from before, but recognizably the "same thing". The same pattern but replacing Gb for F# sounds like this: ~~~~~~~~~~~~~~ It's completely different, and quite wrong for that "major scale" sound we all know and love. Why would it turn out that F# sounds so much better in this context than Gb? Well, the decision to define our scales by unbroken chains of fifths is not arbitrary. Scales created by continuous unbroken chains of one given interval have perceptually relevant properties in music, where intervals sizes are regular (with only two sizes per interval class, aka major and minor), and where the different step sizes are distributed evenly throughout the scale. (For example, the diatonic scale has the minor seconds as far apart as possible between the major seconds.) Such scales are called "Moment of Symmetry" (or MOS) scales. Replacing a note like F# with another note like Gb, though they may be close in pitch, disrupts this regularity, and voids the definition of MOS. You imply this difference between the sharps and flats even when using standard western tuning with a fifth of 700 cents, where F# and Gb are the same pitch. *This*, ladies and gentlemen, is the difference between the sharps and the flats. They differ in their location in the spiral of fifths, which describes their difference in function, which is audible in tunings where the fifth is not 700 cents. Curious about playing and composing music in different tunings? Check out some of these links!

It's interesting that they pick one tuning (72-edo, a tuning with very good approximations of high limit Just Intonation) and base all of their ear training exercises off of that. The school of thought I'm coming from is one more melodically and scale based than harmonically and Just Intonation based, where scale structures are supported by many different possible tunings (along the lines of meantone tunings, but generalized.) Neat that they provide a scholarship for students composing microtonal music!

Since many people don't know where to start when composing microtonal music, I put up a short tutorial on how to get microtonal playback using either and AXiS-49 keyboard or your QWERTY keyboard and some freely available software, with a few examples at the end of different scale structures and chord progressions that are only possible in microtonal music: Is this helpful? It goes into a *very* shallow description of regular temperament theory and provides more links in the description box, and mostly focusses on getting you playback. Any critiques are appreciated!

I know of one program that seems pretty neat called "fractal tune smithy" though I've never tried it. It looks like when you manipulate the parameters it does most of the composing, and it can also do microtonal for nonwestern fractal stuff too. There are some examples on this page: http://robertinventor.com/software/tunesmithy/music.htm John M