in-tune music and nomenclature

Since entering the world of microtonal music many years ago, I have found that there are many different perspectives on “what is microtonality”, “what is not microtonality”, and what other things we can this thing that some people are calling microtonality but hesitate because the word is not accurately describing the work they are doing.

On the wikipedia page for Xenharmonic music (another term used often synonymously with microtonal music, but implying “different” harmony rather than “micro” or small toned harmony, as often intervals in microtonal music are “bigger” than notes they would be compared to, such as the super-major third) it reads: “John Chalmers, author of Divisions of the Tetrachord, writes: “[sic] music which can be performed in 12-tone equal temperament without significant loss of its identity is not truly microtonal.””  While I do not fully disagree, this puts an air of subjectivity into this realm which from one point of view could combine the entire plethora of baroque tunings and 1/x-comma meantones as “not microtonal”, as well as diatonic 19 and 31 tone music, or even n-edo music which approximates diatonic progressions.  This is where the loss-of-identity element comes in, because one could argue that if the same piece was played in 12edo or 19edo, that both tunings have a very obvious and clear unique identity in relation to one another and that one could not switch tunings without an obvious loss of identity.  However, without careful listening, one might not know that an adjustment in tuning was made (as equal temperaments are in a sense artificial constructs anyways, and both 12 and 19 equal can be playing in a 5-limit meantone based system which has inherent functionality outside of the specifics of the tuning of a fixed pitch instrument).  So from this standpoint, these tunings are in fact the same because we are supposedly processing the information in the same functional manner.  But in the end, they are not, because the timbral aspects of the instrument and the technical aspects of the musician also affect the tuning and perception of the tunings of an instrument.  In addition, these ambiguous boundaries can be used as compositional devices to explore this fuzzy boundary.

This can be extended a step further when playing diatonic progressions in 15-equal for example, such as in the Bad Canada song: Hot Mary, which is essentially an A F#m E D progression, and can be played as such in 12 or 19 equal as well.  However, it really really does not sound that same.  I have had people argue to me that it is silly to play this song in this tuning because it is just an out-of-tune 12 progression.  And yes — it is possible to hear it as such, but it is also something else: it is a progression of triads with significantly sharp fifths, which have a beating near in sync with the sharp thirds (which are the same as in 12 equal), it also is a progression which share 3 roots which are a part of the 5 equal scale, and carry a certain odd symmetrical imbalance between the I IV and V chords, it also have a 160¢ step between the E and the F#, a xenharmonic or microtonal step by any accounts.  Neither 12 nor 19 equal can effectively replicate this combination of properties  — and yes this combination is somewhat arbitrary and definitely not an optimization of anything other than exactly itself, but it is not played in 12 without a loss of identity either.

So, while I agree that the pursuit of the integration and usage of higher complexity and higher limit harmonic structures in music is great and interesting goal, the usage of tuning as a compositional device for a variety of purposes and effects is at the heart of what I strive to do, as a mis-tuning, and alternate-tuning, or as an exploration of truly new tones and relations.

With this I propose to call this, as I have been doing so already, “intentionally tuned” music, or “int-tuned” music”, or “in-tune” music.  This last bit of word play will again bring some challenges with performance and it must avoid being used as an excuse for music play unintentionally out of tune.

In the end, however, the tuning of a piece is generally only part of the composition, as are all of the other factors, and so I hope we will soon get to a point where it is realized that 12-tone equal tuning is neither optimal nor unique in utility, and it really more of a colonial remnant which is not accurate in describing more current music and is a weak tool for creating more, and instead of relying on electronic tuners to make our instruments equal arbitrarily tuned so that we do not need to consider the sound of the tuning, we actually learn to tune our instrument how we would like to hear them, and continue to explore the beautiful patterns and alternate timbres that we can create, and just call it “music” which by nature should include intentional tuning.

But for now we will call things many things as we do.

Sonido 13

To whom may be concerned,

The pianos of Sonido 13 have long been unmaintained, and are currently in a state of (mild) disrepair and are not in-tune.  In 2014, I spent 2 months in an investigation of the pianos and the collection is really only in need of minor repairs.  To restore the collection to original tuning and performance condition, a small team of a piano technician, a specialist in microtonal tunings (I could volunteer myself), and someone acquainted with the restoration of historical / cultural artefacts.  In my city of Vancouver, old pianos are abundant and I have worked on retuning pianos (including to microtonal tunings) which are twice the age of the Sonido 13 pianos; there is no reason that these pianos should not able to be played, for the works of Carrillo, and new works.  There is a growing international interest in the possibilities for such a collection.  It is a great shame that after the beginning of world recognition of the work of Carrillo, he died, leaving a majority of the pianos having never been performed in a concert setting.
So, I propose to you all that I can begin a crowdfunding campaign to raise the funds needed to bring the pianos back to working condition.  I believe that $10,000 USD would be practical to raise, and also much much more than would actually be needed for this job.  What remains could be used as a call for scores, for works that use some of all of the pianos in the collection, and a performance / celebration / inauguration of the Sonido 13 collection.
I ask for your support, in writing of the restoration and reinauguration of the Sonido 13 pianos as a prequel to a fund-raising campaign.  I also ask for any advice or suggestions in this process, if differs from what I have proposed.  In the case in which you disagree that the pianos should be restored and played in the future, please leave me with reasons why.
All the best,
 Noah Jordan

two locrians

last week i posted a new tune on patreon called Cloud Rhyme featuring a locrian tuning (not that I stayed in that mode for a majority of the piece or anything)..  but you can hear it here:

and the tuning is as follows to make things easy.

19/18 (93.6¢),

95/81 (276¢) __ 5/3 * 19/27

4/3 (498¢)

7/5 (582.5¢)

128/81 (792.2¢) __ 2*2*2*2*2*2/ 3*3*3*3

16/9 (996.1¢)

2/1 (1200¢)


anyways, I was asked by Cam Taylor for a version in Pythagorean Locrian:


which shares only the fourth, b6 and b7

do these 3 different notes make an impact?  In what way?  Which ones and where?

Listen to this one here:


or alternatively, both are here:


sooom, many options

many words

so, I mistakenly released an album which was already online “in some loung” by posting the photo/link to my portfolio on here, which made a notification to the Noah Jordan Music facebook page.  But it sold me a copy which was cool.   The album is here if you want to hear: .  It is a solo piano exploration of a variety of tunings (and also a couple rhodes tracks).  It was not really meant to be an album per se, but more of an alternate place to store some files, since I was releasing tracks individually through Patreon.  –Which, if you would like to know more detailed information about the tunings of the tracks, they are on the Patreon posts (, they are free to listen and such, but your support is always much appreciated.

Hopefully, I will start to write on here a bit more, the Patreon “blog” section took over my ideas flow because I was posting weekly, usually specifically about the piece that was being posted, but also would go on tangents.  I found it challenging to also write something here as I felt like I would be repeating myself.  But perhaps I can find their own identities moving forwards.

Last weekend was the Microtonal Adventures Fest in Bellingham, Washington.  Outside of it being a great time and being able to meet in person many who I’ve spoken with online, it provided a number of interesting ideas and concepts with which to address, and a number of them are connected.  For example, in the challenge of notation: do we base the “major third” name and notation on what is nearest to 12tone, what is nearest to 5:4, what is nearest to 81:64, or what is generated with 4 fifths and octave reduced.  This is challenging, for example, because in 15edo, 4 fifths is a fourth (and the 400¢ 3rd exists), in 22edo, it is the “large/supermajor” 3rd (and a near 5:4 third exists).  Tall Kite has been working on a notation scheme to be comprehensive (please check it out and bring feedback) (  Comprehensive and complete and generalizable might not be easy for musicians who have not already be trained to play microtonal music, but the easiest notation might not communicate the intuition and intention of the composer (for instance in systems written in 12tone notation with cent deviations marked).  It was discussed also the idea that simultaneous notations might be ideal, as a sort of key that can be used for general interpretations. — Personally, I am fond of the usage of an “adaptive” modular approach, as have used such in most “microtonal” projects I have been a part of.  This is based on knowing which tonic, scale degree, and chord tones are present at a given time and notating as such.  This mean in 15edo, for example, there can be two G’s, these would be the minor 3rd from E, and the perfect 5th from C, assuming this C is the minor 6th from E.  There are other ways to describe this, given a certain system, and these are generally easiest to comprehend when still in a tonal frame of mind, but somewhat familiar with the interval structure of the tuning system.  One of the great challenges in notation is that of familiarity — to which concepts will we be most familiar with and have the easiest time adapting to, and how does this change depending on the training of a musician. 

Another challenge that we are faced with, is that of the perception of microtones, tuning systems, and music in general.  I speak about this in relation to a new tune I have posted: Cloud Rhyme (, if you are interested in a specific discussion.   Essentially, how to we hear comma pumps ((  Is this, as Aaron Wolf hypothesized, a syntax error when the commas are tempered / removed?  And especially, to which commas could we parse this as a syntax error?  The syntonic commas ( and pythagorean comma ( are quite comfortable to western ears, would these be parsed as such?   What about to cultured of a different music tradition?  What about the difference between two 11/9s and 3/2, this is only a comma of 8 cents, but 11/9s are already a strange interval, so would we parse a comma as a syntax error anyways? 

For equal division tunings, especially of the octave, are we hearing perpetual approximations to just intonation?  Or are we hearing a symmetry instead?  Or a combination?  What about higher limits?  Do we hear 350¢ as a 5-limit, or 11-limit, or what is the range in which we are hearing something, and what is the function of context?  If we hear two subsequent 11/9s, can we hear 121/81 or the 19edo 5th, or 3/2?  or is 121/81 always “3/2” when one of the tones leading to it (or in the harmonic framework) is a 11/9?  Could a higher prime function differently, since, of course 11/9 and 11/9 is 11*11/9*9, is two 11s harder or easier to hear than a higher prime, but if they approach a lower prime (with less multiples), is it always approximated?  How does the 3 as the undertone function though in this situation?  Does 11/9 adapt more easily to 3/2 (as in temper to when in multiples) and 11/8 to 2/1 due to their subharmonics? 11/10 and 11/10 make 121/100 which is 121/120 higher than 6/5, but in this case the 5 is the utonality of both the 11/10 and the 6/5.  And for 7 — 11/7 * 11/7 = 121/49 which is the neutral third range, so this falls apart.  But these utones are all different forms, in 11/9 the utone is 3*3, in 11/8 it is 2*2*2, in 11/10 it is 2*5 and in 11/7 it is 7, these are all quite different types of composition.  In 5/4 * 5/4 = 25/16, it is near to 8/5 which would fit this pattern somewhat, but it is hard to analyze these patterns without going into multiples or much higher harmonics much quicker, and then it is a challenge to what the acceptable range of error is from an approximate harmonic.  IS 20¢ reasonable from anything?  But what if it is a superparticular distance?  These questions come into significance which dealing with EDOS as if we are stating that the interval’s nearest just approximation is 15¢ from the 11th or 13th harmonic, is it really possible to hear this in this way? 

I will hopefully test some of these questions more rigorously in pieces to come — and by that I mean compositionally – or through improvisation.  

Motivation for Scale Construction

Recently I finished a recording for the first piano piece I was commission to perform.  It was called Persian Glances by Jesse Maw.  Hear it here (  And check out some of Jesse’s other music too (

The piece uses a scale generated by 3/2s and 7/4 (octave reduced 3rd and 7th harmonics).  On the Patreon page I give some examples of how to derive the scale, but here I will give a quick example of how this could create a scale before I go onto the main topic.

(( for reference:

“Persian Glances” JI tuning-

1:1 F @ 344 hz



Let’s start with the unison 1/1, this can be any note, as long as it is some note that we will use as the reference point for the scale.

Now let’s decide to make the scale repeat at the octave, so now we have pitch set 1/1, 2/1

Since, it is an octave repeating scale, it is convenient to express all notes between 1/1 and 2/1 (between one and two).

Let’s decide (as is the case for the Persian Glances scale) that we want to focus on combinations of the 3rd and 7th harmonics.

So lets triple our initial frequency and we get 3/1 (or 3).  Now, to put it within 1/1 and 2/1, we need to put some power of 2 on the bottom.  In this case it is just 2, so we have 3/2, which is 1.5 and is between one and two.  So now we have 1/1, 3/2, 2/1 (the root, P5 and octave).

Lets do the same for the 7th harmonic: multiply the initial frequency of vibration by 7, and divide by 4 (2 to the power of 2) to put it within the octave and we get 7/4.  Now we have: 1/1 3/2 7/4 2/1

Now lets combine these 3/2 * 7/4 = 3*7/2*4 = 21/8 … octave reduce by dividing another 2 we get 21/16.

We could combine two 7/4s to get 49/16 .. octave reduced to 49/32, but we see this scale doesn’t have it.  Instead we combine two 3/2s to get 9/4 = 9/8 which is in the scale, and then add a 7/4 to get 9*7/8*4 = 63/32 .. so on we can repeat this process.

Anywho, the central idea I wanted to discuss was that of motivation for scale design.  Persian Glances was both a good starting point for an example here, and also part of a conversation I had with Faras Almeer that inspired me to make this post.  We had been speaking recently about scale design and he had given me his specs for possible tunings for the Arabic Maqam of Rast and Hijaz.  He asked about the motivation for the Persian Glances tuning, and Jesse told me it was to explore the comma of 64/63 which is the interval between 63/32 (9/8 and 7/4 combined) and the octave, and this applied various times throughout.

Let’s show how this works:  If we apply another 3/2 to 9/8, following the Pythagorean method of scale construction, we get 3*9/2*8 = 27/16.  Let’s add the 7/4 again — 27/16 * 7/4 = 27*7/16*4 = 189/64 = 189/128.  This interval on the scale list is one step below 3/2.  This makes sense as 27/16 is a kind of major sixth and 7/4 is a kind of minor seventh, a flat one: the harmonic 7th.  So it is intuitive, as based on 12tone music that the 7th of the major 6th will be the 5th, and so a flat one would be flat of that.  And in just intonation, with the 7th harmonic, this does end up being true.  And lets test if this comma is there.  If it is then 189/128 * 64/63 should = 3/2.   We can see that 63*3 = 189 and 64*2= 128 and so this comma of 64/63 appears between 189/128 and 3/2 or between three perfect fifths + a harmonic 7th and a perfect 5th.  This pattern is repeated on each of the pythagorean fifths that are generated, essentially creating a 7/4 from each of these 5ths and then the respective 64/63 commas between each new part of the pattern.

Another perspective of how this works is to focus on the 3/2 and the 64/63 comma.  This was the method of interpretation of Faras Almeer.  You can take the octave and subtract the 64/63 comma and generate the tone of 63/32.  Then you can take the 3/2 and subtract the 64/63 to generate the 183/124 and so on.  This is an example of commas and generators being a different way to fundamentally express the same thing.  This is of music advantage to the composer and performer who can use these functions to their musical and creative advantage.

I have spoken with Faras Almeer a number of times recently about representations of maqam through just intonation.  What follows is not meant to be an accurate portrayal of maqam, not as historically or culturally accurate, but a framework in which some of the methods of scale construction may be analysed.

In Rast maqam, the third is often described as a neutral third (in just intonation, it is more simple to write this as 11/9 .. 347¢), or as sharp from a minor third, or flat from a major third.  These last two definitions both can describe a neutral third but also create an amount of flexibility or ambiguity.  In 12tone music, the minor third is 300¢ and the major third is 400¢, in 5-limit just intonation (ratios using up to the 5th harmonic), the minor third is 6/5 which is about 314¢ and the major third is 5/4 which is about 384¢ — sharp of the minor third or flat of the major third from the reference point of 12tone.

In addition, we can consider a pythagorean method or deriving the major or minor third: 32/27 (three 4/3s) for the minor third at 294¢ or the 81/64 (four 3/2s) for the major third at 404¢.  These don’t quite work for this definition.

Another approach is to approach this by melodic combinations of tones.  Two 3/2s will give us a 9/8 as a whole tone, and two of these will give us 81/64, which is expected as we just derived this interval as four of the 3/2s.  The minor whole tone of 10/9 combined with 9/8 will give us 5/4 which we already considered.  What about two 10/9s?  This gives us 100/81 at 365¢.  This could work.

What about a smaller representation of a tone?  The neutral third or neutral tone is often spoke of in Rast and the neutral sound often has correspondence with the 11th harmonic.  We have thus considered the 11/9 neutral third of 347¢,  what about the 11/10 neutral second (at 165¢)?  Two of these give us 121/100 which is 330¢, which could work.  With 11/10 and 10/9 we get 110/90 = 11/9, hey the neutral third (good to note that this is a way this interval can be generated).  What about 11/10 and 9/8, we get 99/80 which is 376¢.

We can also use another method of finding a note between a note.  Say between 6/5 and 5/4 we can calculate this mean as 6+5 / 5+ 4 = 11/9 which we already know to be between these.  We can continue the process with 6/5 and 11/9 as 6+11 / 5+9 = 17/14 which is 336¢, and another option.  We can also do this with 11/9 and 5/4 as 11+5 / 9+4 = 16/13 as 356¢, another option.

If you can imagine this many senses in which you can generate types of 3rds, you can imagine the spectrum of possibilities that exist in total.

For quick ratio to cent conversion, this place is great:

Hope to write more on this soon.

Stay tuned ❤

twenty three x

The harmonic series can be thought, in one perspective, of as the sequence of real numbers.

1 2 3 4 5 6 7 8 9 10 11 …. ∞

Mathematically, the ratios of the frequencies between any two of these harmonics are exactly the ratio between the numbers.  For example, the ratio between the “first” harmonic (the unison) and the “second” — 2, the octave, is exactly 1:2.  The ratio between the 7th and the 11th harmonic is 7:11.  As just intonation ratios this is also notated as 11:7 or 11/7.

Powers of 2 are significant.  2 4 8 16 32 64 128… Powers of 2 are also each sequential octave (in the harmonic series, or otherwise).

This means we get these harmonic “realms”

1 2

2 3 4

4 5 6 7 8

8 9 10 11 12 13 14 15 16

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

I colour-coded harmonics as they show octave equivalence in the next realm.

Composite numbers such as 9 = 3*3 are both generated as a member of the harmonic series, being part of the sequence of whole numbers.  And are the combination of two or more harmonics.  In this case, the 9th harmonic to a frequency is the same as the 3rd harmonic applied twice to that frequency.

We can re-write segments of the harmonic series to be within an octave.  This is useful because it is a familiar and practical and perceptually relevant way to categorize a group of pitches.

Octave equivalence is created by multiplying or dividing any frequency by 2.  The 6th harmonic, 3*2, is an octave higher than the 3rd harmonic.

Since these harmonic realms are all in higher octaves, we can octave reduce all the frequencies by powers of 2 (some amount of octaves) to bring them all within the root octave or home octave.

1 2 just becomes 2 unisons

2 3 4 becomes 2/2, 3/2, 4/2 = 1/1, 3/2, 2/1 — this is the unison, perfect 5th, and octave

4 5 6 7 8 = 4/4, 5/4, 6/4, 7/4, 8/4 = 1/1, 5/4, 3/2, 7/4, 2/1 — this is a major triad with the harmonic 7th (a bit lower and more resonant / less beating than the minor 7th)

8 9 10 11 12 13 14 15 16 =

8/8, 9/8, 10/8, 11/8, 12/8, 13/8, 14/8, 15/8, 16/8 =

1/1, 9/8, 5/4, 11/8, 3/2, 13/8, 7/4, 15/8, 2/1

You can continue this pattern to the next harmonic series segment utilizing the denominator of 16, 32, 64, and so on.

But what about denominators that are not powers of two?  What do they mean?

Notice that the difference between the 11th harmonic and the 13th harmonic is 13/11.

So if we want to build a set related to the harmonic series, in particular the 11th harmonic, say starting from the 11th harmonic.  Lets define it at the unison 1/1.

Then the relationship between the unison (the 11th harmonic) and the 12th harmonic is 12/11.

So lets build a set

1/1, 12/11, 13/11, 14/11, 15/11, 16/11, 17/11, 18/11, 19/11, 20/11, 21/11, 22/11 = 2/1

So this set is exactly the set of harmonics from the 11th to the 22nd defined in relation to the 11th harmonic.  This is an overtone series.

We can also build a set of a constant numerator.  For example 23.  Say the 23rd harmonic.  Then 23/22 will be the distance from the 22nd harmonic to the 23rd harmonic.  And 23/11 will be the distance from the 21st harmonic to the 23rd harmonic.

Lets build a set.

1/1, 23/22, 23/21, 23/20, 23/19, 23/18, 23/17, 23/16, 23/15, 23/14, 23/13, 23/12, 2/1.

This is an undertone series of the distance between the 23rd harmonic and the harmonics from 12-22.  This is not a descending series however as 23/22, 23/21 etc are increasingly large intervals, but the description of the relationship is that of harmonic relations which are below 23.

At this set is described as 23/x (with x between 12 and 22), it can be spoken as twenty three x.

Take a listen to a recent piece composed in this 23 set 🙂

Please consider supporting if you like the music and/or way of sharing and explaining ideas presented here 🙂 ❤

169/144 and 169/128

169/144 is the difference between 16/13 and 13/9.

This is 277¢, which happens to be very close to 7/6 (which is 267¢)

6/5 (minor 3rd) is the “next” higher interval (at this resolution) at 314¢

Normally the range is expanded by 8/7 (231¢) and 9/8 (204¢ major 2nd), as opposed to between 7/6 and 6/5, with which 13/11 is usually common (as 7/6 < (7+6)/(6+5) < 6/5 and (7+6)/(6+5) = 13/11.

13/11 is 289¢

and then the different between 16/13 and 13/8 is 169/128 which is 481¢, a nicely flat 4, quite sharp from the supermajor 3 of 9/7 that is 495¢

What makes this interesting more is that 16/13 and 13/8 are inverses as 16/13 * 13/8 = 2

On the Rhodes I have retuned, this 1/1 is D, and the 16/13 is the F#, the 13/9 the G# and the 13/8 the A#.   This is based of the idea that this pentatonic scale of F# (major pentatonic scape) has a very tonicizing property to it, even though the 1/1 is not played in the set.

The full note set being 1/1, 14/13, 15/13, 13/11, 16/13, 13/10, 13/9, 3/2, 13/8, 22/13, 26/15, and 13/7.

So, the F# pentatonic set of F# G# A# C# and D# are

14/13, 16/13, 13/9, 13/8, 13/7

D#,      F#,      G#,      A#,   C#

between D# and F# is 8/7

so the step sizes are, sequentially: 8/7, 169/144, 9/8, 8/7, 196/169 (between C# and D#)

196/169 is 256¢, also very much between 8/7 and 7/6

Hear this pentatonic scale at the end of the end of the 13-limit demo posted recently at and also the full scale, this time in “”Cminor”” at the Tribute to Phillip Glass recording (which will eventually be completed more fully with this scale)


Instruments – Musicians – Chicken – Egg

So as the chicken was an ancestor to the dinosaur, so the egg predated the chicken, thus perhaps the vocal cords predated any intentional construction of a pitched instrument — though as natural sounds contain harmonic spectra, our control over pitch and timbre and faculty has been a growing endeavour.

We hear often natural ability on instruments, as well as the idea of music coming from the heart of the musician, and that many great musicians do not or need not “understand” the workings of music theory or the chords that they play.  Firstly, whether or not a great musician can understand an instrument as past scholars have described it, obviously they understand the workings of it deeply.  Natural ability and heart manifest and can be described in many ways – but one aspect of this particularly is what I want to discuss.

If musicians are meant to be vessels for which music can be expressed, is there an innate sense of musicality in these people – something which must transcend our European tuning system of modern history .. as of course this type of musicality transcends this time and cultural constrain.  Or is the instrument the vessel, with which a person with sufficient expressiveness or creativity, and sufficient technical competence will generate beautiful patterns from a frequency set through rhythmic space?

Evidently, the role of instrument designers has been incredible in the shaping of musics around the world.  The inter-relationship between the builders, the theorists, the musicians, and the composers is very intricate.  I am excited to these the development of this in the decades to come.  More thoughts on this eventually.

Where does microtonality start?

In response to comments of David Dornig of Dsilton,

“…That is certainly important to explain to people who think that microtonality is only valid, if you could not approximate it in 12EDO or meantone temperaments. But than we must ask, where dose “microtonality” start. Whats the reference? Can it be objective or is it only a matter of the concept behind the music. Is a flat intonation of a blue note microtonal or not if one thinks in reference to 12 notes per octave? If those questions are settled, it would be easier to discuss.”

Where does microtonality start is a very interesting question.  Maybe lets start with a sub-question of this, which is “where is a microtone” defined.  In some of the discussions at miCROfest 2017 in Zagreb, it was said that a microtone is anything less than 100¢.  I believe that this definition is based on the visualization of 12-equal as a basis of tuning, which is subsequently a consequence of the atonal and serial conceptions of music.  This is not so much the size of the semitone, but the semitone being used as a measuring tool, as opposed to a difference tone between intervals with a small tone between them.

So there are a few places that microtonality could start.

  1. Music using steps smaller than 100¢
  2. Music that utilizes a greater variety of difference tones / enharmonics than are available in 12 equal music.  – this may include some historical tunings and well-temperaments
  3. Music that utilizes a higher harmonic spectra or limit, for example: utilizing the 7th harmonic functionally within the music


We also have the question of note bends in reference to 12edo and if they are microtonal or not.

I think that this goes back to the idea that “microtonality” implies a subcategory or music, where as 12 tone music can be interpreted as one of many choices of tuning of music.

Therefore, I don’t really believe that microtonality actually exists outside of an atonal sense, or, to extend, outside of a harmonically functional sense.  For example, in quarter-tone music, microtonality may exist in many compositions as the quarter-tones are being used as non-harmonic or non-functional sounds in the music.  I believe, in a strict sense of nomenclature, true “microtones” must be functional only by means of symmetry or serialism.

I believe that the broader, often “xenharmonic” principles, reflect more the aesthetic of fully studying the properties of the pitch spectrum and the totality of their relations.  In the set of the pitch spectrum, 12 equal is one approach to tuning which fulfils a set of useful properties.  These properties can also be fulfils by a vast number of other tunings, and also, each tuning will not overlap with another tuning completely in all categories.  So, therefore, we must choose a number of properties that we would like to fulfil, and decide to which degree we want to accommodate each property.

This leads to some follow up questions:

  1. To what degree can a major triad be “out of tune” before it is no longer a major triad, and how context dependent is this?  For example, we accommodate a 400¢ major third quite nicely (and this is made more difficult by the claim that the 81/64 pythagorean major third is in fact the functional third and not 5/4), and in 15edo we have the same third but a similarly sharp 5th (at 720¢), why is this 5th so much more undesirable than this 3rd?
  2. To further this, is accuracy of tuning importance to follow order of harmonics?  ie. octaves pure, 5ths near pure, 3rds close, 7ths existent, 11ths + irrelevant.
  3. How much is “out of tuneness” perceived relative to the system in place?  ie. the blue note in 12ed2 vs the 240¢ interval in 15ed2.

until next time


please check out some of the 31-tone music of Dsilton