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: https://www.patreon.com/posts/cloud-rhyme-18940728

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:

1/1-256/243-32/27-4/3-1024/729-128/81-16/9-2/1,

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:

https://soundcloud.com/noahdeanjordan/cloud-rhyme-pythagorean-locrian-version

 

or alternatively, both are here:

https://noahdeanjordan.bandcamp.com/album/in-some-loung

 

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: https://noahdeanjordan.bandcamp.com/album/in-some-loung .  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 (https://www.patreon.com/noahjordansounds), 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) (http://tallkite.com/).  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 (https://www.patreon.com/posts/cloud-rhyme-18940728), if you are interested in a specific discussion.   Essentially, how to we hear comma pumps ((http://xenharmonic.wikispaces.com/comma+pump))?  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 (https://en.wikipedia.org/wiki/Syntonic_comma) and pythagorean comma (https://en.wikipedia.org/wiki/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 (https://www.patreon.com/posts/persian-glances-17963263).  And check out some of Jesse’s other music too (jessemaw.com).

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

567/512
9/8
5103/4096
81/64
21/16
729/512
189/128
3/2
1701/1024
27/16
7/4
15309/8192
243/128
63/32
2/1

))

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:

http://robertinventor.com/software/tunesmithy/help/cents_and_ratios.htm

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 🙂

https://www.patreon.com/posts/among-dreams-23-16966657

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 https://www.patreon.com/noahdeanjordan 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

Rant on Optimization and microtonal misconceptions

So I have found a train of comments of “microtonal” music that have a certain tendency, that while it is coming from a good place, I believe really just shares a different aesthetic of music than the goal of many composers with interest in tuning.

I will try to explain these two points of view, and they are easiest described as the optimization path and the colour path / free path.

The optimization best often hails 12edo to be the best for most circumstances, with the sometimes allowance for 19/31/53 etc, other high accuracy/resolution tunings, so long as the music uses these new intervals in a way dictated the the classical meantone tradition.  This train of thought believes that tuning to near Just Intonation / tuning accurately to harmonic approximations is the goal of a tuning system.

There are a few problems with this.  Firstly, of all the well-tempered systems for classical era music, there is obviously no “best” system, and each has its own way of dealing with progressions, especially in the context of the music of the composer and piece.  Secondly, if a perfect system is truly desired, we have adaptive just intonation, which is exactly that; the problem being the difficulty of application in many settings.  Thirdly, any high-resolution tuning is really just a combination of the two previous problems.

On the other hand, to choose tuning systems not on an overall optimization, but on compositional preference of the notes and patterns existent in a tuning, has no need to fulfil any requirements other than to be as it is.  This method also denies the optimization is really a possibility, as essentially each system has its own parts which are optimized.  Which do you want to be optimal?  Is this always in order from lowest to highest harmonics?

Yes, many things in 15 or 17 etc can be played in 12, and in 12 is sounds debatably “better”, but only if lack of beating is what qualifies as “better”.  Each tuning has a quality of its own, and “improves” certain notes, and really always at the expense of others.  This was the whole idea of well-temperaments in the past, and why there were so many.  The ability to choose the tuning of a piece, allows the composer to choose which colours to use and how they will relate.  We no longer need to base our aesthetic judgements on tempering the 5th to the 3rd.  Of course we can, and we can expand this as we wish.  But we are asking for infinities now.

St Joe and 2

I’m going to be posting today a cover of the song Saint Joe on the School Bus by Marcy Playground.

The song’s verses have a simple F#m A E progression.  But when playing it in 15 you have to decide which F# to use (if you visualize it like me and take the open E as a reference point).  So in this tuning there are 3 types of 2nd, and essentially two types of major 2nd, one at 160¢ and one at 240¢, conveniently located exactly opposite the 200¢ of 12edo.

And actually, in the case of this song, the small (140¢) major second between the E and the F# is the choice I use because it shares the C# chord tone with the A.

What is more interesting also is that in just intonation, the big and small major seconds are 9/8 and 10/9, and there are combined to make the just major 3rd of 5/4  – two unequal step sizes.  This is in fact a property of 15edo: that you need a 240¢ and a 160¢ to get your 400¢ major 3rd.  In this case, it is quite exaggerated from the 9/8 and the 10/9 (~204¢ and 182¢ respectively).

In 15edo as in the just system, the bigger whole tone (240¢ and 9/8 in these cases) is the difference between the unison and the major 2nd, and the distance between the fourth and the fifth, and the distance between the major sixth and major 7th.  The small tone (160¢ and 10:9 in these cases) is the distance between the major 2nd and major 3rd, and the fifth and the major 6th.

Here is the song.  For more please see the Patreon page

https://www.patreon.com/noahdeanjordan

Some thoughts on resolution

There are 3 types of 7th in 15edo, and as such, with the major 3rd and the lower 2 of these 7ths, we have 2 types of tritone. Both of these 7ths can resolve down 1 or 2 steps to a major or minor 3rd, as in 12 tone music.

Harmonically, when the neutral / middle 7th resolves down one step, the new root is 1 step higher than the previous. This means that the V is going to the augI instead of the I. This is balanced a bit by the fact that the V in this tuning is quite sharp.

I have made a video demo of this accessible from my page: https://www.patreon.com/noahdeanjordan

If all this is agreeable to you , we can then suggest that some of the following may be true:
1. the step size from the 7 of the V to the 3 of the I can be of a variable size, whether it resolves to a i min or a I maj.

2. the V chord could resolve to places other than the I or the vi

3. different tritones may have different tendencies

4. the just harmonic approximation of the tritone or the 7th may determine in part the quality of the resolution

– this last part begs a question:

In 15edo, the lower tritone is a 7/5 or 11/8 perhaps,

and the higher tritone a 16/11 or a 10/7, the undertone versions.

That means that in this tuning, the distinction between these tritone is not 7th harmonic vs 11th harmonic but overtone vs undertone.

On the other hand, the lowest of the 7ths in 15edo is a near 7/4, and the middle 7th a 11/6 or a 9/5 perhaps. And in any event, these are not inverses of each other, but definitely of different harmonic qualities.

So the questions are: does the harmonic limit of an interval affect the quality of the resolution? will the over/undertone nature of an interval affect its quality or functionality? how are these related? particularly if the tritone and the 7th within a resolution are of different harmonic limits. — furthermore, in this 15edo tuning we have a few possibilities for the interpretation of these senses:

1. In the 4:5:6:7 chord (major triad with the lowest 7th (7:4)) – we can interpret it as a 7:5 and a 7:4 or as a 11:8 and a 7:4… will this 7:4 predominance force us to hear the lower tritone as a 7:5? — YES, if this is a 4:5:6:7 chord. But if we don’t analyze this as a full form (because we are not in just intonation), the other approach could be possible.

side note:

to read 4:5:6:7, we are looking at a series of ratios

the ratio between the first two notes, the root and the major 3rd is 4:5 (5:4… 5/4)

the ratio between the 2nd two notes, the M3 and the fifth is 5:6 (6:5, min 3), and the ratio between the root and the fifth is 4:6 or 2:3 (3/2, P5)

the ratio between the final two is 6:7, between the 2nd and the final is 5:7 (7:5, the tritone in question),

and the ratio between the root and the final is 4:7, or 7:4 -> the note we define.
2. In the 12:15:18:22 chord with the neutral 7 (as a 11:6) on the major chord, we get the tritone defined in this sense as 15:22. If we want the large tritone to be an undertone, any of out close approximations of 10:7 or 16:11, we cannot fully analyze the chord this way.  The combination of an overtone and an undertone being analyzed in the same chord has difficulties.

This difficulty is shown in the different of complexity between writing a major and a minor triad.

4:5:6

and

10:12:15..

what happens is you want to write 5:6 (for the minor third), but must make the 2:3 as well, and as 5 is not a multiple of 2, we must multiply 2 to all.

 

More thoughts into this to come..