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What I Learned Writing an Album in Just Intonation

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Re: What I Learned Writing an Album in Just Intonation

#71

Post by bwanab »

elevation wrote: post #48

For practical performance of alternate intonations, you can tune a guitar so that a particular chord shape presents perfect roots and 5ths, while arranging for a single string to be a perfect third (e.g, a few cents "flat" compared to equal temperament.) When playing this same chord shape (or an appropriate derivative) up the neck, the relative intervals is maintained for other chords, allowing perfect intonation not…

Aren't you still fighting the fact that the frets themselves are tuned as 12TET?

Re: What I Learned Writing an Album in Just Intonation

#72

Post by pierrec »

pclmulqdq wrote: post #69
pierrec wrote: post #3

I'd love to get some feedback on this, because I'm not sure I've hit the original goal of this article. For a long time, the topics of just intonation, harmonic lattices, and of their relationship with temperaments were difficult for me to understand, despite reading the theory on them. It's not until I applied the concepts that could say I understood them. I tried writing the "missing article" that would have given…

If I may provide some constructive criticism as someone who also has a background in tuning and tuning theory: I believe the article is pretty cool and you clearly have a good background on modern microtonal music. However, it is also clear that you do not have a good background on historical tunings (there are some factual errors in this regard). Modern sources often implicitly assume that you understand tuning theo…

Thanks for bringing in your understanding of historical tunings. Maybe a simpler correction would be to describe the example as "a Pythagorean tuning" rather than "THE Pythagorean tuning", which would be (according to some) technically correct despite not being historically representative. The goal of that example isn't to provide historical background, it's to provide one of the simplest possible derivations of 12 usable tones from just intervals. I believe that that the 3 being a multiplier or a divisor is a matter of perspective: if you start from the last tone in my example, the same scale is derived from reversed fractions. Starting from the middle might be closer to what you're describing, hopefully I'm getting that right.

Grounding the whole thing in the harmonic series on would certainly make sense.

Re: What I Learned Writing an Album in Just Intonation

#73

Post by pclmulqdq »

pierrec wrote: post #72
pclmulqdq wrote: post #69

Earlier quoted context omitted.

If I may provide some constructive criticism as someone who also has a background in tuning and tuning theory: I believe the article is pretty cool and you clearly have a good background on modern microtonal music. However, it is also clear that you do not have a good background on historical tunings (there are some factual errors in this regard). Modern sources often implicitly assume that you understand tuning theo…

Thanks for bringing in your understanding of historical tunings. Maybe a simpler correction would be to describe the example as "a Pythagorean tuning" rather than "THE Pythagorean tuning", which would be (according to some) technically correct despite not being historically representative. The goal of that example isn't to provide historical background, it's to provide one of the simplest possible derivations of 12 u…

You are correct about the ratios not mattering that way, but it is jarring for someone who knows about this stuff to see 4/3 (the fourth) instead of 3/2 and to see the positioning of the wolf being the I-V relationship in the scale. I would suggest that if you want to keep it, you should be explicit that the frequency relationship on this fifth is far off, and I would suggest drawing a circle of fifths (which will be a lot more clear after seeing a fifth in the harmonic series). Instinctively, that section feels very wrong despite being (mostly) right. I would personally suggest dropping it entirely - you don't need the history.

Edit: I was thinking about your existing diagram a bit more, and here is the rework I might suggest without doing anything too major:

Do an up-a-fifth-down-a-fourth path through the chart, and that is easier to rationalize and understand than "mostly 4/3 with 2/3 ratios interspersed when we need to drop an octave":

C * 3/2 -> G

G * 2/3 -> D

D * 3/2 -> A

A * 2/3 -> E

E * 3/2 -> B

B * 2/3 -> F#

There is only one break from the pattern right here with two descending perfect fourths in a row:

F# * 2/3 -> C#

C# * 3/2 -> G#

...

You can also restart from C going down to F and Bb once you reach F# and that gives you the typical Pythagorean tuning, but then you will have to explain why that happens to readers who are unfamiliar with tuning.

Re: What I Learned Writing an Album in Just Intonation

#74

Post by cjbprime »

wrs wrote: post #67
cscheid wrote: post #64

Earlier quoted context omitted.

(In case you didn’t know, it’s fun that) Jacob Collier plays a 5-string guitar in a tuning that takes advantage of this trick exactly. IIRC, from lowest to highest he tunes them a fifth, a fifth, a fourth and a fourth apart. By flatting the middle string a dozen cents or so, you can get a root-fifth-perfect tenth voicing of a major chord. Neat trick.

Speaking of Jacob Collier, this is a good time to mention the song where he “hacks” equal temperament to modulate to G-half-sharp without anyone noticing. https://www.youtube.com/watch?v=Xd54l8gfi7M&t=3m17s

One more Jacob Collier mention: also generally releases albums using just intonation, I think, and especially does hand-tuning of thirds etc. I like his description of having to "monkey swing" when an e.g. flattened third needs to become a new root note.

Re: What I Learned Writing an Album in Just Intonation

#75

Post by cjbprime »

jfengel wrote: post #54
wonger_ wrote: post #7

Earlier quoted context omitted.

I am very unfamiliar with the topic. Here's my takeaways from the article: - almost all music I've heard uses 12TET - JI is microtonal / an alternative tuning to 12TET - you're the author of the lovely ambient.garden, which uses JI - the audio widgets and visualizations were very helpful - JI has perfect harmonies, and 12TET approximates those enough to still sound good - JI opens up cool chord options - but there's…

The key to the article, and to all of Western music, is about a set of weird mathematical coincidences involving the twelfth root of two. In particular, that twelfth-root-of-two to the fourth power is almost exactly 4/3, and twelfth-root-of-two to the fifth power is is almost exactly 3/2. And, of course, twelfth-root-of-two to the 12th power is exactly 1. For some reason, not entirely well understood, when you play f…

I think you're maybe missing the main motivation, which is that "correctly tuned" overtones are in nature, in physics, in all sounds -- they aren't a social construct, unlike everything else about music. I think the argument from JI advocates is that our body's perception knows very well what a justly-tuned chord sounds like, etc.

Re: What I Learned Writing an Album in Just Intonation

#76

Post by klik99 »

banana-lambda wrote: post #60
klik99 wrote: post #41

Earlier quoted context omitted.

I think you need some info in the start of the article about why 12tet was invented, what problems it was solving and how it fails. Like - In Bachs day you'd have to retune the instrument if you changed key signatures and you could only modulate a few steps away before the ratios sounded really bad. So 12TET opens up the possibilities to modulate far away from your original and requires less retuning, at the cost of…

> each key signature in 12TET has a unique sound (Like CMaj vs EMaj) due to inconsistent ratios in perfect fifths I thought this wasn't true in 12TET, because in that case a perfect fifth interval is always a ratio of the twelfth root of two to power seven, or almost 3/2, no matter the starting point, C or E.

Sorry, that was an incorrect thing I was taught and I never checked it... The ratio of a perfect fifth in 12TET is always 1.498307, not exactly 1.5, but it's close enough to 3/2. The deviation from "simple ratios" is more or less depending on the interval (P5(3/2): -0.001692923, M3(5/4): 0.00992105, m3(6/5): -0.010792885).

Re: What I Learned Writing an Album in Just Intonation

#78

Post by naltroc »

riemannzeta wrote: post #77

I feel like I can hear beat frequencies in the 12TET chords that I don't hear in the JI chord examples. Anybody else hearing that? Or is that my imagination?

Yes!

We don't even have to listen to the audio examples to know that yes, ET chords produce beat frequencies. This is an artifact of the non-integer relationships created in ET chords that don't exist in the JI examples.

A piano will produce similar beat frequencies when playing a perfect fifth. Some ears are more sensitive to hearing it than others.

Re: What I Learned Writing an Album in Just Intonation

#79

Post by riemannzeta »

Readers who enjoyed this essay might also enjoy this one, which goes into further depth on the mathematics of musical intervals, the connection between the "blue note" in Equal Tempered jazz music, and more:

https://iopscience.iop.org/article/10.1088/0034-4885/72/7/07...

As an added bonus, the author draws an extended analogy between musical intervals and energy levels in quantum mechanics toward the end of the article, and indeed the study of the patterns in intervals between and among energy levels in the simple harmonic osciallator remains an active area of reasearch!

Re: What I Learned Writing an Album in Just Intonation

#80

Post by riemannzeta »

pierrec wrote: post #3

I'd love to get some feedback on this, because I'm not sure I've hit the original goal of this article. For a long time, the topics of just intonation, harmonic lattices, and of their relationship with temperaments were difficult for me to understand, despite reading the theory on them. It's not until I applied the concepts that could say I understood them. I tried writing the "missing article" that would have given…

I will admit that I already had some familiarity with the topic, but I feel you did an amazing job at explaining specifically the constraints you're under as a composer, and in elucidating how having digital tools doesn't magically solve the problems you face in harmonizing in just intonation over the course of a piece. I had not really understood that before reading your article, and at least suspected that we might transition to hearing mostly Just Intonation eventually as digital tools became more widely used.

Of course you could make it more accessible by starting from some even more basic principles and elaborating further on those foundations, but then you risk losing some of our audience that's already familiar with those. For me at least you struck a good balance. (No pun intended!)

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