## The Golden Ratio as a musical interval

The golden ratio, also known as φ (phi) or approximately 1.618, is a number with some trippy properties. It’s no wonder that many people treat the golden ratio with a great deal of mysticism, because (here’s the cliche part) it appears repeatedly in nature and also crops up in many fields of mathematics. And we all know that mathematics is the language of the universe. Duuude.

So I want to talk a little about the golden ratio and how it might sound if we used it as a musical interval.

You may have seen or heard some youtube videos which exclaim “this is what the golden ratio sounds like!!” — These are almost always based on the decimal expansion of phi, where each digit is assigned to a note of the major scale. I won’t give an example here because it’s such an unimaginative Western-centric interpretation of nature’s finest ratio. Just imagine the background music for a shitty Google commercial, children and mothers playing, young trendy people smiling while looking into their laptops, and you get the idea.

Let’s have none of this. The truth is that the golden ratio, as a musical interval, is gritty, dirty, dissonant, inharmonic, and not remotely like you’d expect. And it’s explicitly microtonal.

Take for example John Chowning’s Stria, an important electronic work from 1977, using his then-new discovery of FM synthesis. The golden ratio is used as the interval between carrier and modulator, such that the resulting timbre is an inharmonic cloud of golden-ratio-related partials. To get a sense of what the golden ratio may sound like as a musical interval, start from here and let the sounds slowly work their way into your brain.

### Which golden ratio is the real golden ratio?

Here is something that causes confusion time and time again. There are two musical intervals which both claim to be the golden ratio! How is this possible?

Consider the octave in terms of cents. Cents are a musical measurement of pitch which divide the octave into 1200 logarithmically equal parts. This means that 200 cents is equivalent to an equal-tempered whole tone, 100 cents is equivalent to a semitone, 50 cents is equivalent to a quartertone, etc. So our first version of phi will simply divide 1200 by phi:

`1200 / φ = 741.6407865 cents`

~741 cents sounds like a horribly sharp fifth. Well, we already know that the golden ratio sounds dissonant, so this could be the golden ratio that we’re looking for. But first let’s look at the other contender:

`1200 * log2(φ) = 833.090296357`

While both of these intervals could be used to generate scales that you could play interesting music with, only one of them gives me that one-with-the-universe vibe. To find out why, we need to look at a psychoacoustic effect known as combination tones.

### Combination tones

It is a simple fact of psychoacoustics that any two tones that you play will produce additional combination tones. There are two types of combination tone: difference tone and sum tone. The sum tone is calculated by summing the frequencies of the two tones. The difference tone is calculated by subtracting the frequency of one tone from the other.

Let’s play A (440Hz) and E (660Hz) on the keyboard and work out these combination tones for ourselves.

```Difference tone = 660Hz - 440Hz = 220Hz

Sum tone = 660Hz + 440Hz = 1100Hz```

The difference tone sits an octave below the root note of A. This is the reason why you hear the missing fundamental being “filled in” by your brain when you play a power chord or a just major triad.

As for the sum tone, it is tuned to the 5th harmonic above that missing fundamental of 220Hz, because 220Hz*5=1100Hz. The 5th harmonic is a just major third plus two octaves. The sum tone is perhaps one of the reasons why the just major triad is such a stable and pleasing sonority.

Just to recap: If I play TWO tones (440Hz and 660Hz), your brain hears FOUR tones, (220Hz, 440Hz, 660Hz, 1100Hz). The effect is subtle, and the combination tones are heard a lot more softly than the real tones, but the effect can be perceived. Bearing that in mind, let’s work out the combination tones that appear when using the golden ratio interval.

We shall play a 1kHz tone and a ~1.618kHz tone. The interval between these two tones is the golden ratio of ~833 cents.

```Difference tone = ~1.618kHz - 1kHz = ~0.618kHz

Sum tone = 1kHz + ~1.618kHz = ~2.618kHz```

What’s interesting about these combination tones is that they are themselves related to the original tones by the golden ratio. This is easy to demonstrate:

```1kHz / ~0.618kHz = φ

~2.618kHz / ~1.618kHz = φ```

All four of these tones are related by the golden ratio! If we calculate the 2nd order combination tones, 3rd order, and so on, we’ll find the same thing again and again. Every combination tone is connected to some other tone by the golden ratio. This is exactly the recursion effect that we expect to find when we use the golden ratio properly.

However this landscape of recursive inharmonic partials can best be described as a chaotic and complex mess. This is an extreme contrast to the use of the golden ratio in visual proportions, paintings, architecture, flowers and nautilus shells, which most people would agree appear harmonious and pleasing. But that’s just the truth of it. The ear works in weird ways. Not all visual forms have an analogue as sound.

### Generating a scale based on the golden ratio

By now it should be clear that golden ratio music has some interesting properties that the Western equal-tempered scale could scarcely hope to reproduce. If you have some highly accurate micro-tunable instruments and perhaps some scale-designing software such as Scala, alt-tuner or LMSO, then your next step is to design a scale that features the golden ratio.

There are many possibilities when making your own phi-based scale, but today I will only give you one. Developing your own golden ratio scales is left to you as a rainy day exercise.

Almost all scales contain an interval of equivalence, and for most scales that interval is the octave. In octave-based systems, a note such as C appears many times, and each C is separated by one or more octaves.

By the way, an octave is a ratio of 2. If you play an A at 440Hz and then play the A’ one octave higher, the higher pitched note would be 880Hz.

For our scale, we want to pack in as many golden ratios as possible. There’s no room for any boring numbers such as 2. Indeed, our interval of equivalence will be the golden ratio itself. This means the scale will repeat after approximately 833 cents. This scale would fit snugly into the frequency space between 1kHz and ~1.618kHz.

Our next step is to divide this interval into smaller intervals (a & b). Each time we divide an interval into two intervals, those intervals should have the golden proportions.

Then take the biggest interval that remains in the scale and break that down into 2 intervals with the golden proportion. Repeat this process until you have enough pitch material to write music with.

I’ve done all the calculations, and the resulting scale (taken to 8 steps) is presented below in Scala format so that you may try it out for yourself:

```! sevish_golden.scl
!
Scale based on the golden ratio
8
!
121.546236174916
196.665941335636
318.212177510552
439.758413685468
514.878118846189
636.424355021105
711.544060181825
833.090296356741```

Some interesting notes about the above scale… When generating the values, I noticed that the scale was a Moment of Symmetry (MOS) after 3, 5, 8 steps. Those are Fibonacci numbers. I stopped after 8 steps because it will map very nicely to a piano keyboard with linear mapping. With this mapping, you will always hear the golden ratio when you play a minor sixth on the keyboard. Coincidentally, this tuning has some fair approximations to the fifth, fourth, supermajor third, minor third, and major second.

If you want to find more tunings with the golden ratio, you might look at Heinz Bohlen’s 833 cent scale for inspiration.

### The golden ratio as a spectral microtuning

When using scales that are based on the golden ratio, you will find that timbre makes a huge difference to the result. If you use an instrument that produces bright and clear harmonic overtones such as the violin or a sawtooth wave, you will hear an unruly clash of a harmonic timbre against an inharmonic scale.

Instead, you could take a similar approach to John Chowning and bake the golden ratio right into the timbre itself by using inharmonic overtones based on the golden ratio. Then suddenly your timbres and your scale will align, resulting in a much smoother sound. It’s just one more demonstration of the fact that tuning and timbre are deeply intertwined.

If you’re a computer musician, download a free copy of the Xen-Arts synth plugins. These instruments can produce inharmonic partials which can be controlled by the user. By default, they include a setting where each partial is tuned to some multiple of the golden ratio. These instruments could be a quick way to explore the topics I’ve been discussing in this post.

### Exercises

• Jam and improvise in the above golden ratio tuning until your ears tune into it.
• Apply the above tuning method to sequence your own golden ratio rhythms.
• Write a piece of music that combines both of the above.
• Make sarcastic YouTube comments saying that John Chowning’s music heals DNA.
• Change your desktop wallpaper to a picture of a spiral galaxy.
• Get your head around microtonal tunings by listening to my new album Harmony Hacker.

### 28 thoughts on “The Golden Ratio as a musical interval”

1. Chris Vaisvil

Hi Sean, with some help from Jake Freivald I took a different approach – taking the 17th root of Phi. To my ears the tuning is useful and fitted my DIY 17 note controller well (the reason to take the 17th root)

! E:\cakewalk\scales\17th_root_of_phi.scl
!
17th_root_of_phi
17
!
49.00531
98.01062
147.01594
196.02125
245.02656
294.03187
343.03718
392.04249
441.04781
490.05312
539.05843
588.06374
637.06905
686.07436
735.07968
784.08499
833.09030

1. Sevish Post author

What at first glance looks like a stretched 24-EDO is actually way out there, and tunes the golden ratio perfectly. :) Thanks for sharing this Chris, I’ll have to jam in this one as soon as I start my next creative cycle.

2. Other Paul

How about, instead of generating a polytonic scale by stacking Pythagorean fifths (i.e. keep multiplying frequencies by 3/2 as in 1, 3/2, 9/4, 27/8, 81/16 etc and then dividing these by 2s to get these rationals back into the octave range lying between 1 and 2 – i.e. 1, 3/2, 9/8, 27/16, 81/64 etc), generate one by stacking phiths (in much the same way as the western chromatic scale may be generated by successive multiples of 27/12).

Thus the root frequency is 1, the phith is φ, the phith of the phith is φ2/2, the phith of the phith of the phith is φ3/4 etc. Numbering from 0, the n-th ratio is thus φn/2m where m = floor(n log2(φ)) = integer part of (0.69424191363 n)

If you generate (say) 7 such pitches and re-order them, then the 7 ascending frequency ratios are [1, φ3/4, φ6/16, φ2/2, φ5/8, φ, φ4/4] which is approx [1, 1.059, 1.122. 1.309, 1.386, 1.618, 1.714]. If 1 represents C, then these pitches are closest to [C, C#, D, F, F#, A, A#] but with noticeably flat F#, A and A# (each about a quarter tone down).

Of course there’s no need to stop at 7, but it’s only after you’ve made the decision where to stop that you get to sort your sequence, not before.

1. Roee Sinai

A very nice idea indeed. You can also use that to approximate high limit JI, as you can use φ as an approximation to ratios of successive elements of the Fibonacci sequence. If you set φ ~ 13/8 ~ 21/13 ~ 34/21 ~ … you get every Fibonacci number starting from 8, and also every Lucas number starting from 18, because Ln = F2n/Fn (18 = 144/8, 29 = 377/13 etc.)
However, because 3^2 = 9 = 18/2 ~ φ^6 / 2, it is be a good idea to repeat not at the octave but at the tritone = 2^(1/2). Also, to add the prime 5 notice that the ratio between matching Lucas and Fibonacci numbers approaches the square root of 5. In this temperament this ratio is always φ^6 / 8 which means we can use its square as the approximation for 5.
This may give scales of varying sizes that despite being based on the dissonant golden ratio and tritone can create pretty consonant intervals (although I’d suggest not playing the golden ratio itself or its square in isolation, but octave equivalent intervals and octave complements should be fine). The 10 note MOS is quite similar to Bohlen’s 833cent scale, and even in the 26 note MOS every interval between 20/19 and 19/10 is an approximation of a 19-limit consonant. You can go to higher and higher limits with more and more notes up to 31-limit using the following map, for what can be thought of as 31-limit Semimothra:
⟨2 -1 -12 7 18 6 4 -4 -9 0 -22|
⟨0 3 12 -1 -8 1 3 9 13 7 23|
that was generated solely by looking at Fibonacci and Lukas numbers. You can also approximate 41 as φ^7 * 2^(1/2) since L10 = 123 = 41 * 3, but other paimes such as 37 and 43 cannot be deduced from the Fibonacci sequence alone. (This is a corollari of Zigmondi’s theorem on ℤ[φ])
You can also repeat at the octave to get a subgroup temperament for 2.9.5.21.11.13.17.19.69.29.31.

3. Jake Freivald

Interesting that you choose the 833-cent phi instead of the 742-cent phi. I’ve done a little work (very little work, none of it particularly fruitful) with a phi-based tuning, and I accepted the octave as an interval of equivalence to do it. Octave equivalence is natural, after all — although some people say they can get, e.g., the tritave to be an octave of equivalence, I can’t hear an octave and not perceive it as “the same not only higher” over the root. Phi-based tuning is odd enough on its own, so I didn’t feel the need to add the additional complexity of using a non-octave scale.

So I thought of the 742-cent phi dividing the octave up into proportional sections, and then repeating at the octave. Three octaves together might then be analogous to three panels on a wall, each one equivalent to the other, but the divisions inside each panel being related using the golden section.

One challenge with this way of thinking is in creating music where this “golden section of the octave” is obvious. I think five or eight notes is better than 13 or 21, in that case — using too many notes means we fall back on the ratios that the phi-based tuning approximates, like 5/4, 11/9, 13/11, and so on.

And by “we” I mean “I”, of course. Your mileage may vary. :)

4. Randy S

One such consideration could be to view the graphs of the constituent parts of the golden ratio [phi^1=0.5*(1+sqrt(5)) and phi^-2=0.5*(3-1*sqrt(5)); so for x=1 plot points at y=+1 and y=+sqrt(5), and for x=-2 plot points at y=+3 and y=-sqrt(5), continuing on with powers of phi being the independent variable] and see that the two sets of points produce two separate functions, y=(phi^x+phi^-x) and y=(phi^x-phi^-x), with each point set having alternating (even-odd) points lying on these two graphs.
This is very similar to how y=e^x can be drawn as the addition of y=cosh(x) and y=sinh(x) (respectively).
This shows that, because of the fact that negative powers of phi contain negative constituent parts [such as phi^-3 = 0.5*(2*sqrt(5)-4) and phi^-4 = 0.5*(7-3*sqrt(5)] similar to e^x, the midpoint of these two graphs produces the graph of y=phi^x. Noticeably, the graphs of y = cosh(x) and y = sinh(x) are described as y=0.5*(e^x+e^-x) and y=0.5*(e^x-e^-x) (respectively). This indicates that, similar to e^x, phi^x would actually be the sum of the two graphs y=0.5*(phi^x+phi^-x) and y=0.5*(phi^x-phi^-x).
If that hasn’t dissuaded anyone from continuing to see the point of all this yet, I’ll continue by just adding in that I had taken the time to look at the lines connecting the two different point sets [one set being whole number y-values, the other set being y-values that are whole multiples of sqrt(5)] to notice how they interact with one another and “cancel out” each other to “create” the tangible graph of phi^x.
The whole purpose of this, is to view the graph(s) as the creation principle (since all values of each of the powers of phi are sums of the two values of the powers before them, if that makes any sense to those who don’t know what phi is).
My own concept was to see where on the graph of y=phi^x the value would be 2, since the entire premise of creation is reaching powers of 2. This turns out (not surprisingly) to be equal to log(2)/log(phi)=1.4404200904…, so that would be the time(t) at which one cycle was complete. This means that, for every whole number multiple “n” of x=1.4404200904.. on the graph of y=phi^x, y=2^n.
This could be seen as similar to the division of cells over time, or the reproduction of a species with no limiting factors as proposed by Fibonacci when he derived his sequence from studying nature. However, I provide a disclaimer as to having no knowledge of IF cells divide every 1.4404200904.. seconds, that claim is not part of my entire claim.
What, then, would one do with this number (1 cycle/1.4404200904.. seconds = 0.6942419136306.. Hz)? What I did, was use it in an equation to produce a function which gives frequencies similar to (but sharper than) standard concert pitch. The resulting A4 note ended up being about 447.84128…Hz, which provides an uplifting feeling when playing harmonies.

5. Stephen Boyes

How about golden meantone! Where the ratio between a tone and a semitone equals phi (ie 1.618/1 instead of 2/1)
This produces a tempered diatonic scale, that continues in a spiral of fifths, that fill in the gaps with smaller and smaller intervals, in the proportion of phi.

This is related to the sequence
7 EDO
12 EDO
19 EDO
31 EDO
50 EDO

1. Roee Sinai

Golden meantone also includes the 742cent φ as it is its wolf fifth.
Another very nice fact about golden meantone that I noticed but didn’t find mentioned anywhere is that the sequence of smaller and smaller step sizes each of which is a golden ratio smaller than the one before it – a perfect fourth, a minor third, a whole tone, a diatonic semitone and a chromatic semitone approximate the ratios 4/3, 6/5, 9/8~10/9, 16/15 and 25/24, all of which are superparticular ratios whose numerator and demominator are products of close Fibonacci numbers – 4/3 = (2*2)/(1*3), 6/5 = (2*3)/(1*5), 9/8 = (3*3)/(1*8), 10/9 = (2*5)/(3*3), 16/15 = (2*8)/(3*5) and 25/24 = (5*5)/(3*8).
That is no coincidence, because these are in fact the ratios between ratios of consecutive Fibonacci numbers – 2/1, 3/2, 5/3 and 8/5 (except for 9/8 which is (6/5)/(16/15) = (3/1)/(8/3) = (2/1 * 3/2)/(5/3 * 8/5)), and the logarithmic ratios between these ratios of ratios indeed approach the golden ratio (a lot of ratios to keep track of) when using larger and larger Fibonacci numbers. This also means that golden meantone can be extended to include an approximation for the prime 13 by adding 13/8 to the list, and subsequently making 26/25 ~ 25/24 and adding 40/39 and 65/64 to the series of approximated decreasing steps with logarithmic φ between them. However, continuing to add 7 by adding 21/13 makes a 7 that is more complex and less accurate than the well known C-A# of septimal meantone. It breaks like that because the ratios of Fibonacci numbers approach the golden ratio, but the intervals you get from the golden meantone octave (2/1), fifth (3/2), major sixth (5/3), minor sixth (8/5), 13th harmonic (13/8) etc. generated by starting from the octave and then removing and adding every second step size (octave – fourth = fifth, fifth + tone = major sixth, etc.) approache an interval of 6/5 golden meantone fifths or 835.457368745468… cents, which is more than two cents sharp of the true golden ratio which is of course 833.090296356740… cents.
To get a scale that uses more of these infinite progression of decresing intervals you must make this series approach φ, and the way to do that is to not temper out the syntonic comma and start the base series of Fibonacci numbers’ ratios from 3/2 instead of from 2/1. This way the basic scale contains 5 tones of 10/9, 2 diationic semitones and three syntonic commas, which I like to arrange in the order tone-comma-tone-semitone-tone-comma-tone-tone-comma-semitone, to approximate the scale 1/1, 10/9, 9/8, 5/4, 4/3, 40/27, 3/2, 5/3, 50/27, 15/8, 2/1 that includes the JI major scale. Using the logarithmic geometric progression of ratios of Fibonacci ratios you can extend the segments between the commas to make the scale include the 13th, 7th, 17th and finally the 11th harmonic. I like to call these regions “flowers”, because like many flowers that have a Fibonacci number of petals, these regions in every iteration have a Fibonacci (or double Fibonacci) number of steps in them, and hence I call these scales “three flower scales”.
These scales in fact always have three more notes than the MOSes of golden meantone (and the EDOs that approach it), because of the three commas between the flowers. The 10 note scale suffices to approximate 5 limit harmony, it can be extended to 15 note 5-limit scales, with 22 notes you can get some 13 limit intervals but the 13th harmonic itself comes with 34 notes, 53 notes give you the 7th harmonic, 134 give you the 17th harmonic and 346 give you the 11th. I won’t suggest to go further since 89 is a quite large prime and 144 gives a mapping for 9 that contradicts the already existing one. (in fact a try to reconcile them brings you back to an imperfect golden ratio and makes a terrible and complex temperament that has, similarly to my try to extend golden meantone using 21, simpler and more accurate approximations for primes like 3 and 11 that means it was pointless to make a temperament that satisfies these constrains about them)
For the curious minds who want to continue beyond 55 to 144 and after that to deduce 19 from 2584 etc. you’ll need again to start your series one ratio further – with 5/3 instead of 3/2. This will give you 10 flower scales that should be able to approximate JI pretty accurately up to the 31st limit, with more primes such as 41 also available, somewhat similarly to the scales in my comment to Other Paul (albeit in a much more complex, though more accurate, way). However because of the large size and hence limited practicality of these scales I haven’t spent studying them yet.

1. Roee Sinai

I forgot to mention a method I found to tune these three flower scales that ensures the ratios of approximated Fibonacci numbers will indeed approach the golden ratio. This method relies on the fact that every element in that series is φ^2 times closer to φ logarithmically than the one before it. In particular, if we say that A = (M6th)/φ, then φ/(P5th) = A^(φ^2) and φ/(m6th) = A^(φ^-2). This means that if we multiply the frequency ratios of a perfect fifth, 3 major 6ths and a minor 6th we get φ^5 * A^(3-φ^2-φ^-2). It’s easy to check that φ^2 + φ^-2 = 3, and therefore a perfect fifth, three major sixths and a minor sixth shoud equal φ^5 exactly. This approximates the ratio of 3/2 * (5/3)^3 * 8/5 = 100/9, which is conveniently two octaves and two major sixths, which means the major sixth should be exactly φ^(5/2) / 2. This gives us the value of A which is φ^(3/2) / 2, and we can use it to find the correct approximation to 3/2, which, with the 5/3 we already have, is enough to deduce all the intervals in these scales, although I’d suggest to also calculate the expected value for 8/5, 13/8 etc. and compare that to the value you got to make sure we didn’t make any mistake. Also note that φ and A are used to tune the scale but are themselves not found in the scale, though their 5th powers are.
The 10 flower scales can be tuned similarly, setting the approximation of 832/75 (which also approximates 3549/320, 7497/676 etc.) to φ^5 instead of 100/9.

6. Douglas Blumeyer

Your scale is intriguing. It’s a hybrid of acoustic phi and logarithmic phi. I’d like to know if this was your intention when designing it. Let me try to clarify my perspective:

Bohlen’s 833 cent scale is purely acoustic, functioning like a JI scale except using phi instead of rational ratios. It focuses the recursive power of phi acoustically, i.e in the realm of frequency, on a cloud of combination tones.

Golden meantone (like all other golden moment of symmetry scales) is purely logarithmic, iterating a generator in order to divide a period up into step sizes which are in the golden proportion to each other. It focuses the recursive power of phi logarithmically, i.e. in the realm of pitch, on a scale for which every general step size is in the same proportion with other general step sizes.

But your scale includes a bit of both effects. Because you repeat the scale at acoustic phi instead of the octave, you get the the acoustic effect outside the scale, between notes of the same pitch class (up or down repetitions of the scale). But then you get the logarithmic effect within the scale, between notes of different pitch classes.

Furthermore, yours is also the only phi-based scale I’ve seen which entirely eschews the octave. There’s nothing binding them to the octave, but other logarithmic-based phi scales typically choose the octave as the period because they are not after combination tones, but rather spicing up the typical tempering objective to approximate low-limit JI intervals like 2/1 and 3/2 with a little fractally interval spice. And even Bohlen’s 833 cent scale involves the octave; while it repeats at phi instead of the octave, strangely the octave is still involved because the steps are created by stacking 833 cent intervals and octave-reducing them.

I have one more question: since you only discovered after the fact that your scale satisfied the conditions of a moment of symmetry scale, it seems you created your scale by manually breaking down intervals. Or did you use rabbit sequences to achieve this?

7. Charlie

The Xen-Arts website linked is down, and I can’t seem to find the new site or any mirrors of their instruments :(

1. Stewart Miller

Xen-Arts has been shut down for a little while now. If you’re looking for a synth that can manipulate harmonics in a similar fashion you can try ZynAddSubFx.

8. cat

dear sir have not read any of this except.yes. this is what am looking for some.geniiouss music. comment oppinion is ”we must conform to the universal in spite of our primate container.” ok let there be happy ballads … not.my thing to say.what is.music. area.of the brain similar.to.coca . contribute.here the idea of phi music as Time and light connected science what little is known about brain waves. proof arriving from better measuremnt of tools that measure.that. point pulses as.light in time. making.a.sustem as am. able.to.do that kaliedoscope. ie dodecqheron. right… tnx good enrrgy to yall

9. Ellen Corene Jones

This has all ben interesting, as one who is poor at mathematics but rich in reading & comprehension, I find I cannot comprehend the values each of you are explaining. Ironic the term comprehension becomes a general term with multi-meanings. The general term, ” Hurt your brain” comes to mind, I arrived at this saved bit of communication in the vast sea of information saved in the cloud of memory broken down into zeroes & ones for as long as the system retaining it is maintained, infinite, probably not as a flare from a greater force the Sun may infinitely end what is here, I applaud those of you who can understand these musings, and thank you all for taking me to a place in my thoughts I may not have explored! I hope to one day see mention of this perplexing issue of the true sound of Phi to be solved, for as the Universe is vast & everything no matter how infinitesimally small has beauty within, may hear phi as a beautiful sound to my ears, how can it not be harmonious, to think it would be gritty, harsh, grating and so unharmonious to hear with one’s ears simply cannot be…

10. xenon

Well, after studying some psychoacoustics, and how intervals like the octave, fifth and fourth are almost universal, with
Octave being ”the most” universal among different cultures and eras of earth’s musical history, my approach also had to include in some way the octave, so indeed the important interval here was 1200/φ= 741.6.. cents.

Jake, I don’t know what was your approach, after accepting this as your ‘φ’ generator, but I kept dividing by 1/φ up to an interval,
that was (my) fitting to some psychoacoustic thresholds (not so minute as to be almost unheard, or unimportant in the resulting, tuning/scale), So by keeping dividing with 1/φ, Ι came across to these intervals

(1200 cents= 12 semitones)

1200/Φ= 741.6,../Φ..=458.3…,283.2,,,175…,108.2,,,,66,8…

(In reverse order).

As the golden section is closely related to the Fibonacci sequence, which is also related to a simple division of numbers,.

it follows that each interval within the octave cycles, is the sum of the previous intervals :

Thus : 66.8…+108.2=175, 175..+108.2=283.2…so forth

Every interval is the sum of the two previous intervals, as in the Fibonacci series. (but without skipping any intervals in-between, like some other examples of the above φ tunings, (which of course use Fibonacci series but superimposed-making the scales a little more complex)

thus we have a heptatonic scale/tuning system with (i believe) has many possibilities of potential relations, usable for formulating a larger system with modulations, inversions, symmetries, etc

For anyone interested, this was my first effort to make a piece based on the ‘1200/phi-tuning, order’ :

https://phideusmusic1.bandcamp.com/track/agnoia-paranoia-agonia

The piece is free of course, and this is by no means a form of spam, into a page of a musician/programmer that I respect very much -and without his Sevish scale workshop, I couldn’t make anything of this.

SEVISH, your program, is what I was always trying to convince my programmer friends to make, but, much much better! Thanks for everything!
Any remarks by people here about how sound is the result or the theory behind this, I am all ears! (and eyes), and hopefully, if it is interesting enough to anyone like you Jake, i would be happy to modify, enhance this tun system/sound/scale etc

cheers!

11. David

This is the result… After some experimenting and improvising.

Produced in Reaper on Debian 11. 3 tracks are Surge XT, 1 track is sfizz. All tracks using the sevish_golden.scl tuning presented here.

https://soundcloud.com/oddy-o-lynx/eternal-golden-bow

12. Peter Vodden

After stumbling upon this posting, I began my own journey into the wonders of Golden Ratio intervals. Thank you so much for sharing your knowledge, Sevish. I use algorithmic and network-based MIDI generators connected to micro-tonal synths to create ambient soundscapes, often using the “Sevish” 833 tuning.

Playlist: Lovecraft Ambience 2

13. Mouseborg

If I may ask, how exactly was sevish_golden.scl derived? I know the basic method was stated but I think I am missing some detail.

14. empleat

It sounds inharmonic so? I don’t know anything about music, because i have chronic pain, but has there to be harmony for music to be considered a good music, isn’t it subjective??? I think it is amazing, I never heard something so mysterious/existential in awhile! It brings my reality to another level! I Am a philosopher with lots of existential feelings… Also hollyshit: it sounds so dark/sinister/dreamlike/mysterious and so on and so on…

15. Zephram

While 36-EDO provides a suitable approximation of Heinz Bohlen’s 833-cent scale, 49-EDO provides one for Sevish’s 833-cent scale.
As a coincidence, 49-EDO is also the optimal patent val for Superpyth, a temperament supported by 22-EDO, one of Sevish’s favorite EDOs!

16. william wesley

Keeping it simple since a minor 6th is involved in the major and minor authentic and plagal cadences and since the interval of 833 cents is a 33 cent sharp minor 6th what happens if the bass note in these cadences is dropped 33 cents? In just intonation the minor 6th is 8/5 which is 814 cents which is only a 19 cents flat minor 6th so same question, what happens if the bass note of the minor 6th is 19 cents flat in just intonation?
I’d like to hear an example.

17. Carl

You can mathematically prove that the golden ratio is the *least* pleasant sounding interval, starting from the premise that intervals are pleasing iff they approximate whole number ratios.

Proof: any ratio can be expressed as a continued fraction. (for example, pi = 3+1/(7+1/(15+1/(1+1/(…))))). We denote an arbitrary continued fraction as a0+1/(a1+1/(a2+1/(a3+1/(…)))), where ai is an positive whole number for any i≥0. The larger the value of any ai, the closer that continued fraction is to a whole number ratio (for example 1/2 = 1/(2+1/∞)), and the more pleasant the interval. Therefore, lower values of ai correspond to less pleasant intervals, and since ai is a positive whole number, it has a lowest possible value, that being 1. Therefore the least pleasant interval is the continued fraction where ai=1 for any i≥0. This is, by definition, the golden ratio.

Obviously not everyone will have the patience to follow along with the explanation above, there’s a video that can be easier to digest, it doesn’t discuss music but it might make it easier to see the reason why: https://www.youtube.com/watch?v=sj8Sg8qnjOg