Concrete, Laplace Resonance and Heptatonic Musical Scales

My son turned 36 today while my granddaughter is aged 18 and I'm aged 72. I couldn't help noticing that the ratio of these ages 18 : 36 : 72 reduces to 1 : 2 : 4 in its simplest form.

This set me on a course to find out what was interesting, mathematically and otherwise, about this ratio. If you search for these proportions, the first thing that pops up is, oddly enough, concrete. See Figure 1.

Figure 1: source

Yes, it turns out that 1 : 2 : 4 is the ratio of cement : sand : aggregrate, by volume, required to make M15 concrete. It is also known as PCC (Plain Cement Concrete) and can be used in construction of Levelling course, bedding for footing, concrete roads, etc. The site from which the table in Figure 1 was taken has lots of calculations for figuring out weights etc. but I'm not going to go into here as it's rather boring.

When I searched for 1 : 2 : 4 planetary motion, I came up with something more interesting: 

A Laplace resonance is a three-body resonance with a 1 : 2 : 4 orbital period ratio (equivalent to a 4 : 2 : 1 ratio of orbits). The term arose because Pierre-Simon Laplace discovered that such a resonance governed the motions of Jupiter's moons Io, Europa, and Ganymede. It is now also often applied to other 3-body resonances with the same ratios, such as that between the extrasolar planets Gliese 876 c, b, and e. Three-body resonances involving other simple integer ratios have been termed "Laplace-like" or "Laplace-type".

Figure 2 shows a simulation of the Jovian moons' Laplace resonance (source):

Figure 2: The three-body Laplace resonance exhibited
by three of Jupiter's Galilean moons. Conjunctions are highlighted
by brief color changes.There are two Io-Europa conjunctions
(green) and three Io-Ganymede conjunctions (grey) for each
Europa-Ganymede conjunction (magenta). This diagram is not to scale.

The Laplace resonance is a particular type of orbital resonance and the eponymous Wikipedia article has a lots of examples of this and other types of resonances.

Intrinsic to the 1 : 2 : 4 proportions is the number 7. In fact (1, 2, 4) represents one of the 15 partitions of 7. This immediately to mind heptatonic musical scales:

A heptatonic scale is a musical scale that has seven pitches, or tones, per octave. Examples include:

• the major scale or minor scale in C major: C D E F G A B C

• the relative minor, A minor, natural minor: A B C D E F G A 

• the melodic minor scale, A B C D E F♯G♯A ascending, 

• the melodic minor scale, A G F E D C B A descending 

• the harmonic minor scale, A B C D E F G♯A 

• the Byzantine or Hungarian, scale, C D E♭ F♯ G A♭ B C.

So any partition of 7 can be viewed in terms of such a scale. Applying this to 1 : 2 : 4, there are six possible divisions. Figure 3 shows two of these:

Figure 3

I've written about musical scales more generally in my previous post titled Equal Temperament Tuning.