Tag Archives: vibration

Deadly screw sizes, avoid odd numbers and UNF.

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The glory of American screws and bolts is their low cost ubiquity, especially in our coarse thread (UNC = United National Coarse) sizes. Between 1/4 inch and 5/8″, they are sized in 1/16″ steps, and after that in 1/8″ steps. Below 3/16″, they are sized by wire gauges, and generally they have unique pitch sizes. All US screws and bolts are measured by their diameter and threads per inch. Thus, the 3/8-16 (UNC) has an outer diameter (major diameter) of 3/8″ with 16 threads per inch (tpi). 16 tpi is an ideal thread number for overall hold strength. No other bolt has 16 threads per inch so it is impossible to use the wrong bolt in a hole tapped for 3/8-16. The same is true for basically every course thread with a very few exceptions, mainly found between 3/16″ and 1/4″ where the wire gauges transition to fractional sizes. Because of this, if you stick to UTC you are unlikely to screw up, as it were. You are also less-likely to cross-thread.

I own one of these. It’s a tread pitch gauge.

US fine threads come in a variety of standards, most notably UNF = United National Fine. No version of fine thread is as strong as coarse because while there are more threads per inch, each root is considerably weaker. The advantage of fine treads is for use with very thin material, or where vibration is a serious concern. The problem is that screwups are far more likely and this diminishes the strength even further. Consider the 7/16″ – 24 (UNF). This bolt will fit into a nut or flange tapped for 1/2″- 24. The fit will be a little loose, but you might not notice. You will be able to wrench it down so everything looks solid, but only the ends of the threads are holding. This is a accident waiting to happen. To prevent such mistakes you can try to never allow a 7/6″-24 bolt into your shop, but this is uncomfortably difficult. If you ever let a 7/6″-24 bolt in, some day someone will grab it and use it, in all likelihood with a 1/2″ -24 nut or flange, since these are super-common. Under stress, the connection will fail in the worst possible moment.

Other UNF bolts and nuts present the same screwup risk. For example, between the 3/8″-24 and 5/16″-24 (UNF), or the #10-32 (UNF) and also with the 3/16″- 32, and the latter with the #8-32 (UNC). There is also a French metric with 0.9mm — this turns out to be identical to -32 pitch. The problem appears with any bolt pair where with identical pitch and the major diameter of the smaller bolt has a larger outer diameter (major diameter) than the inner diameter (minor diameter) of the larger bolt. If these are matched, the bolts will seem to hold when tightened, but they will fail in use. You well sometimes have to use these sizes because they match with some purchased flange. If you have to use them, be careful to use the largest bolt diameter that will fit into the threaded hole.

Where I have the option, my preference is to stick to UNC as much as possible, even where vibration is an issue. In vibration situations, I prefer to add a lock nut or sometimes, an anti-vibration glue, locktite, available in different release temperatures. Locktite is also helpful to prevent gas leaks. In our hydrogen purifiers, I use lock washers on the ground connection from the power cord, for example.

I try to avoid metric, by the way. They less readily available in the US, and more expensive. The other problem with metric is that there are two varieties (Standard and French — God love the French engineering) and there are so many sizes and pitches that screwups are common. Metric bolts come in every mm diameter, and often fractional mm too. There is a 2mm, a 2.3mm, a 2.5mm, and a 2.6mm, often with overlapping pitches. The pitch of metric screws and bolts is measured by their spacing, by the way, so a 1mm metric pitch means there is 1mm between threads, the the equivalent of a 24.5 pitch in the US, and a 0.9mm pitch = US-32. Thread confusion possibilities are endless. A M6x1 (6mm OD x 1mm pitch) is easily confused with a M5x1 or a M7x1, and the latter with the M7.5×1. A M8x1.25 is easily confused with a M9x1.25, and a M14x2 with an M16x2. And then there is confusion with US bolts: a 2.5mm metric pitch is nearly identical to a US 10tpi pitch. I can not rid myself of US threads, so I avoid metric where I can. As above, problems arise if you use a smaller diameter bolt in a larger diameter nut.

For those who have to use metric, I suggest you always use the largest bolt that will fit (assuming you can find it). I try to avoid bringing odd-size bolts into their shop, that is, stick to M6, M8, M10. It’s not always possible, but it’s a suggestion. I get equipment with odd-size metric bolts too. My preference is to stick to UNC and to avoid odd numbers.

Robert Buxbaum, January 23, 2024. Note: I’ve only really discussed bolt sizes between about #4 and 1″, and I didn’t consider UNRC or UNJF or other, odd options. You can figure these issues out yourself from the above, I think.

Musical Color and the Well Tempered Scale

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by R. E. Buxbaum, (the author of all these posts)

I first heard J. S. Bach’s Well Tempered Clavier some 35 years ago and was struck by the different colors of the different scales. Some were dark and scary, others were light and enjoyable. All of them worked, but each was distinct, though I could not figure out why. That Bach was able to write in all the keys without retuning was a key innovation of his. In his day, people tuned in fifths, a process that created gaps (called wolf) that prevented useful composition in affected keys.

We don’t know exactly how Bach tuned his instruments as he had no scientific way to describe it; we can guess that it was more uniform than the temper produced by tuning in fifths, but it probably was not quite equally spaced. Nowadays electronic keyboards are tuned to 12 equally spaced frequencies per octave through the use of frequency counters.  Starting with the A below “middle C”, A4, tuned at 440 cycles/second (the note symphonies tune to), each note is programmed to vibrate at a wavelength that is lower or higher than one next to it by a factor of the twelfth root of two, 12√2= 1.05946. After 12 multiples of this size, the wavelength has doubled or halved and there is an octave. This is called equal tempering.

Currently, many non-electric instruments are also tuned this way.  Equally tempering avoids all wolf, but makes each note equally ill-tempered. Any key can be transposed to another, but there are no pure harmonies because 12√2 is an irrational number (see joke). There is also no color or feel to any given key except that which has carried over historically in the listeners’ memory. It’s sad.

I’m going to speculate that J.S. Bach found/ favored a way to tune instruments where all of the keys were usable, and OK sounding, but where some harmonies are more perfect than others. Necessarily this means that some harmonies will be less-perfect. There should be no wolf gaps that would sound so bad that Bach could not compose and transpose in every key, but since there is a difference, each key will retain a distinct color that JS Bach explored in his work — or so I’ll assume.

Pythagoras found that notes sound best together when the vibrating lengths are kept in a ratio of small numbers. Consider the tuning note, A4, the A below middle C; this note vibrates a column of air .784 meters long, about 2.5 feet or half the length of an oboe. The octave notes for A4 are called A3 and A5. They vibrate columns of air 2x as long and 1/2 as long as the original. They’re called octaves because they’re eight white keys away from A4. Keyboards add 4 black notes per octave so octaves are always 12 notes away. Keyboards are generally tuned so octaves are always 12 keys away. Based on Pythagoras, a reasonable presumption is that J.S Bach tuned every non-octave note so that it vibrates an air column similar to the equal tuning ratio, 12√2 = 1.05946, but whose wavelength was adjusted, in some cases to make ratios of small, whole numbers with the wavelength for A4.

Aside from octaves, the most pleasant harmonies are with notes whose wavelength is 3/2 as long as the original, or 2/3 as long. The best harmonies with A4 (0.784 m) will be with notes with wavelengths (3/2)*0.784 m long, or (2/3)*0.784m long. The first of these is called D3 and the other is E4. A4 combines with D3 to make a chord called D-major, the so-called “the key of glory.” The Hallelujah chorus, Beethoven’s 9th (Ode to Joy), and Mahler’s Titan are in this key. Scriabin believed that D-major had a unique color, gold, suggesting that the pure ratios were retained.

A combines with E (plus a black note C#) to make a chord called A major. Songs in this key sound (to my ear) robust, cheerful and somewhat pompous; Here, in A-major is: Dancing Queen by ABBA, Lady Madonna by the Beatles; Prelude and Fugue in A major by JS Bach. Scriabin believed that A-major was green.

A4 also combines with E and a new white note, C3, to make a chord called A minor. Since E4 and E3 vibrate at 2/3 and 4/3 the wavelength of A4 respectively, I’ll speculate that Bach tuned C3 to 5/3 the length of A4; 5/3*.0784m =1.307m long. Tuned his way, the ratio of wavelengths in the A minor chord are 3:4:5. Songs in A minor tend to be edgy and sort-of sad: Stairway to heaven, Für Elise, “Songs in A Minor sung by Alicia Keys, and PDQ Bach’s Fugue in A minor. I’m going to speculate the Bach tuned this to 1.312 m (or thereabouts), roughly half-way between the wavelength for a pure ratio and that of equal temper.

The notes D3 and E3 will not sound particularly good together. In both pure ratios and equal tempers their wavelengths are in a ratio of 3/2 to 4/3, that is a ratio of 9 to 8. This can be a tensional transition, but it does not provide a satisfying resolution to my, western ears.

Now for the other white notes. The next white key over from A4 is G3, two half-tones longer that for A4. For equal tuning, we’d expect this note to vibrate a column of air 1.059462 = 1.1225 times longer than A4. The most similar ratio of small whole numbers is 9/8 = 1.1250, and we’d already generated one before between D and E. As a result, we may expect that Bach tuned G3 to a wavelength 9/8*0.784m = .88 meters.

For equal tuning, the next white note, F3, will vibrate an air column 1.059464 = 1.259 times as long as the A4 column. Tuned this way, the wavelength for F3 is 1.259*.784 = .988m. Alternately, since 1.259 is similar to 5/4 = 1.25, it is reasonable to tune F3 as (5/4)*.784 = .980m. I’ll speculate that he split the difference: .984m. F, A, and C combine to make a good harmony called the F major chord. The most popular pieces in F major sound woozy and not-quite settled in my opinion, perhaps because of the oddness of the F tuning. See, e.g. the Jeopardy theme song, “My Sweet Lord,” Come together (Beetles), Beethoven’s Pastoral symphony (Movement 1, “Awakening of cheerful feelings upon arrival in the country”). Scriabin saw F-major as bright blue.

We’ve only one more white note to go in this octave: B4, the other tension note to A4. Since the wavelengths for G3 was 9/8 as long as for A4, we can expect the wavelength for B4 will be 8/9 as long. This will be dissonant to A4, but it will go well with E3 and E4 as these were 2/3 and 4/3 of A4 respectively. Tuned this way, B4 vibrates a column 1.40 m. When B, in any octave, is combined with E it’s called an E chord (E major or E minor); it’s typically combined with a black key, G-sharp (G#). The notes B, E vibrate at a ratio of 4 to 3. J.S. Bach called the G#, “H” allowing him to spell out his name in his music. When he played the sequence BACH, he found B to A created tension; moving to C created harmony with A, but not B, while the final note, G# (H) provided harmony for C and the original B. Here’s how it works on cello; it’s not bad, but there is no grand resolution. The Promenade from “Pictures at an Exhibition” is in E.

The black notes go somewhere between the larger gaps of the white notes, and there is a traditional confusion in how to tune them. One can tune the black notes by equal temper  (multiples of 21/12), or set them exactly in the spaces between the white notes, or tune them to any alternate set of ratios. A popular set of ratios is found in “Just temper.” The black note 6 from A4 (D#) will have wavelength of 0.784*26/12= √2 *0.784 m =1.109m. Since √2 =1.414, and that this is about 1.4= 7/5, the “Just temper” method is to tune D# to 1.4*.784m =1.098m. If one takes this route, other black notes (F#3 and C#3) will be tuned to ratios of 6/5, and 8/5 times 0.784m respectively. It’s possible that J.S. Bach tuned his notes by Just temper, but I suspect not. I suspect that Bach tuned these notes to fall in-between Just Temper and Equal temper, as I’ve shown below. I suspect that his D#3 might vibrated at about 1.104 m, half way between Just and Equal temper. I would not be surprised if Jazz musicians tuned their black notes more closely to the fifths of Just temper: 5/5 6/5, 7/5, 8/5 (and 9/5?) because jazz uses the black notes more, and you generally want your main chords to sound in tune. Then again, maybe not. Jimmy Hendrix picked the harmony D#3 with A (“Diabolus”, the devil harmony) for his Purple Haze; it’s also used for European police sirens.

To my ear, the modified equal temper is more beautiful and interesting than the equal temperament of todays electronic keyboards. In either temper music plays in all keys, but with an un-equal temper each key is distinct and beautiful in its own way. Tuning is engineering, I think, rather than math or art. In math things have to be perfect; in art they have to be interesting, and in engineering they have to work. Engineering tends to be beautiful its way. Generally, though, engineering is not perfect.

Summary of air column wave-lengths, measured in meters, and as a ratio to that for A4. Just Tempering, Equal Tempering, and my best guess of J.S. Bach's Well Tempered scale.

Summary of air column wave-lengths, measured in meters, and as a ratio to that for A4. Just Tempering, Equal Tempering, and my best guess of J.S. Bach’s Well Tempered scale.

R.E. Buxbaum, May 20 2013 (edited Sept 23, 2013) — I’m not very musical, but my children are.