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Strähle construction

A geometric construction that approximates fractional powers to lay out sounding lengths for equal-diameter, equal-tension strings in a rational tempered tuning.

Version
v1 · 2026-09-28 · History
Domain-specific #
12302
Domain group
Arts & Aesthetic Practice
Origin domain
Music & Musicology
Subdomains
Tuning Theory, Instrument Design → Music & Musicology
Aliases
Stråhle construction

Core Idea

Stråhle's construction converts a tuning problem into projective geometry. Given endpoint values and a mean proportional, an auxiliary line is divided into equal steps and rays from a constructed point project those divisions onto a line of sounding lengths. Similar triangles make the resulting points rational approximations to the fractional powers required for a tempered scale.

For uniform strings, frequency varies inversely with sounding length, so the projected ratios can guide keyboard stringing or fret placement. The method approximates equal temperament rather than calculating it exactly, and its bounded usefulness depends on the root range. Its history also demonstrates a crucial distinction between the construction and Faggot's erroneous numerical table, which used the wrong trigonometric lookup.

Structural Signature

Sig role-phrases:

  • endpoint lengths — encode the larger and smaller reference values whose fractional interpolation is sought It is essential. Counterfactual: Without calibrated endpoints, projected divisions have no tuning meaning.
  • mean proportional — fixes an intermediate geometric relation used to construct the approximation It is essential. Counterfactual: Removing it breaks the similar-triangle ratio on which the formula depends.
  • divided auxiliary line — represents equal steps of the exponent or scale It is essential. Counterfactual: Arbitrary division positions no longer correspond to successive pitch steps.
  • projection center and rays — map auxiliary divisions to approximate string-length locations It is essential. Counterfactual: The scale cannot be transferred to the string-length line without the projective mapping.
  • uniform string conditions — allow length ratios to stand for frequency ratios It is essential. Counterfactual: Different tension or linear density would invalidate the length-to-pitch conversion.
  • error comparison — distinguishes the rational construction from exact exponentiation and detects historical miscalculation It is diagnostic. Counterfactual: Ignoring error can misidentify a faulty table as the intended temperament.

What It Is Not

  • It is not an exact construction of equal temperament.
  • It is not Faggot's erroneous table of calculated pitches.
  • It is not valid for strings with uncorrected differences in tension or linear density.
  • It is not every geometric fret-layout method.
  • Closest near-miss. Equal-temperament fret placement is the closest practical neighbor, but Stråhle's output is an approximation with its own error pattern.

Scope of Application

  • Historical instrument making. Geometric layout supports clavichord and related string scales.
  • Tuning theory. Interval errors are compared with equal and rational temperaments.
  • Fractional-power approximation. Similar triangles yield a rational approximation to N raised to an intermediate exponent.
  • Fretted instruments. Projected positions can be adapted to fret spacing under uniform string assumptions.

Clarity

State the endpoints, division count, geometric points and lines, string assumptions, scaling convention, target temperament, and maximum pitch error. Keep Stråhle's instructions separate from Faggot's later calculation and from Barbour's generalized construction.

Manages Complexity

The construction replaces repeated root or logarithm calculations with a small geometric apparatus. That makes workshop use possible, but hides systematic approximation error and the physical assumptions relating length to frequency. Modern interpretation must restore both the algebra and the acoustics.

Abstract Reasoning

  1. Choose the endpoint ratio and number of equal exponent steps.
  2. Represent the two endpoint quantities as lengths on the base line.
  3. Construct the required mean proportional and auxiliary intersections.
  4. Divide the auxiliary segment into equal parts and project rays through them.
  5. Read the projected points as approximate string-length ratios.
  6. Convert lengths to pitches under uniform string conditions.
  7. Compare resulting intervals with the target temperament and report approximation error.

Knowledge Transfer

The projective approximation transfers from musical string lengths to fractional-power estimation when the same similar-triangle formula and error range are preserved. It stops at exact exponentiation or at physical strings whose tension and density vary without compensation. The cargo is a rational geometric approximation, not historical naming alone.

Examples

Applied / In Practice

A builder uses projected divisions to mark a sequence of sounding lengths for one tempered octave of uniform strings.

Mapped back: geometry → Rays transfer equal exponent steps into unequal length ratios.; acoustics → Uniform strings make inverse length track pitch..

Applied / In Practice

Barbour's similar-triangle formula approximates N^m for intermediate exponents between chosen endpoints.

Mapped back: approximation → The construction realizes a rational function rather than exact exponentiation..

Applied / In Practice

Faggot's published table uses the wrong trigonometric-table column and produces lengths inconsistent with Stråhle's instructions.

Mapped back: boundary → Historical association does not make erroneous output part of the construction..

Structural Tensions

T1 — Workshop Simplicity versus Mathematical Accuracy. A readily drawn construction avoids logarithmic calculation but introduces systematic tuning error.

Diagnostic: Quantify interval deviations before accepting the convenience as musically adequate.

T2 — Historical Source versus Later Reconstruction. The original instructions and later numerical tables diverged because of a calculation error.

Diagnostic: Reconstruct from the geometric operations and compare independently computed lengths, rather than copying the appended table.

Structural–Framed Character

The projected ratio and error curve are structural; judging the temperament acceptable is framed by musical practice and instrument tolerances. Historical transcription and calculation errors further show that a diagram's provenance does not guarantee mathematical fidelity.

Structural Core vs. Domain Accent

The skeleton is equal parameter steps mapped nonlinearly by projection. Tuning supplies pitch, cents, string length, tension, and temperament. Without that acoustic reading, the device is a fractional-power approximation rather than Stråhle's musical construction in use.

  • Approved root. Frozen DAG review leaves the construction unparented.

  • Related — equal temperament and mean proportional. They supply the target comparison and a key geometric element, but neither is the construction itself.

Neighborhood in Abstraction Space

Strähle construction sits in a crowded region of the domain-specific corpus (40th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Coordinate Systems & Spatial Measures (29 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With

  • Equal-temperament formula. Tell: Computes exact exponential ratios rather than Stråhle's rational approximation.
  • Faggot table. Tell: A historically appended but miscalculated output.
  • Rule of 18. Tell: A different practical fret-spacing approximation.
  • Geometric mean. Tell: One component of the construction, not its full projective sequence.

References

  • Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Str%C3%A4hle_construction (revision 1361638862).
  • Preserved source candidate: https://www.digitale-sammlungen.de/en/view/bsb10598999?page=187,188
  • Preserved source candidate: https://www.musikforskning.se/stm/STM1977/STM1977_1HeleniusOberg.pdf
  • Preserved source candidate: http://www.royalcourt.se/royalcourt/theroyalpalaces/stromsholmpalace/thepalace/history.4.396160511584257f21800017339.html
  • Preserved source candidate: http://www.musikmuseet.se/samlingar/detalj.php?l=sv&iid=172&str=
  • Preserved source candidate: https://books.google.com/books?id=yyIQAAAAYAAJ&pg=PA132&dq=polhem+orgel&lr=&as_brr=3&ei=5AL9Sq_qD5u0zASZ4LSjDw&client=firefox-a#v=onepage&q=polhem%20orgel&f=false–
  • Preserved source candidate: http://www.musikforskning.se/stm/STM1979/STM1979_1HeleniusOberg.pdf
  • Preserved source candidate: https://runeberg.org/nfbg/0678.html
  • Preserved source candidate: http://musikforskning.se/stm/STM1987/STM1987HeleniusOberg.pdf

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.