Major Limma¶

A just-intonation interval of frequency ratio 135:128, approximately 92.18 cents, derivable as the difference between a 9:8 ditone and a 6:5 minor third.

Version: v1 · 2026-09-28 · History
Domain-specific #: 7691
Domain group: Arts & Aesthetic Practice
Origin domain: Music & Musicology
Subdomain: Tuning Theory → Music & Musicology
Aliases: Greater Limma

Core Idea¶

The Major Limma is a musical interval with the just-intonation frequency ratio 135:128.^[1] It spans approximately 92.18 cents. Its identity is the exact ratio, while the cents value is a logarithmic approximation useful for comparing it with intervals in other tuning systems.

If the lower pitch has frequency \(f\), the upper pitch has frequency \(135f/128\). Transposing both pitches by the same factor preserves the interval because their ratio remains constant. Absolute pitch, instrument, register, and timbre are not part of the mathematical identity.

The interval can be derived as the difference between a ditone made from two 9:8 major tones and a 6:5 just minor third.^[2] The ditone is \((9/8)^2=81/64\). Dividing it by \(6/5\) gives \((81/64)/(6/5)=405/384=135/128\). “Difference” in interval arithmetic means ratio division, not subtraction of frequencies.

Cents translate a ratio \(r\) through \(1200\log_2(r)\). For 135:128 the result is about 92.18 cents. The cent number is not a second definition: rounding can vary, while the rational interval remains exact.

The major limma is smaller than a 100-cent semitone in twelve-tone equal temperament.^[3] The approximately 7.82-cent discrepancy matters in sustained tones, beating relations, adaptive tuning, and notation systems that preserve just ratios. It may be negligible or intentionally tempered out in other contexts.

“Limma” has multiple historical and tuning-theoretical uses. The Pythagorean limma is 256:243, about 90.22 cents, and is not the major limma.^[4] Qualifiers and ratios are therefore indispensable. A source saying only “limma” may refer to a different interval.

The adjective “major” does not mean a standard major interval such as a major second under common-practice classification. It distinguishes this limma from neighboring small intervals in particular taxonomies. The exact ratio should accompany the name whenever ambiguity is possible.

The major limma can also be described through the syntonic comma and a just chromatic semitone. The syntonic comma is 81:80, about 21.51 cents. Multiplying it by 25:24, about 70.67 cents, yields \((81/80)(25/24)=2025/1920=135/128\). This decomposition shows how the interval relates to 5-limit just intonation.

Prime-factor notation exposes its structure: \(135/128=3^3\times5/2^7\). The interval lies in 5-limit just intonation because its prime factors do not exceed five.^[5] This is a tuning-theory classification, not a claim that performers consciously calculate factors while playing.

Intervals compose multiplicatively. Reversing the major limma gives 128:135. Adding it repeatedly multiplies ratios and adds cents. Octave equivalence may reduce a compound result by powers of two. Calculations must state whether ratios are directed and octave-reduced.

Notation is conventional. Composer Ben Johnston's extended just-intonation notation uses accidentals that can combine comma adjustments with familiar staff positions.^[6] The frozen article reports a composite accidental for raising or lowering by the major-limma amount. That symbol belongs to Johnston's system rather than constituting the interval itself.

Other microtonal notations may spell the same sounding ratio differently or use the same visual accidental for a tempered approximation. A notation–tuning mapping must be declared. Staff spelling alone does not prove the realized frequency ratio.

On fixed-pitch equal-tempered instruments, the exact ratio may be approximated by available steps. In twelve-tone equal temperament the nearest chromatic semitone is 100 cents. In other equal divisions, a different number of steps may approximate 92.18 cents more closely. Approximation error should be quantified rather than calling every near-semitone the major limma.

In flexible-pitch vocal or instrumental performance, realization depends on reference pitch, harmonic context, listening, and ensemble practice. A score can specify 135:128 while performance fluctuates. The abstraction distinguishes intended tuning relation from every instantaneous measurement.

The audible character of a small interval depends on spectrum, duration, register, level, context, and training. Two complex tones at a 135:128 fundamental ratio can create beating patterns unlike pure tones. The mathematical interval does not determine one universal perceptual quality.

The node is useful in composition and analysis because it preserves a reusable transformation. A pitch can be adjusted by the ratio, a harmony can be decomposed into interval relations, and tuning discrepancies can be traced through comma arithmetic. These operations recur across works and instruments.

Structural Signature¶

Sig role-phrases:

the directed pitch pair — two tones whose upper-to-lower frequency relation is evaluated in a declared orientation.
the exact interval ratio — 135:128 as the identity preserved across changes of absolute pitch, register, instrument, and timbre.
the transposition invariance — common scaling of both pitch frequencies that leaves their ratio unchanged.
the ditone-minus-third derivation — interval division of 81:64 by 6:5, yielding 135:128 rather than subtracting frequencies.
the logarithmic image — approximately 92.18 cents obtained from the exact ratio for additive comparison with other tuning steps.
the 5-limit factorization — the prime structure 3³ × 5 / 2⁷ that locates the interval in 5-limit just intonation.^[7]
the directed inverse — 128:135 before any octave-equivalent reduction of the descending relation.
the notation mapping — a declared microtonal system's symbol for the adjustment, kept distinct from the interval it represents.
the practical realization — the performed or tempered pitch relation assessed against the rational target under a stated tolerance.
the approximation error — the signed cent difference between 135:128 and an available equal-tempered or performed substitute.
the exact-identity boundary — the exclusion of the Pythagorean limma, a 100-cent semitone, or an unqualified near-92-cent interval.^[8]

What It Is Not¶

Not the Pythagorean limma. The major limma is exactly 135:128, whereas the Pythagorean limma is 256:243; proximity in size and shared use of “limma” do not erase the ratio identity.
Not the equal-tempered semitone. Its approximately 92.18-cent logarithmic image is distinct from the 100-cent step of twelve-tone equal temperament, even when the latter is used as a practical approximation.^[9]
Not a quarter tone, syntonic comma, or Breedsma. Each neighboring microinterval has a different exact ratio or tempered definition; a generic “small interval” description is insufficient.
Not defined by one composer's accidental. Ben Johnston's symbol is a notation mapping for the interval in a particular system, not the constitutive frequency relation itself.
Not obtained by subtracting absolute frequencies. Musical interval composition and the ditone-minus-third derivation operate on frequency ratios multiplicatively, with cents providing the corresponding logarithmic difference.
Not every performance measurement near 92 cents. The named interval preserves 135:128; a realized or tempered pitch relation qualifies only as an approximation under an explicitly stated tolerance.

Scope of Application¶

The Major Limma is a tuning-theory identity bounded by the directed pitch-frequency ratio 135:128.^[10] Its literal habitats are musical settings that preserve that ratio exactly or declare a measured or tempered approximation to it; a bare numerical ratio outside pitch organization is not the interval.

Just-intonation analysis. The interval is located in 5-limit tuning through its factorization and can be derived as an 81:64 ditone divided by a 6:5 minor third.
Interval arithmetic. Tuning calculations compose, invert, and octave-reduce the ratio, with cents supplying an additive comparison while the exact rational identity remains primary.
Microtonal composition. Composers can use 135:128 as a directed pitch adjustment, harmonic relation, or fine structural difference when the score or tuning plan declares the intended ratio.
Microtonal notation. Johnston's or another declared notation system may encode the interval, but the symbol qualifies only through an explicit mapping to 135:128 rather than by its graphic form alone.
Temperament and tuning-system comparison. Analysts compare the approximately 92.18-cent interval with the 100-cent twelve-tone semitone or steps in other equal divisions and report the signed approximation error.
Instrument tuning and performance realization. Fixed- and flexible-pitch practice can target or approximate the ratio under a stated reference pitch, harmonic context, direction, and tolerance.
Performance and psychoacoustic analysis. Measured frequency pairs, beating, and perceptual salience may be assessed against the rational target, while spectrum, duration, register, and listening context remain separate conditions.

Clarity¶

Naming the major limma as 135:128 removes an ambiguity that the word limma alone cannot. It distinguishes this exact 5-limit interval from the Pythagorean limma (256:243), the 100-cent twelve-tone semitone, and any performed pitch that merely falls near 92 cents. It also separates identity from representation: 92.18 cents is an approximate logarithmic measure of the ratio, while a Johnston accidental is one notation system's instruction for realizing it.

The distinction makes interval arithmetic auditable. “The difference between a ditone and a minor third” means division of their frequency ratios, not subtraction of pitches or hertz, so ((9/8)^2/(6/5)=135/128). The better tuning question is: which exact ratio is intended, how is it derived or notated in the declared system, and what error is introduced by the instrument's or performer's actual realization?

Manages Complexity¶

Major limma packages a recurring tuning discrepancy into the exact unit 135:128, so a theorist need not rederive the relation from every scale spelling or frequency pair. The analyst tracks the directed ratio, octave reduction, and the tuning system in which intervals are composed. Multiplication and division then expose equivalent constructions—such as an 81:64 ditone divided by a 6:5 minor third—while the logarithmic value of about 92.18 cents makes the same unit comparable with tempered steps. From that compact representation one can read whether a transformation is exact, inverted, octave-equivalent, or only approximate, and calculate that substituting a 100-cent twelve-tone semitone introduces about 7.82 cents of error.

The compression also keeps three layers from proliferating into separate identities: the rational interval, a notation that instructs it, and a performed or tempered realization. A Johnston accidental or another microtonal spelling can encode 135:128 without defining it, and different instruments can realize the target with different tolerances. The ratio does not by itself determine audibility, beating, harmonic function, or expressive effect, which depend on spectrum, duration, register, context, and training. Nor may the unqualified word “limma” replace the ratio, because historically distinct limmas occupy nearby but nonidentical points in the tuning lattice.

Abstract Reasoning¶

Reasoning with the major limma starts from an exact relation between two pitches and carries it through interval arithmetic. Given a lower frequency (f), the upper frequency is (135f/128); given a chain of intervals, multiplying or dividing their ratios determines whether the chain contains the same interval. Thus two 9:8 tones followed by removal of a 6:5 minor third yield ((9/8)^2/(6/5)=135/128). A different absolute register preserves the identity because common transposition changes both frequencies but not their ratio.

The logarithmic conversion supports comparison without replacing the ratio. Computing 1200 log₂(135/128) gives about 92.18 cents, from which a tuner can predict that substituting one 100-cent step of twelve-tone equal temperament creates about 7.82 cents of discrepancy. The sign of that error depends on the direction of substitution, and repeated composition requires stating whether intervals are directed and reduced by octaves. Prime-factor form 3³ × 5 / 2⁷ independently locates the interval in 5-limit just intonation.

Diagnostic reasoning separates identity from representation and realization. A score symbol establishes the major limma only when its declared notation system maps that symbol to 135:128; an unqualified “limma,” a nearby cent value, or a measured performance deviation does not. Conversely, two differently spelled pitches may instantiate the same ratio. The operative question is therefore whether the derivation, notation mapping, or measured frequency pair preserves 135:128 exactly, approximates it within a stated tolerance, or instead denotes a neighboring interval such as 256:243. Audibility and musical function remain downstream questions conditioned by spectrum, duration, register, and context.

Knowledge Transfer¶

Within tuning theory, the major limma transfers literally across just-intonation analysis, microtonal composition, notation systems, instrument tuning, and performance measurement whenever the intended directed pitch ratio is 135:128. Its exact ratio, inversion, 5-limit factorization, derivation by interval division, and approximate 92.18-cent representation carry unchanged. So do the principal diagnostics and interventions: state the notation-to-tuning mapping, compare measured or tempered realization with the rational target, quantify cent error, and distinguish the unqualified limma, Pythagorean 256:243, and the 100-cent equal-tempered semitone. Instrument, register, and absolute pitch may change without changing the interval.

Beyond music, (B) shared abstract mechanism is supported only through the parent Ratio: preserving, inverting, composing, and comparing an exact multiplicative relation are general operations, but they do not preserve this interval's identity. Converting a ratio to a logarithmic additive value and recording an exact ratio separately from its rounded representation is a portable measurement-and-notation component technique; because it does not carry the named interval, it transfers that technique rather than the Major Limma itself. Descriptions of a small residual gap or fine adjustment outside tuning are merely (A) analogy unless the objects are pitch frequencies related by 135:128. Pitch roles, octave equivalence, interval composition, cents as 1/1200 of an octave, just-intonation limits, accidentals, and harmonic or melodic function remain home-bound. A bare 135:128 relation outside a declared pitch context is therefore simply a ratio, not a major limma; that is the stopping boundary.

Examples¶

Canonical¶

Begin with a 256 Hz pitch and form a ditone by applying two 9:8 major tones: \(256(9/8)^2=324\) Hz. Removing a 6:5 just minor third means dividing the interval ratio by 6:5, so the resulting pitch is \(324/(6/5)=270\) Hz. Relative to the original pitch, \(270/256=135/128\), exactly the major limma. Its logarithmic image is \(1200\log_2(135/128)\approx92.18\) cents.^[11] The calculation therefore reaches the named interval through ratio multiplication and division, not by subtracting frequencies or rounded cent values.

Mapped back: The 256 Hz and 270 Hz tones are the directed pitch pair; their reduced relation is the exact interval ratio. The two 9:8 steps followed by division by 6:5 implement the ditone-minus-third derivation, and the 92.18-cent result is the logarithmic image. If the final ratio were 256:243 or merely a nearby measured span, the exact-identity boundary would exclude the result from the major limma.

Applied / In Practice¶

In Ben Johnston's extended just-intonation notation, a composite accidental can instruct an approximately 92-cent raising or lowering.^[12] An interpreter must read that mark through Johnston's declared notation system rather than treating its visual form as universal. The mark supplies a performance instruction for the 135:128 adjustment; a realized pitch can then be compared with the rational target, while any deviation is reported as approximation rather than silently redefining the interval.

Mapped back: Johnston's convention supplies the notation mapping, and the sounded pitch supplies the practical realization. Comparing that realization with 135:128 yields the approximation error. The same-looking mark in a different notation system, or a 100-cent equal-tempered semitone used without qualification, fails the exact-identity boundary even if it is musically close.

Structural Tensions¶

T1: Exact ratio identity versus performed tolerance. The interval is fixed by 135:128, while voices and instruments realize a target with finite tuning variability. Treating every deviation as a different interval makes performance analysis unusably brittle; treating every near-92-cent span as exact erases the ratio that defines the name.

Diagnostic: Is the observation an exact construction, an explicitly tolerated realization of 135:128, or merely a nearby measured pitch span?

T2: Rational precision versus cent legibility. Ratio notation preserves exact factorization and interval arithmetic, whereas cents allow immediate additive comparison with temperaments and other intervals. Rounding the logarithmic image improves readability but can conceal distinctions between close rational intervals.

Diagnostic: Does the account retain 135:128 as primary while stating enough cent precision for the comparison being made?

T3: Stable interval identity versus local notation. The same frequency ratio can be written with different accidentals and spellings across tuning systems, and one symbol can denote different adjustments under different conventions. Notation makes the interval performable only by adding a system-specific mapping that must not replace the ratio itself.

Diagnostic: Is the score symbol explicitly mapped to 135:128 in the declared notation system, rather than inferred from graphic appearance?

T4: Just-intonation exactness versus tempered integration. Realizing 135:128 preserves the major limma's rational relation, while a fixed equal temperament may substitute an available step so that the interval works coherently within the instrument's tuning. Practical integration can justify approximation, but not silently redefine the exact interval.

Diagnostic: What tempered step is substituted, what signed cent error does it introduce, and is the result labeled as approximation rather than identity?

T5: Named specificity versus historical ambiguity. “Major limma” disambiguates one ratio, while the unqualified word “limma” has been used for nearby but nonidentical intervals such as 256:243. Historical continuity makes the shared vocabulary meaningful, yet omission of the ratio can make a source impossible to classify securely.

Diagnostic: Does the source state a ratio or derivation that selects 135:128 from the competing limma usages?

T6: Octave equivalence versus directed realization. Octave reduction lets interval calculations compare pitch classes compactly, but the performed relation still has an orientation and register. Inverting or reducing a ratio changes its representation and may change its musical use even when the interval-class calculation remains related.

Diagnostic: Are direction, inversion, and any power-of-two octave reduction stated before the ratio is identified as the major limma?

T7: Mathematical magnitude versus contextual audibility. The 7.82-cent difference from a twelve-tone semitone is exact in the tuning calculation, while its perceptual salience varies with spectrum, duration, register, beating context, and listener. Failure to hear the difference in one setting does not collapse the ratios, and exact distinction alone does not guarantee a fixed musical effect.

Diagnostic: Is the claim about mathematical tuning error or about perception under a specified acoustic and performance context?

T8: Major Limma autonomy versus reduction to Ratio (Ratio). Every Major Limma is a strict kind of the parent Prime: the upper-pitch frequency is the numerator, the nonzero lower-pitch frequency is the denominator, ordered division yields 135:128, and their common frequency units cancel within one directed pitch-pair scope. Scaling both frequencies by the same nonzero factor preserves the quotient, whereas changing the denominator or reversing the orientation changes the interval relation. Reduction therefore preserves Ratio's complete ordered-comparison structure, but it loses the Major Limma's exact 135:128 identity, approximately 92.18-cent logarithmic image, octave-aware interval arithmetic, 5-limit classification, notation mapping, and distinction between specified and realized tuning. Treating the interval as wholly autonomous hides why transposition preserves it; treating it as nothing but Ratio erases the musical residual.

Diagnostic: Does the case establish a scope-aligned upper-frequency-to-nonzero-lower-frequency quotient with the units, orientation, common-transposition invariance, and denominator sensitivity required for a Ratio, and then preserve 135:128 plus the tuning-specific residual required for the Major Limma?

Structural–Framed Character¶

Major Limma is framed-leaning. Its vocab_travels is low: the name, cents, limma taxonomy, octave equivalence, and just-intonation notation belong to tuning theory even though the underlying quotient is ordinary mathematics. Its evaluative_weight is moderate because 135:128 fixes exact identity while judgments about acceptable tempered or performed deviation require a declared musical tolerance. Its institutional_origin lies in theorized and notated tuning practice. Its human_practice_bound is substantial: frequencies can stand in the ratio without a listener, but identifying that relation as this interval and distinguishing it from neighboring limmas depends on a musical system. On import_vs_recognize, one recognizes the quotient directly, yet imports the interval name, pitch orientation, cent representation, and tuning conventions.

The smallest portable skeleton is Ratio: an ordered numerator and nonzero denominator yield a quotient invariant under common scaling. Portable and cross-domain reach belongs to that Prime. The structure explains why transposition preserves 135:128 and why inversion produces 128:135. What it does not carry is the selection of 135:128 as a named pitch interval, the ditone-minus-third derivation, octave-aware interpretation, 5-limit classification, notation mapping, or the boundary against 256:243 and a 100-cent semitone. Those conditions keep the Major Limma distinct rather than reducing it to a bare numerical ratio.

Its character: framed-leaning because an exact ratio supplies a strong portable core, while musical interval identity, tuning arithmetic, notation, and realization conventions supply the decisive frame.

Structural Core vs. Domain Accent¶

This decomposition explains why Major Limma is a domain-specific abstraction rather than a Prime.

What is skeletal (could lift toward a cross-domain prime). Two ordered, like-dimensional magnitudes occupy numerator and nonzero-denominator roles; division yields a dimensionless quotient that common scaling preserves and reversal changes. The invariant is that ordered quotient, while recognition fails if orientation, denominator, or scope changes. Major Limma is therefore a strict specialization of Ratio: Ratio supplies the full ordered-comparison structure, and the child fixes the quotient at 135:128 in a directed pitch relation.

What is domain-bound. The carriers are an upper and lower pitch frequency, and the identity is their exact 135:128 relation. Tuning practice adds octave-aware interval multiplication and division, the ditone-minus-third derivation, 5-limit factorization, an approximately 92.18-cent logarithmic image, notation-to-tuning mappings, and explicit tolerances for tempered or performed realizations. A bare 135:128 quotient, an unqualified limma, a 256:243 Pythagorean limma, or a nearby measured interval does not preserve this complete musical identity.

Why this does not clear the prime bar. The complete directed-pitch, exact-135:128, octave-aware, just-intonation, cent-image, notation-mapping, and realization-boundary signature does not recur literally across at least three unrelated domains with the same recognition and failure conditions. Knowledge Transfer accordingly gives ordered division and common-scale invariance to Ratio; logarithmic conversion elsewhere is a portable component technique, and a small residual gap elsewhere is only analogy, not a Major Limma. Removing the tuning accent leaves the dimensionless ratio 135:128 but not the named interval, while removing the ordered numerator–denominator relation leaves pitch labels, cents, or notation without the invariant that makes them representations of this interval.

This entry is a kind of Ratio.

Instantiates — Ratio (Ratio). For an upward interval, the upper pitch frequency is the numerator and the nonzero lower pitch frequency is the denominator; ordered division fixes their quotient at exactly 135/128. The terms share frequency units, so the quotient is dimensionless, while the musical scope declares that this number compares a directed pitch pair. Multiplying both frequencies by the same nonzero transposition factor leaves the quotient unchanged, and interval inversion reverses the ordered roles to 128/135. If the relation is only a rounded value near 92 cents, an unqualified limma, or a pitch pair with another quotient, the Major Limma identity and this Ratio instantiation collapse. Ratio remains broader because it does not require pitch, octave-aware interval arithmetic, just intonation, cents, or musical notation.

Relationships to Other Abstractions¶

Current abstraction Major Limma Domain-specific

Parents (1) — more general patterns this builds on

Major Limma is a kind of Ratio Prime

For an upward interval, the upper pitch frequency is the numerator and the nonzero lower pitch frequency is the denominator; ordered division fixes their quotient at exactly 135/128.

Hierarchy path (1) — routes to 1 parentless root

Major Limma → Ratio → Comparison → Self Checking

Neighborhood in Abstraction Space¶

Major Limma sits in a moderately populated region (43^rd percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Musical Intervals & Voice Texture (8 abstractions)

Nearest neighbors

Musical Interval — 0.92
Consonance — 0.89
Atonality — 0.86
Transposition (Music) — 0.86
Musical Key — 0.86

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

Not to Be Confused With¶

Pythagorean limma. The Pythagorean limma is the distinct interval 256:243, approximately 90.22 cents; its shared noun and nearby size make it a sibling, not another name for the 135:128 major limma. Tell: Does the source or tuning calculation specify 256:243 or 135:128?
Twelve-tone equal-tempered semitone. A semitone in twelve-tone equal temperament is one exactly 100-cent step and can approximate the major limma on a fixed-pitch instrument, but it is not the exact just ratio 135:128. Tell: Is the interval fixed by one 12-TET step, or by the rational upper-to-lower frequency ratio 135:128 with about 7.82 cents of difference?
25:24 chromatic semitone. The exact 25:24 chromatic semitone is a smaller neighboring 5-limit interval that can combine multiplicatively with the 81:80 syntonic comma to yield 135:128; it is a component of that decomposition, not an alias for the result. Tell: Does the reduced ratio equal 25:24, or does multiplying it by 81:80 produce the named 135:128 interval?
Syntonic comma. The syntonic comma is the distinct 81:80 interval, approximately 21.51 cents, used as a comma adjustment in just-intonation arithmetic; it is another component in one decomposition of the major limma. Tell: Is the measured or derived adjustment 81:80 alone, or the composite product (81:80)(25:24) = 135:128?
Ben Johnston accidental. A Ben Johnston accidental is a notation mark whose pitch adjustment is determined by that composer's declared just-intonation system; it represents an interval but does not constitute the interval's ratio. Tell: Does the notation system explicitly map the displayed mark to 135:128, or is only the graphic symbol known?

References¶

[1] Algorithms for Tunings and Temperaments registry ↩

[2] Unverified encyclopedia synthesis; claim-specific authoritative support was not established in this verification pass. ↩

[3] Unverified encyclopedia synthesis; claim-specific authoritative support was not established in this verification pass. ↩

[4] Unverified encyclopedia synthesis; claim-specific authoritative support was not established in this verification pass. ↩

[5] Unverified encyclopedia synthesis; claim-specific authoritative support was not established in this verification pass. ↩

[6] Unverified encyclopedia synthesis; claim-specific authoritative support was not established in this verification pass. ↩

[7] Unverified encyclopedia synthesis; claim-specific authoritative support was not established in this verification pass. ↩

[8] Unverified encyclopedia synthesis; claim-specific authoritative support was not established in this verification pass. ↩

[9] Unverified encyclopedia synthesis; claim-specific authoritative support was not established in this verification pass. ↩

[10] Unverified encyclopedia synthesis; claim-specific authoritative support was not established in this verification pass. ↩

[11] Unverified encyclopedia synthesis; claim-specific authoritative support was not established in this verification pass. ↩

[12] Unverified encyclopedia synthesis; claim-specific authoritative support was not established in this verification pass. ↩