Calculations for your workbook
Find a calculation, inspect its inputs and see how you could use it. Excel formulas use the proposed NF. prefix; Sheets formulas use NF_.
Proposed spreadsheet interfaces. Existing nfield calculations supply part of the library; additional arithmetic utilities and both spreadsheet adapters are in development.
36 proposed functions · 108 documented workflows
NF_FIELDArithmetic
List the fractions and digit-field records for a denominator.
NF_FIELD(denominator, base)
Inputs
denominator- Positive integer denominator. Small bounded pilot examples recommended; the adapter must enforce the selected engine command’s domain.
base- Integer numeral base, at least 2; supply explicitly.
Result
columns: fraction numerator, denominator, base and field records. Final normalized digit-column schema needs adapter design.
Precision: Exact rational / integer.
When to use it
- Prepare a classroom worksheet comparing fractions with denominator seven
- Compare the same fraction field in bases ten and twelve
- Inspect a composite-denominator field beside a prime field
Proposed interface. The spreadsheet adapter and its output contract are under development.
NF_ALIGNAlignment
Compute the synchronized alignment of a fractional field.
NF_ALIGN(denominator, base)
Inputs
denominator- Positive integer denominator. Small bounded pilot examples recommended; the adapter must enforce the selected engine command’s domain.
base- Integer numeral base, at least 2; supply explicitly.
Result
columns: metric, exact value, numerical display.
Precision: Exact rational / integer.
When to use it
- Rank denominators by alignment within one base
- Compare a chosen denominator across numeral bases
- Add exact alignment results beside a teaching table
Proposed interface. The spreadsheet adapter and its output contract are under development.
NF_DECOMPOSEAlignment
Return the named alignment components rather than a single score.
NF_DECOMPOSE(denominator, base)
Inputs
denominator- Positive integer denominator. Small bounded pilot examples recommended; the adapter must enforce the selected engine command’s domain.
base- Integer numeral base, at least 2; supply explicitly.
Result
columns: component name, exact value, numerical display.
Precision: Exact rational / integer.
When to use it
- Explain which component changes between two denominators
- Reconstruct a reported alignment from its component output
- Compare component contributions in a base sweep
Proposed interface. The spreadsheet adapter and its output contract are under development.
NF_ALIGN_MATRIXAlignment
Build the synchronized pairwise alignment matrix at one common depth.
NF_ALIGN_MATRIX(denominator, base, depth)
Inputs
denominator- Positive integer denominator. Small bounded pilot examples recommended; the adapter must enforce the selected engine command’s domain.
base- Integer numeral base, at least 2; supply explicitly.
depth- Positive common synchronization depth; affects all pairwise entries together.
Result
matrix: row/column field labels plus exact alignment entries; certificates in a separate summary table.
Precision: Exact rational / integer.
When to use it
- Create a heatmap of pairwise fractional alignment
- Compare pairwise structure before changing the common depth
- Inspect whether a scalar summary hides distinct row patterns
Proposed interface. The spreadsheet adapter and its output contract are under development.
NF_ALIGN_FOURIERAlignment
Read the numerical Fourier spectrum of the exact alignment blocks.
NF_ALIGN_FOURIER(prime, base)
Inputs
prime- A prime satisfying the selected command’s additional hypotheses.
base- Integer numeral base, at least 2; supply explicitly.
Result
columns: block, mode, numerical eigenvalue, check residual.
Precision: Numerical.
When to use it
- Identify dominant modes of a prime alignment matrix
- Compare exact matrix structure with its numerical spectral summary
- Teach how circulant blocks lead to Fourier eigenvectors
Proposed interface. The spreadsheet adapter and its output contract are under development.
NF_DEFICITTuning
Return exact alignment deficit and check the supported-depth hypotheses.
NF_DEFICIT(prime, depth, base)
Inputs
prime- A prime satisfying the selected command’s additional hypotheses.
depth- Positive common synchronization depth; affects all pairwise entries together.
base- Integer numeral base, at least 2; supply explicitly.
Result
columns: metric, exact value; include domain checks, depth and deficit.
Precision: Exact rational / integer.
When to use it
- Compare supported depths 24 and 36
- Demonstrate why an unsupported depth is rejected
- Plot exact deficit against supported refinement depth
Proposed interface. The spreadsheet adapter and its output contract are under development.
NF_DEFICIT_RATIOTuning
Compare two deficits with the exact ratio, limiting ratio and error certificate.
NF_DEFICIT_RATIO(prime, depth1, depth2, base, factor)
Inputs
prime- A prime satisfying the selected command’s additional hypotheses.
depth1- Positive supported depth: every prime factor must divide the base.
depth2- Positive supported depth: every prime factor must divide the base.
base- Integer numeral base, at least 2; supply explicitly.
factor- Positive integer refinement factor. The scaled depths must remain supported by the base.
Result
columns: finite exact ratio, limiting ratio, exact error/certificate values; final labels need schema normalization.
Precision: Exact rational / integer.
When to use it
- Study convergence toward the ratio three over two
- Compare a finite-depth interval with its limiting interval
- Find a supported refinement meeting a chosen arithmetic error tolerance
Proposed interface. The spreadsheet adapter and its output contract are under development.
NF_MAJOR_SCALETuning
Produce the existing base-thirty arithmetic major-scale table.
NF_MAJOR_SCALE(factor)
Inputs
factor- Positive integer refinement factor. The scaled depths must remain supported by the base.
Result
table: named interval, supported depths, finite ratio, limiting ratio, reported error bounds.
Precision: Exact rational / integer.
When to use it
- Prepare an exact arithmetic scale worksheet
- Compare scale tables at two refinement factors
- Document finite-depth errors alongside target scale ratios
Proposed interface. The spreadsheet adapter and its output contract are under development.
NF_CHROMATIC_SCALETuning
Produce the existing base-thirty arithmetic chromatic-scale table.
NF_CHROMATIC_SCALE(factor)
Inputs
factor- Positive integer refinement factor. The scaled depths must remain supported by the base.
Result
table: named interval, supported depths, finite ratio, limiting ratio, reported error bounds.
Precision: Exact rational / integer.
When to use it
- Compare neighboring chromatic interval readings
- Build a chromatic-scale lesson with explicit finite errors
- Compare major-scale entries with their chromatic counterparts
Proposed interface. The spreadsheet adapter and its output contract are under development.
NF_PYTHAGOREANTuning
Reproduce the existing base-twelve Pythagorean-comma calculation.
NF_PYTHAGOREAN(factor)
Inputs
factor- Positive integer refinement factor. The scaled depths must remain supported by the base.
Result
table: calculation term, exact ratio, finite-depth comparison and error.
Precision: Exact rational / integer.
When to use it
- Explain the arithmetic comma in an interval worksheet
- Compare the comma calculation under refinement
- Keep exact ratios next to approximate musical displays
Proposed interface. The spreadsheet adapter and its output contract are under development.
NF_COLLISIONCollision
Return a lag-one digit-bin collision count and residue fingerprint.
NF_COLLISION(integer, base)
Inputs
integer- Integer input from 2 to 10000000 for direct collision-table inspection.
base- Integer numeral base, at least 2; supply explicitly.
Result
labeled result: input, residue fingerprint, exact collision value.
Precision: Exact rational / integer.
When to use it
- Calculate a collision fingerprint for a prime input
- Compare a composite input with a prime input
- Investigate inputs sharing a residue class modulo base squared
Proposed interface. The spreadsheet adapter and its output contract are under development.
NF_COLLISION_COMPARECollision
Distinguish equal table values from equal complete residue fingerprints.
NF_COLLISION_COMPARE(integer1, integer2, base)
Inputs
integer1- First integer input within the direct-inspection command limits.
integer2- Second integer input within the direct-inspection command limits.
base- Integer numeral base, at least 2; supply explicitly.
Result
labeled result: two fingerprints, two values, fingerprint equality and value equality.
Precision: Exact rational / integer.
When to use it
- Check whether two equal scores represent the same fingerprint
- Find false equivalences caused by reducing a table to one number
- Prepare a residue-equivalence teaching example
Proposed interface. The spreadsheet adapter and its output contract are under development.
NF_COLLISION_TABLECollision
Return the finite unit-cell collision table with reflected partners.
NF_COLLISION_TABLE(base)
Inputs
base- Integer numeral base, at least 2; supply explicitly.
Result
columns: unit residue, collision value, reflected residue, reflected value.
Precision: Exact rational / integer.
When to use it
- Create a complete base-ten collision heatmap
- Compare table symmetry in bases eight and twelve
- Inspect reflected residue pairs with opposite labels
Proposed interface. The spreadsheet adapter and its output contract are under development.
NF_COLLISION_SUMMARYCollision
Return exact extrema, sign counts and mean of the finite table.
NF_COLLISION_SUMMARY(base)
Inputs
base- Integer numeral base, at least 2; supply explicitly.
Result
labeled result: extrema, sign counts, exact mean.
Precision: Exact rational / integer.
When to use it
- Compare sign balance across selected bases
- Check table extrema against individual cells
- Build a compact report of collision distributions
Proposed interface. The spreadsheet adapter and its output contract are under development.
NF_FIELD_SPECTRUMSpectrum
Compute additive spectral power for the arithmetic field.
NF_FIELD_SPECTRUM(prime, base)
Inputs
prime- A prime satisfying the selected command’s additional hypotheses.
base- Integer numeral base, at least 2; supply explicitly.
Result
columns: mode and numerical power; retain source/base metadata.
Precision: Numerical.
When to use it
- Plot a prime fractional-field power spectrum
- Compare spectral concentration across bases
- Contrast two primes with different repetend structures
Proposed interface. The spreadsheet adapter and its output contract are under development.
NF_AUTO_INSPECTSpectrum
Compare a reference repetend orbit with aggregation across all multiplicative orbits.
NF_AUTO_INSPECT(prime, base)
Inputs
prime- A prime satisfying the selected command’s additional hypotheses.
base- Integer numeral base, at least 2; supply explicitly.
Result
tables: reference orbit, all-orbit aggregate and numerical comparison diagnostics.
Precision: Exact with numerical cross-checks.
When to use it
- Show when one orbit fails to represent the entire field
- Compare primitive-root and nonprimitive-root examples
- Separate aggregate correlation from reference-orbit correlation
Proposed interface. The spreadsheet adapter and its output contract are under development.
NF_AUTO_SLICESpectrum
Compare an exact collision count with its numerical Fourier-line reading.
NF_AUTO_SLICE(prime, residue, base)
Inputs
prime- A prime satisfying the selected command’s additional hypotheses.
residue- Nonzero residue modulo the supplied prime; use a unit residue for inverse-based Fourier slices.
base- Integer numeral base, at least 2; supply explicitly.
Result
columns: residue, exact collision count, numerical Fourier reading, residual/tolerance.
Precision: Exact with numerical cross-checks.
When to use it
- Cross-check a residue correlation against its Fourier slice
- Study how a residue multiplier changes a collision count
- Explain the distinction between exact counts and floating transforms
Proposed interface. The spreadsheet adapter and its output contract are under development.
NF_AUTO_HARMONICSSpectrum
Read the harmonic energies of the normalized circulant arithmetic response.
NF_AUTO_HARMONICS(prime, base)
Inputs
prime- A prime satisfying the selected command’s additional hypotheses.
base- Integer numeral base, at least 2; supply explicitly.
Result
columns: mode, numerical harmonic energy and tolerance diagnostics.
Precision: Numerical.
When to use it
- Chart nonnegative harmonic energies
- Compare harmonic distribution across prime examples
- Inspect whether dominant harmonics change after a base change
Proposed interface. The spreadsheet adapter and its output contract are under development.
NF_INTERVAL_CENTSMusic tools
Convert a positive frequency ratio into cents using 1200 log2(ratio).
NF_INTERVAL_CENTS(ratio)
Inputs
ratio- Positive dimensionless frequency ratio.
Result
Single value
Precision: Numerical.
When to use it
- Compare a just fifth with a seven-semitone equal-tempered fifth
- Display the size of a supplied comma in cents
- Convert an arithmetic deficit ratio into a musical display without asserting perceptual equivalence
Proposed interface. The spreadsheet adapter and its output contract are under development.
NF_CENTS_RATIOMusic tools
Convert a cents interval into its positive frequency ratio.
NF_CENTS_RATIO(cents)
Inputs
cents- Finite real cents interval; 1200 cents is one octave.
Result
Single value
Precision: Numerical.
When to use it
- Create ratios for a user-defined temperament
- Convert a cent correction into a playback multiplier
- Build an interval table from measured cent offsets
Proposed interface. The spreadsheet adapter and its output contract are under development.
NF_TRANSPOSE_HZMusic tools
Apply a cents transposition to a positive reference frequency.
NF_TRANSPOSE_HZ(frequency, cents)
Inputs
frequency- Positive frequency in hertz.
cents- Finite real cents interval; 1200 cents is one octave.
Result
Single value
Precision: Numerical.
When to use it
- Transpose an instrument reference by a specified interval
- Generate frequencies for a supplied scale table
- Compare pitch references at 440 and 442 hertz
Proposed interface. The spreadsheet adapter and its output contract are under development.
NF_BEAT_HZMusic tools
Return the absolute difference of two pure-tone frequencies.
NF_BEAT_HZ(frequency1, frequency2)
Inputs
frequency1- First positive pure-tone frequency in hertz.
frequency2- Second positive pure-tone frequency in hertz.
Result
Single value
Precision: Numerical.
When to use it
- Estimate beating between two nearby tuning tones
- Compare reference and measured oscillator frequencies
- Calculate the difference tone of a supplied pair without modeling a complex instrument
Proposed interface. The spreadsheet adapter and its output contract are under development.
NF_EDO_STEPMusic tools
Return the ratio 2 raised to step divided by divisions.
NF_EDO_STEP(divisions, step)
Inputs
divisions- Positive integer divisions per octave; browser demo cap 1000000.
step- Safe integer step, allowed to be negative or larger than the division count.
Result
Single value
Precision: Numerical.
When to use it
- Compare twelve-tone and nineteen-tone divisions
- Generate every step of a thirty-one-tone octave
- Explore negative and octave-crossing step numbers
Proposed interface. The spreadsheet adapter and its output contract are under development.
NF_OCTAVE_REDUCEMusic tools
Reduce a positive ratio into the half-open octave interval from one to two.
NF_OCTAVE_REDUCE(ratio)
Inputs
ratio- Positive dimensionless frequency ratio.
Result
Single value
Precision: Numerical.
When to use it
- Compare intervals differing by whole octaves
- Normalize a harmonic-ratio list before plotting
- Separate octave displacement from interval class
Proposed interface. The spreadsheet adapter and its output contract are under development.
NF_RATIO_APPROXMusic tools
Find a nearest bounded-denominator rational approximation with an error report.
NF_RATIO_APPROX(ratio, maximumDenominator)
Inputs
ratio- Positive dimensionless frequency ratio.
maximumDenominator- Positive bounded search limit; prototype cap 10000.
Result
Single value
Precision: Numerical.
When to use it
- Find a simple fraction near an equal-tempered interval
- Compare approximation quality as the denominator budget grows
- Express a measured frequency ratio as candidate small-integer intervals
Proposed interface. The spreadsheet adapter and its output contract are under development.
NF_RATIONAL_SUMExact arithmetic
Sum rational strings exactly and return a reduced rational string.
NF_RATIONAL_SUM(textFractions)
Inputs
textFractions- Range of exact rational strings such as "1/3".
Result
Single value
Precision: Exact rational / integer.
When to use it
- Sum thirds and sixths without decimal rounding
- Combine exact weights exported by nfield commands
- Check whether rational contributions cancel exactly
Proposed interface. The spreadsheet adapter and its output contract are under development.
NF_RATIONAL_PRODUCTExact arithmetic
Multiply rational strings exactly and return a reduced rational string.
NF_RATIONAL_PRODUCT(textFractions)
Inputs
textFractions- Range of exact rational strings such as "1/3".
Result
Single value
Precision: Exact rational / integer.
When to use it
- Calculate a chain of just-intonation intervals
- Verify closure of a rational interval cycle
- Apply a sequence of exact scale factors to an arithmetic reading
Proposed interface. The spreadsheet adapter and its output contract are under development.
NF_RATIONAL_COMPAREExact arithmetic
Compare two rational strings by integer cross multiplication.
NF_RATIONAL_COMPARE(textFraction1, textFraction2)
Inputs
textFraction1- First exact rational string with a nonzero denominator.
textFraction2- Second exact rational string with a nonzero denominator.
Result
Single value
Precision: Exact rational / integer.
When to use it
- Order nearly equal exact arithmetic results
- Check equality of differently written fractions
- Compare two large-rational certificates without converting to doubles
Proposed interface. The spreadsheet adapter and its output contract are under development.
NF_ORBIT_PERIODArithmetic
Compute the preperiod and eventual remainder-cycle length using exact division.
NF_ORBIT_PERIOD(numerator, denominator, base)
Inputs
numerator- Integer numerator; use text for integers beyond the spreadsheet’s exact numeric range.
denominator- Positive integer denominator. Small bounded pilot examples recommended; the adapter must enforce the selected engine command’s domain.
base- Integer numeral base, at least 2; supply explicitly.
Result
labeled result: preperiod length, cycle length, terminating flag. Termination uses cycle length zero.
Precision: Exact rational / integer.
When to use it
- Separate terminating prefixes from repeating tails
- Compare decimal and duodecimal period lengths
- Find a denominator whose full period exceeds the requested digit preview
Proposed interface. The spreadsheet adapter and its output contract are under development.
NF_PERIOD_SWEEPArithmetic
Tabulate eventual remainder periods across a bounded denominator range.
NF_PERIOD_SWEEP(firstDenominator, lastDenominator, base)
Inputs
firstDenominator- Inclusive positive integer lower endpoint. Proposed resource cap must be finalized in implementation.
lastDenominator- Inclusive upper endpoint at least the first; bounded to the pilot’s row and runtime limit.
base- Integer numeral base, at least 2; supply explicitly.
Result
columns: denominator, preperiod, cycle length, terminating flag.
Precision: Exact rational / integer.
When to use it
- Find long-period fractions for an arithmetic lesson
- Compare period-length distributions between two bases
- Select representative short and long cycles for a demonstration workbook
Proposed interface. The spreadsheet adapter and its output contract are under development.
NF_BIG_ADDArbitrary precision
Add arbitrarily large integers exactly.
NF_BIG_ADD(left, right)
Inputs
left- Exact text input; nonzero divisor required.
right- Exact text input; nonzero divisor required.
Result
Text, preserving all exact digits.
Precision: Exact integers / rationals.
When to use it
- Add integer counts beyond ordinary cell precision
- Combine large exact numerators
- Check cancellation in large integer sums
Proposed interface. The spreadsheet adapter and its output contract are under development.
NF_BIG_SUBTRACTArbitrary precision
Subtract arbitrarily large integers exactly.
NF_BIG_SUBTRACT(left, right)
Inputs
left- Exact text input; nonzero divisor required.
right- Exact text input; nonzero divisor required.
Result
Text, preserving all exact digits.
Precision: Exact integers / rationals.
When to use it
- Compare adjacent large integer readings
- Calculate an exact large balance difference
- Preserve a one-unit difference beyond cell precision
Proposed interface. The spreadsheet adapter and its output contract are under development.
NF_BIG_MULTIPLYArbitrary precision
Multiply arbitrarily large integers exactly.
NF_BIG_MULTIPLY(left, right)
Inputs
left- Exact text input; nonzero divisor required.
right- Exact text input; nonzero divisor required.
Result
Text, preserving all exact digits.
Precision: Exact integers / rationals.
When to use it
- Build a large integer product without rounding
- Multiply exact combinatorial counts
- Check a proposed integer factorization
Proposed interface. The spreadsheet adapter and its output contract are under development.
NF_BIG_DIVIDEArbitrary precision
Divide integer or rational strings and return a reduced exact rational.
NF_BIG_DIVIDE(left, right)
Inputs
left- Exact text input; nonzero divisor required.
right- Exact text input; nonzero divisor required.
Result
Text, preserving all exact digits.
Precision: Exact integers / rationals.
When to use it
- Keep one third exact
- Reduce a ratio of large integer products
- Divide rational interval products without decimal rounding
Proposed interface. The spreadsheet adapter and its output contract are under development.
NF_BIG_POWERArbitrary precision
Raise an exact rational to an integer power.
NF_BIG_POWER(value, exponent)
Inputs
value- Exact text input; nonzero divisor required.
exponent- Integer exponent within the documented resource limit.
Result
Text, preserving all exact digits.
Precision: Exact integers / rationals.
When to use it
- Calculate a long chain of repeated rational intervals
- Compare positive and negative rational powers
- Construct large exact integer powers
Proposed interface. The spreadsheet adapter and its output contract are under development.
NF_DECIMAL_ROUNDArbitrary precision
Convert an exact rational string to a requested number of significant decimal digits with explicit rounding.
NF_DECIMAL_ROUND(value, digits, rounding)
Inputs
value- Exact text input; nonzero divisor required.
digits- Positive integer significant-digit count within the documented resource limit.
rounding- Declared rounding mode, such as HALF_EVEN.
Result
Decimal text with explicit significant digits and rounding mode.
Precision: Rounded decimal; requested significant digits.
When to use it
- Display one third at fifty significant digits
- Compare rounded displays without altering the exact source
- Round an exact result using half-even or another declared mode
Proposed interface. The spreadsheet adapter and its output contract are under development.