# Example 948 — The ear could not hear the difference and could count it `the_ear_could_not_hear_the_difference_and_could_count_it.eml` - A violinist tunes the A string to the piano's A by playing them together and listening for a difference in pitch, hears none, and declares the string in tune while it throbs once a second against the piano. What a one-hertz gap sounds like, as a pitch and as a pulse, is computed below. ## EML ```eml # Self-authored for the EML case corpus (no external origin). A violinist tunes # the A string to the piano's A by playing them together and listening for a # difference in pitch, hears none, and declares the string in tune while it # throbs once a second against the piano. What a one-hertz gap sounds like, as a # pitch and as a pulse, is computed below. # # The reasoning is careful. The two notes were sounded together; the player # listened for a pitch difference with a trained ear; none was audible; and the # intent was exactly 'match the string to the piano'. # # Pitch is heard as a ratio and the ear cannot resolve a ratio closer than # about half a percent, but two tones a hertz apart drift in and out of step # once a second and that beat is counted, not heard as pitch - so the check # that listens for pitch passes a string the check that counts beats fails. 440 => piano_a_hz 441 => string_a_hz 445 => string_a_badly_off_hz 50 => smallest_pitch_ratio_the_ear_resolves_per_myriad 10 => seconds_listened string_a_hz - piano_a_hz => gap_hz int(string_a_hz * 10000 / piano_a_hz) - 10000 => gap_as_a_ratio_per_myriad smallest_pitch_ratio_the_ear_resolves_per_myriad - gap_as_a_ratio_per_myriad => margin_below_what_the_ear_resolves_per_myriad gap_hz * seconds_listened => beats_in_the_time_listened string_a_badly_off_hz - piano_a_hz => badly_off_gap_hz int(string_a_badly_off_hz * 10000 / piano_a_hz) - 10000 => badly_off_gap_as_a_ratio_per_myriad badly_off_gap_hz * seconds_listened => badly_off_beats_in_the_time_listened "piano A : " + str(piano_a_hz) + " Hz" ^0 "string A : " + str(string_a_hz) + " Hz" ^0 "gap : " + str(gap_hz) + " Hz" ^0 "gap as a pitch ratio : " + str(gap_as_a_ratio_per_myriad) + " per ten thousand" ^0 "smallest ratio the ear resolves : about " + str(smallest_pitch_ratio_the_ear_resolves_per_myriad) + " per ten thousand" ^0 "margin below the ear's floor : " + str(margin_below_what_the_ear_resolves_per_myriad) + " per ten thousand - inaudible as pitch" ^0 "beats in " + str(seconds_listened) + " seconds : " + str(beats_in_the_time_listened) + ", one a second, countable by anyone" ^0 "" ^0 "a badly off string : " + str(string_a_badly_off_hz) + " Hz, gap " + str(badly_off_gap_hz) + " Hz" ^0 "its gap as a pitch ratio : " + str(badly_off_gap_as_a_ratio_per_myriad) + " per ten thousand - audible as pitch" ^0 "its beats in " + str(seconds_listened) + " seconds : " + str(badly_off_beats_in_the_time_listened) + ", a rattle rather than a pulse" ^0 "" ^0 # ---- what the player verified ---- "the listen-for-pitch reasoning" ^0 " method : both notes sounded together" ^0 " ear : trained, attentive" ^0 " result : no difference in pitch audible" ^0 " intent : match the string to the piano" ^0 " facts wrong : 0" ^0 " verdict : IN TUNE" ^0 "" ^0 " listening for a pitch difference with a trained ear is the" ^0 " part done right here, and it is why nothing was heard: a" ^0 " gap of " + str(gap_as_a_ratio_per_myriad) + " per ten thousand is below what any ear resolves" ^0 "" ^0 # ---- what a hertz sounds like ---- "two measurements of the same gap" ^0 " as pitch : a ratio, " + str(gap_as_a_ratio_per_myriad) + " per ten thousand, under the ear's" ^0 " floor of about " + str(smallest_pitch_ratio_the_ear_resolves_per_myriad) ^0 " as beats : a difference, " + str(gap_hz) + " Hz, heard as the loudness" ^0 " swelling and fading " + str(gap_hz) + " time a second" ^0 " why the beat has no floor : it is not a pitch to resolve" ^0 " but a count to make, and one a second is easy to make" ^0 " what the tuner does : stops listening for pitch and counts" ^0 " beats, then turns the peg until the count reaches zero" ^0 "" ^0 # ---- what the player got ---- "the string" ^0 " believed : in tune with the piano" ^0 " actual : " + str(gap_hz) + " Hz high, beating " + str(beats_in_the_time_listened) + " times in " + str(seconds_listened) + " seconds" ^0 " was the ear wrong : no; there was no pitch difference to hear" ^0 " was the check wrong : yes; it asked the ear for a ratio" ^0 " finer than the ear can give, when the difference was" ^0 " there to be counted" ^0 "" ^0 # ---- null control ---- # The same string checked by counting beats instead of by listening for a # pitch difference. 0 => nc_gap_the_pitch_check_reports_hz 1 => nc_gap_the_beat_count_reports_hz 1 => nc_hertz_the_beat_count_reveals "null control - count the beats" ^0 " gap, pitch check : " + str(nc_gap_the_pitch_check_reports_hz) + " Hz" ^0 " gap, beat count : " + str(nc_gap_the_beat_count_reports_hz) + " Hz" ^0 " hertz the beat count reveals : " + str(nc_hertz_the_beat_count_reveals) ^0 " no string and no ear changed; the gap was measured as the" ^0 " difference it is instead of the ratio it is too small to be" ^0 "" ^0 # ---- the rule ---- "what a trained ear hearing no pitch difference guarantees" ^0 " the ratio is under the ear's floor : exactly; " + str(gap_as_a_ratio_per_myriad) + " per ten" ^0 " thousand is below about " + str(smallest_pitch_ratio_the_ear_resolves_per_myriad) ^0 " the two notes are the same : not addressed; a " + str(gap_hz) + " Hz gap that no" ^0 " ear can hear as pitch beats " + str(gap_hz) + " time a second, " + str(beats_in_the_time_listened) + " times in" ^0 " " + str(seconds_listened) + " seconds, and the beat is the gap itself" ^0 "" ^0 "a difference too small to hear is not too small to count; the ear that fails" ^0 "to resolve the ratio can time the throb, because the throb is the difference" ^0 "wearing a slower clock" ^0 "" ^0 "The ear heard no pitch difference, and it was right: " + str(gap_as_a_ratio_per_myriad) + " per ten thousand is" ^0 "under its floor. But the " + str(gap_hz) + " Hz gap is audible another way - as " + str(beats_in_the_time_listened) + " beats in " + str(seconds_listened) ^0 "seconds - so the string passed the pitch check and failed the count, until the" ^0 "gap is measured as the difference it is rather than the ratio it is too small to be." ^0 ``` ## Python (deterministic transpilation) ```python piano_a_hz = 440 string_a_hz = 441 string_a_badly_off_hz = 445 smallest_pitch_ratio_the_ear_resolves_per_myriad = 50 seconds_listened = 10 gap_hz = string_a_hz - piano_a_hz gap_as_a_ratio_per_myriad = int(string_a_hz * 10000 / piano_a_hz) - 10000 margin_below_what_the_ear_resolves_per_myriad = smallest_pitch_ratio_the_ear_resolves_per_myriad - gap_as_a_ratio_per_myriad beats_in_the_time_listened = gap_hz * seconds_listened badly_off_gap_hz = string_a_badly_off_hz - piano_a_hz badly_off_gap_as_a_ratio_per_myriad = int(string_a_badly_off_hz * 10000 / piano_a_hz) - 10000 badly_off_beats_in_the_time_listened = badly_off_gap_hz * seconds_listened print("piano A : " + str(piano_a_hz) + " Hz") print("string A : " + str(string_a_hz) + " Hz") print("gap : " + str(gap_hz) + " Hz") print("gap as a pitch ratio : " + str(gap_as_a_ratio_per_myriad) + " per ten thousand") print("smallest ratio the ear resolves : about " + str(smallest_pitch_ratio_the_ear_resolves_per_myriad) + " per ten thousand") print("margin below the ear's floor : " + str(margin_below_what_the_ear_resolves_per_myriad) + " per ten thousand - inaudible as pitch") print("beats in " + str(seconds_listened) + " seconds : " + str(beats_in_the_time_listened) + ", one a second, countable by anyone") print("") print("a badly off string : " + str(string_a_badly_off_hz) + " Hz, gap " + str(badly_off_gap_hz) + " Hz") print("its gap as a pitch ratio : " + str(badly_off_gap_as_a_ratio_per_myriad) + " per ten thousand - audible as pitch") print("its beats in " + str(seconds_listened) + " seconds : " + str(badly_off_beats_in_the_time_listened) + ", a rattle rather than a pulse") print("") print("the listen-for-pitch reasoning") print(" method : both notes sounded together") print(" ear : trained, attentive") print(" result : no difference in pitch audible") print(" intent : match the string to the piano") print(" facts wrong : 0") print(" verdict : IN TUNE") print("") print(" listening for a pitch difference with a trained ear is the") print(" part done right here, and it is why nothing was heard: a") print(" gap of " + str(gap_as_a_ratio_per_myriad) + " per ten thousand is below what any ear resolves") print("") print("two measurements of the same gap") print(" as pitch : a ratio, " + str(gap_as_a_ratio_per_myriad) + " per ten thousand, under the ear's") print(" floor of about " + str(smallest_pitch_ratio_the_ear_resolves_per_myriad)) print(" as beats : a difference, " + str(gap_hz) + " Hz, heard as the loudness") print(" swelling and fading " + str(gap_hz) + " time a second") print(" why the beat has no floor : it is not a pitch to resolve") print(" but a count to make, and one a second is easy to make") print(" what the tuner does : stops listening for pitch and counts") print(" beats, then turns the peg until the count reaches zero") print("") print("the string") print(" believed : in tune with the piano") print(" actual : " + str(gap_hz) + " Hz high, beating " + str(beats_in_the_time_listened) + " times in " + str(seconds_listened) + " seconds") print(" was the ear wrong : no; there was no pitch difference to hear") print(" was the check wrong : yes; it asked the ear for a ratio") print(" finer than the ear can give, when the difference was") print(" there to be counted") print("") nc_gap_the_pitch_check_reports_hz = 0 nc_gap_the_beat_count_reports_hz = 1 nc_hertz_the_beat_count_reveals = 1 print("null control - count the beats") print(" gap, pitch check : " + str(nc_gap_the_pitch_check_reports_hz) + " Hz") print(" gap, beat count : " + str(nc_gap_the_beat_count_reports_hz) + " Hz") print(" hertz the beat count reveals : " + str(nc_hertz_the_beat_count_reveals)) print(" no string and no ear changed; the gap was measured as the") print(" difference it is instead of the ratio it is too small to be") print("") print("what a trained ear hearing no pitch difference guarantees") print(" the ratio is under the ear's floor : exactly; " + str(gap_as_a_ratio_per_myriad) + " per ten") print(" thousand is below about " + str(smallest_pitch_ratio_the_ear_resolves_per_myriad)) print(" the two notes are the same : not addressed; a " + str(gap_hz) + " Hz gap that no") print(" ear can hear as pitch beats " + str(gap_hz) + " time a second, " + str(beats_in_the_time_listened) + " times in") print(" " + str(seconds_listened) + " seconds, and the beat is the gap itself") print("") print("a difference too small to hear is not too small to count; the ear that fails") print("to resolve the ratio can time the throb, because the throb is the difference") print("wearing a slower clock") print("") print("The ear heard no pitch difference, and it was right: " + str(gap_as_a_ratio_per_myriad) + " per ten thousand is") print("under its floor. But the " + str(gap_hz) + " Hz gap is audible another way - as " + str(beats_in_the_time_listened) + " beats in " + str(seconds_listened)) print("seconds - so the string passed the pitch check and failed the count, until the") print("gap is measured as the difference it is rather than the ratio it is too small to be.") ``` ## stdout (executed) ```text piano A : 440 Hz string A : 441 Hz gap : 1 Hz gap as a pitch ratio : 22 per ten thousand smallest ratio the ear resolves : about 50 per ten thousand margin below the ear's floor : 28 per ten thousand - inaudible as pitch beats in 10 seconds : 10, one a second, countable by anyone a badly off string : 445 Hz, gap 5 Hz its gap as a pitch ratio : 113 per ten thousand - audible as pitch its beats in 10 seconds : 50, a rattle rather than a pulse the listen-for-pitch reasoning method : both notes sounded together ear : trained, attentive result : no difference in pitch audible intent : match the string to the piano facts wrong : 0 verdict : IN TUNE listening for a pitch difference with a trained ear is the part done right here, and it is why nothing was heard: a gap of 22 per ten thousand is below what any ear resolves two measurements of the same gap as pitch : a ratio, 22 per ten thousand, under the ear's floor of about 50 as beats : a difference, 1 Hz, heard as the loudness swelling and fading 1 time a second why the beat has no floor : it is not a pitch to resolve but a count to make, and one a second is easy to make what the tuner does : stops listening for pitch and counts beats, then turns the peg until the count reaches zero the string believed : in tune with the piano actual : 1 Hz high, beating 10 times in 10 seconds was the ear wrong : no; there was no pitch difference to hear was the check wrong : yes; it asked the ear for a ratio finer than the ear can give, when the difference was there to be counted null control - count the beats gap, pitch check : 0 Hz gap, beat count : 1 Hz hertz the beat count reveals : 1 no string and no ear changed; the gap was measured as the difference it is instead of the ratio it is too small to be what a trained ear hearing no pitch difference guarantees the ratio is under the ear's floor : exactly; 22 per ten thousand is below about 50 the two notes are the same : not addressed; a 1 Hz gap that no ear can hear as pitch beats 1 time a second, 10 times in 10 seconds, and the beat is the gap itself a difference too small to hear is not too small to count; the ear that fails to resolve the ratio can time the throb, because the throb is the difference wearing a slower clock The ear heard no pitch difference, and it was right: 22 per ten thousand is under its floor. But the 1 Hz gap is audible another way - as 10 beats in 10 seconds - so the string passed the pitch check and failed the count, until the gap is measured as the difference it is rather than the ratio it is too small to be. ``` ## Round-trip `ok: true` — round-trip fixpoint reached (python1 == python2) ## Trace event types eml:run:start · eml:assign · eml:output · eml:run:done